Nightly planning (early days)

One area where Tycho left few details was how he planned each night's observations. At the beginning everything needed to be measured, but, as data accumulated, he needed to avoid excess repetition. He needed repeated measurements to help sort out defective measurements, but too many observations of the same star(s) was wasted effort. In the first few months I planned the nightly measurements largely by hand, help by a few simple tools.

Meridian altitudes are fairly straightforward. Planning is largely dictated by the availability of stars near the meridian during the night, but there are other considerations that need attention for efficiency reasons.

The separation measurements, however, were sensitive to the orientation of the stars to each other and the pole. The goal of these measurements was to determine the difference in ra between the two stars whose separation was being measured. The two stars and the north celestial pole formed a spherical triangle that permitted the delta ra to be calculated if the separation angle and the two declinations had been measured. But if the two stars were roughly north-south of each other, then the resulting triangle was narrow and more susceptible to errors.

In the following diagram, I try to illustrate the effect for planar triangles. The effect is similar for spherical triangles.

'P' represents the delta ra angle (dra) between the two stars being measured. The top point of each triangle is the celestial pole. 'a' and 'c' run from the pole to the two stars under study. 'b' is the separation angle that is being measured. An in measuring 'b' results in an error in the determination of 'p'. The formula for 'p' is given in terms of the three sides of the triangles. In the left-hand triangle, where the separation 'b' lies mostly E-W, a 1% error in 'b' causes an error in 'p' of about 0.7 degrees. In the right-hand triangle, where the separation lies mostly N-S, the same error in 'b' creates a 1.8 degree error in 'p'. So there is value in selecting star pairs that avoid the N-S orientation when measuring separations.

(Note to reviewers - I find this graphical explanation to be clumsy. I will try to rework this page using the formal derivative.)

These sensitivity concerns were minimized in the early days by simply inspecting the separation of possible pairs and avoiding cases where the ras of the two stars were similar. Later I found out that this argument generalized to any 'skinny triangle' no matter what its orientation.

I decided to concentrate on the primary stars at first because they were to be the positional backbone of the final catalog. As I accumulated data, I needed a way to measure progress. I first used a large (5 foot diameter) plot, created by my brother Tom, which had the north celestial pole in the center and the primary stars mapped out according to their ra and distance from the pole (codeclination). This tries to map a sphere onto a flat circle, so it does distort, but not too badly. I added the star names and then covered the poster in plastic so that I could use the surface as a white board. I could then draw in all the separation lines and color code them to indicate multiplicity. The dipper is obvious in the lower left.

This approach worked well for several months as I slowly accumulated confidence, and data. After about 8 months of taking data I began to see how to do a better, more efficient, job collecting data, so I decided to expand my efforts to Tycho's entire catalog. This poster approach was not going to work when covered by hundreds of stars. I finally quit using it in early 2021 and here is the last update I did to it in March 2021.

As you can see here, most of the separations run E-W and not N-S. A red line indicated two (see color code at lower left) separations had been measured, not enough for robust statistics. So, although I had measured all the primary stars, it was clear that my planning was not working well, and that I was not being very efficient.

By the way, you can see the small yellow dots that indicated the solar position on some of the days that I had measured it.

The TPSS acronym is an unserious reference to the SPSS planning system for the HST.

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