The instrument

Tycho's workhorse instrument was his "trigonis sextantis" or triangular sextant. It was used to measure the separation between two stars. It required two people to operate it. It was a sophisticated and ingenious design that incorporated four independent axes of motion. To explain the 4-axis requirement, one needs to discern the method of observation. Because one observer had to sight on a primary star, it needs at least two axes of movement, two degrees of freedom, to meet that requirement. This is standard for most hobby telescopes today. But it is also necessary to point to a second star, which would be offset from the first star by some separation angle, a third axis of movement. This separation angle only describes a circle on the sky around the primary star, so there had to be a fourth degree of freedom, an axis that would permit the second observer to pick a point on this separation circle where the second star was located. Here is Tycho's depiction of his instrument:

Here is a small segment of Tycho's comments on the sextant, as translated from the Latin by Dr. Rosa.

The need for a 4-axis device is clear in the above image. The plane of the triangle has to be in the same plane defined by the observers and the two stars. This is accomplished by rotating the entire instrument, on its 2-axis globe mount, so that the plane of the sextant matches the plane of the stars. Now, a rotation of the triangular framework, about the third axis ('E' in the first diagram) permits the first observer (red sock guy) to sight on the primary star in the pair. Lastly, the second observer now sights on the secondary star in the pair by rotating his sight line about the 4th axis at point 'A' in the first diagram.

A slightly more abstract explanation is in this hand-annotated sketch:

In the above diagram, the 'w-axis' is moved by rotating the globe base in two axes to put the plane of the triangular framework into the plane of the two stars. The triangular framework is then rotated about this axis, by the angle epsilon, to bring the primary star into the sights of the first observer. The sights of the second observer are rotated by a further angle, theta, to find the secondary star in the pair. Four different rotations needed.

I needed a 'trigonis sextantis' of my own. How to design a 4-axis optical system that could point to two different stars simultaneously, track them both, and measure the angle between them?

I took a 5" aperture telescope that I had purchased twenty years earlier when computer-controlled telescopes first became available, and took it apart to see how it worked. Having built computer-controlled telescopes of my own before, I had a good idea of what to expect. I had investigated earlier to see if I could use this scope's electronics 'as-is' but found them to be too limited for my needs.

Here is a picture of the unmodified scope:

These hobby telescopes have two characteristics that make them difficult for measuring large angles. The altitude and azimuth positions are determined by encoders that are mounted on the drive motors. This permits fine resolution of the motor rotation, but imperfections in the gear trains causes the actual position accuracy on the sky to be much poorer, typically ten arc minutes in this model. The other problem is what is called 'mirror flop' where the primary mirror changes its orientation slightly between high altitudes (stars near the zenith) and low altitudes (stars closer to the horizon). This effect can be a significant fraction of a degree. Neither of these problems is much of a concern for backyard viewing of the sky, but they limit the absolute position accuracy of the readouts. Tycho achieved a final accuracy of two arc minutes, 1/30 of a degree. I needed to find a way, with limited budget, to get close to Tycho's accuracy.

To fix the accuracy problems, the motor encoders either have to be moved and attached directly to each axis of motion, or additional encoders need to be added. To fix the 'mirror flop' , the source of the problem needs to be found and eliminated.

I chose to add additional encoders to the axes, and I decided to epoxy the primary mirror in place so that it could no longer move. To add additional encoders, one has to consider how to mount them and their cost. By epoxying the primary, I could no longer use the primary to achieve focus. So I needed a way to focus without the motion of the primary mirror.

In addition, the telescope would need to be modified to permit the measurement of the angle between two stars.

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