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Separation measures are more complex than meridian altitudes because of the need to observe two stars simultaneously. This complicates the planning and the observing, but there is one simplification in that a separation measurement can be made whenever the two stars are at reasonable altitudes - not just at one time in the night like meridians.
The separation angle between two stars is, to fairly high accuracy, a constant, so to track two stars simultaneously only three axes need to track. Pitch remains constant during the observation, while roll, az, and alt change constantly. This is another effect of working with an alt/az mount, equivalent to the third axis camera rotation, needed for an imager on an alt/az mount.
The roll angle (measured from the direction to the zenith) varies in a complex way, so I found it easiest to simply recompute the required roll angle at 16Hz and use roll position commands to accomplish the tracking. Details of the roll mechanism are on an earlier page. Of course, alt and az are changing constantly, so their recalculation takes place at 16Hz also. It took a while to sort out the roll calculations, and I had two errors that proved difficult. The first was the use of full-angle calculations for the roll angle. This turned out to fail as the roll angle passed through 90 degrees. Instead of passing through the 90 deg point smoothly, the calculation 'bounced' and the roll reversed itself. An investigation revealed that I had to use 'half-angle' formulas to avoid this behavior. I still have an intermittent error in my roll calculation that sometimes yields a roll with the correct magnitude but wrong sign. I have not been able to track down this error but it is easy to recognize, and I simply move the roll manually to the correct value as a work-around.
For efficiency, it makes sense to place a primary star in the direct FOV and the secondary stars in the pitch FOV. This allows the entire group of pairs associated with that primary to be sequentially observed. Once that group of pairs is finished, I can move on to the next primary and its group of secondaries. This offered insight into what Tycho might have considered when planning his separation measures.
The ordering of the secondary stars in the pairs is important to minimize the amount of roll shell slewing. Slew velocity is limited by the use of simple stepper motors, and I made no attempt to speed them up. My typical roll velocity was 2.7 deg/sec, which meant that maximum slews could take up to two minutes. I had a cable limitation at a roll of -40 degrees, so if one secondary star fell at -35 degrees roll and the next secondary star at -45 degrees roll, just on the other side of the cable limit, the required slew would be 350 degrees in length.
What was needed was an ordering whose roll would be monotonically increasing from the lowest roll angle in the group to the largest roll angle in the group. What I found was, that just taking a map of the stars near the primary lead to a natural ordering in roll. Here is a small section of Norton's map, centered approximately on betaPeg. The heavy blue line gives the direction to the pole, and the roll angle of a secondary star is given by the angle clockwise from the polar direction to the secondary star. So, in this case, the correct order to observe pairs related to betaPeg would be etaPeg, iotaPeg, alphaPeg, gammaPeg, and alphaAnd. The first secondary star to be observed depends on the season and the time of night, but the list order remains the same. I expect that Tycho used similar logic in planning his nightly separation observations. Note that the roll angle here is measured from the celestial North Pole; sort of 'pole_roll'.
Unfortunately, what I needed was different from Tycho. His instrument, the trigonal sextant, was not tied to any particular coordinate system. In my instrument, the roll angle is measured from the zenith, not the north pole; sort of 'zenith_roll'. I needed an ordering whose zenith roll would be monotonically increasing from the lowest roll angle in the group to the largest roll angle in the group. The problem here is that the zenith roll angle to each of the secondary stars is constantly changing, whereas the 'pole_roll' remains a constant. Ordering the secondaries by right ascension was often acceptable, but if the secondary declinations stretched across the zenith then the roll sequence became more chaotic. I did not ever find an algorithm that maximized the efficiency of these separation groups.
To illustrate, here are plots of segments of the roll history on two successive nights, where the sequence of secondaries was in ra order. The first is with gammaTau as the primary, the second plot is with betaPer(Algol) as the primary. Gamma Tau is near dec 16 degrees, and its secondaries are mostly away from the zenith and fall into two groups needing only one long roll slew. Beta Per is near dec 41 deg, so its secondaries were passing close to the zenith, and the roll angles vary significantly, necessitating more roll slews. The red data points show that the ras of the secondary stars were ordered sequentially. The blue data points show the required zenith roll needed to reach that secondary star.
The actual tedious measurement technique can be sampled with this video.
Part of the data reduction process was to average certain pairs of measures to a single measure. This was because the pitch encoder that determined the separation angle had coarse resolution (one arc minute). I sometimes could not get the pitch star to align to the primary star as close as I wanted, so I would 'bracket' the primary star with two measurements, close in time, one on either side of the primary, with the intent of averaging these two together in a later processing step. This reduced the 6334 raw separation measures to 5405 independent measures. At the end of data collection, I condensed the separation data from the list of all measurements to produce a list of all the star pairs observed, along with the averaged separation and the standard deviation of those measures. This list of measures includes all that went into the catalog plus other supporting measurements such as in the zenith star determinations and test runs.
This was then input to the final catalog assembly process, where it was integrated with the altitude data. The condensed separation list can be found here. This list is very succinct. It contains only the 2020_ra of each star (which is used as the ID of the star), number of separation measurements, the average separation, and the sigma. No other star info. The list of stars (here) was later used as a look-up table to find the names of these stars, their modern catalog magnitude and declination, and to guide the integration of this separation data into the catalog formation.
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