Astrolabes
By Spencer Connor
When looking back at the history of astronomical instruments, there is perhaps nothing that that looks quite like an astrolabe. There is something undeniably artistic about these device and yet they are clearly scientific in basis; it’s clear that there is a high level of craftsmanship associated with them beyond just functionality. But what are they used for and how? Astrolabes are often quoted as having more than 1000 uses but fundamentally they really only do two, like a modern calculator may only have a handful of functions but each can serve a multitude of uses.
Those functions are:
Measure an angle from vertical, by sighting down the alidade while the instrument is suspended by the ring. This is a common feature of many instruments.
Perform a coordinate transform between the celestial sphere and the observer reference frame. This is the really unique function that makes the astrolabe so useful.
The celestial sphere can be imagined as a transparent shell around the earth upon which all the stars are fixed. In antiquity this was believed to have have been a physical construct floating out around the globe, the true form of the heavens. But even in modern day we use it as a convenient analog, as it accurately represents the results of calculations for astrolabes and similar devices to the precision required.
Building on Armillary Spheres
The image of the celestial sphere shown here may look familiar, it is reminiscent of another ancient astronomical device called an armillary sphere. Armillary spheres are simply a three-dimensional model of the earth and celestial sphere, a most direct representation. While more intuitive to understand, the armillary sphere suffers from two major impracticalities. First, armillary spheres are not particularly portable. A large, delicate sphere would quickly become a annoyance for travel let alone being used for navigation. Secondly and likely most importantly, it is difficult or impossible to take meaningful measurements off of them.
The astrolabe solves both of these issues by being a two-dimensional representation of an armillary sphere. It is flat and robust, making it well suited for travel and adverse conditions. And the method by which the 3D to 2D “flattening” is achieved makes angular measurements very straightforward.
Stereographic Projection
That method of representing a three-dimensional system in a two-dimensional model is key. Those familiar with cartography will know the difficulty in representing the surface of a sphere in a flat format for a map. Various projections like Mercator and Boggs eumorphic can be used to optimize certain properties, but nothing works for all cases. Areas get distorted, straight lines become curved, or some combination of the two result. And that’s just for a single static sphere! Our armillary is showing two spheres, and one is rotating relative to the other. There is a type of projection that works perfectly well for this, as it happens. That is a stereographic projection. Like other projections it doesn’t preserve all parameters (notably the size of features is drastically varied by their location), but those things have little use compared to the parameters that ARE preserved. Specifically, circles on the sphere (such as equator, tropics, meridians, ecliptic) remain circles when projected. They just get scaled and offset, for example the ecliptic (tilted at 23.5° from the equator) simply becomes an offset circle.
Let’s take a look at constructing the two coordinate frames: celestial and observer
The Celestial Frame
In the celestial frame, we have the celestial sphere (just a collection of stars at this point) all relatively fixed and uniformly rotating around the earth.
We chose a projection point (the south celestial pole) and a plane (the equator) that our stereographic projection will use.
We then project the brightest stars onto the plane using the projection point. Northern stars end up closer to middle, and southern stars at the perimeter.
When projected in the same manner as the stars, the tilted circle of the ecliptic becomes an offset circle.
And here is our final 2D representation of the celestial frame, which rotates once per sidereal day.
The Observer Frame
Next, we consider our current place on the earth (our latitude and longitude). We’ll mark our location on the surface of the earth
We then consider the frame around our location in basic terms: the horizon around us and the point directly overhead (called zenith)
Let’s add to that the meridians (cardinal directions, North-South/East-West) and almucantar (angles above the horizon) to make a grid.
And now let’s project that spherical grid onto the equatorial plane using the same projection point to get the stereographic 2D grid.
And here is our final 2D representation of the observer frame, which is static (although will be different based on our latitude).
Putting it all together
If we overlay the two we start to get something starting to resemble the astrolabe, but it’s still missing the skeletonized rete to represent the ecliptic circle and the floating star points. This is really where the artistic portion comes in, as really any shape with a circle and something indicating the star locations work. This can vary from simple spikes to ornate foliage and animal carvings. But if you look across the retes of astrolabes you’ll find those spikes in the exact same place (within the workmanship of the item.
Precession of the Equinoxes
One thing to note is that while the star positions are fixed, the ecliptic circle (oriented with the black crosshair) appears at slightly different angles. This is due to an effect called axial precession, where the Earth’s pole wobbles around in ~26,000 years. The ecliptic is the plane normal to the pole, so it rotates correspondingly. The equinoxes are where the ecliptic intersects the celestial equator, so this effect is often called the precession of the equinoxes. In fact, this rate (which works out to about a degree every 72 years) can be used to date when the astrolabe was designed since the rete will show a “snapshot” of where the ecliptic was at the time.
