Hierarchical reductions: the starter stars

The approach to compiling the catalog was to use three steps: the selection of a 'starter set' of stars; the use of those stars to estimate the relative ra positions of the 100 primary stars; and then the use of the primary star's relative ra positions to assign relative ra positions to the remaining 600+ secondary stars. All the ras were still relative at this point because I had not yet added in my determination of the right ascension of alphaAri.

The separation data had a slightly unusual quality in that this one positive number, the relative separation, was assigned to a pair of stars at the endpoints of the separation. The position of the stars in the pair was not known, and the separation measure itself would have its own internal measurement error.

The altitude data, in comparison, was absolute, not relative. The altitude measures did have an internal error, but the error in any one altitude did not influence other measured altitudes.

One can use the separation information in two directions, depending on which star one wants to use as the reference. The optimum approach would be to toss all the data into a giant simultaneous interative least-squares fit and stop the iterations when the fit no longer improves. I had no such tool, nor did Tycho.

One knows ahead of time that no data point is completely accurate. But the reflexive nature of the separation (the two end points influence each other) means that the stars can all interact to provide a better solution than the initial one. The catalog starting points were guaranteed to have an error, so the hope was that the ensemble of stars would help correct the start point error.

With the final datasets of individual altitudes and separations at hand, the path to the catalog involved several steps.

• The altitude and separation data needed to be averaged and combined

• Sets of east-west linkages needed to be found among the primary stars, which span the entire celestial sphere

• The best of these sets needed to be converted into a 'starter' set, which would form the basis for the catalog

• The starter set would then be used as reference points, to determine the ras of any other primary stars that were linked to the starter set of stars

• As more primary stars are assigned ras, the linkages could be reversed and new, modified, estimates of the starter stars could be made

• This enlarged set of ras would be used again, iteratively, until all the primary stars had been linked, forward and backward, to the starter set

• The quality of the primary ras thus determined would be checked against modern values and the modified starter set would be re-evaluated

• Other starter sets would be tried until the best quality solution was achieved

• Once an acceptable set of primary ras had been found, the remaining secondary stars were linked to the primaries to complete the catalog

Tycho seems to have used several sets of primary stars. The first group had 7 stars, the second group 21, and the final group of about 100. He built these carefully because the quality of each following group depended on the quality of the prior smaller set. I adopted the slightly different strategy of observing the 100 primary stars densely and then, later, picked out specialized subsets.

My first step in the hierarchy was to find candidate starter sets of stars that linked complete around the globe. I was able to find, by hand, 6 semi-independent sets of 12 to 18 stars that could be linked to each other completely around the celestial sphere (globe). Here is how their modern ra/decs look: ( I apologize for the plotting of ra increasing to the right - it was just easier.)

To evaluate the appropriateness of each of these starter sets, I calculated how well they met the requirement that their delta ras sum to 360 degrees and the noisiness of the stars in each set. To do that I plotted, from the starter set page, the accumulating measurement error (c13) for each starter set.

Of the six starter sets, two sum to more than 360, three sum to less than 360, and one sums to almost exactly 360 degrees. I selected group F as my first starter set because it had the best sum and also started with the same star (alphaAri) that Tycho selected.

I decided to perform a partial build of the catalog that would obtain just the ras of all the primary stars. I found that the error in the larger set seemed to reflect the errors in the starter set. Here is the comparison:

It seemed clear that I needed a better way to select my starter set. Going back to the sextet of error plots above, I tried to make sense of the differing summations to near 360 degrees. Why were some slightly in excess of 360 while others were slightly deficient?

It seemed that it might be in the way that the individual errors were accumulating. If I had a starter set with a dozen stars with similar separations, then their errors, if random, should add together in a root-sum-squares way. If each separation had an error of, say, two arc minutes (0.032 deg), then the sum should have an error of sqrt(12)*2 arc minutes, or about 0.1 degrees, which is about the right magnitude for what I saw. This does make me wonder about Tycho's claim to have summed around the globe with errors less than an arc minute.

If this is the case, then it might be acceptable to simply force the accumulated sums to be 360 deg. These sets start and end with the same star, so the sums necessarily needed to sum to 360. Perhaps I could simply scale all the sets to sum correctly. If so, then how to select the best starter set? Forcing the sums to 360 is equivalent to fitting a linear curve to each error set and then use the slope to normalize the sum. Perhaps the next criterion would be to ask which normalized starter set had the least noise.

Reminding myself that maximum accuracy was not my goal in assembling this catalog, I narrowed down the starter sets and decided to reduce the catalog with starter set E. Its cumulative ra wandered away from the modern catalog accumulated ra, but the deviation was pretty linear and the data points held pretty close to the trend. I liked this starter set too because it also started with alphaAri. Here was my process, starting with the 5th starter set in the above sextet plot:

This forcing was equivalent to taking all the measured delta ras in group E and multiplying them by 360/359.824.

With a reasonable starter set in hand, plus a preview of the noise level of the ra position, it was time to build the catalog proper. The next level in the catalog construction hierarchy was to determine the position of all the primary stars relative to the starter set. The final level in the catalog construction hierarchy would be to use those 100 primary positions to determine the position of the remaining secondary stars.

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