Catalog construction for the primary stars
Having found a starter set of stars, the task was to use them to estimate the relative ras of the primary stars, and later, the relative ras of all the remaining secondary stars. The declinations come directly from the meridian altitude measures, so there is no 'starter set' needed for them. The construction of the catalog starts with the condensed (averaged) data sets mentioned earlier.
I don't know what Tycho did at this stage. I relied heavily on the sorting tools in my spreadsheet program. Sorting is an arduous manual task, particularly with large dataset, so I don't know how he handled dealing with these datasets that had thousands of elements.
For each separation pair, the averaged meridian altitude of each star was inserted into a spreadsheet via a small external program that then also took the separation and altitude data and calculated the expected difference in ra between the stars in that pair. That equation is described on an earlier page. I use the shortcut 'mdra' to stand for the ' measured delta right ascension'. These mdras were the first step in the preparation of the catalog. Note that these mdras were all positive values because the separation cannot be a negative value. This integrated, condensed, data file had all the pair separations and all the altitudes in it. For this reduction, that file is TYOUT_2022-02-06.txt, which can be found here. It has quote marks in it that may need to be deleted.
The first task is to separate out the primary pairs from the total list of measurements. The primary star names all had a tilde(~) appended to them for later identification. I had to learn some fairly obscure uses of the Linux utility 'grep' to achieve this extraction.
To isolate just the primary pairs:
grep '~.*~' TYOUT_2022-02-06.TXT > tyout_primaries_2022-02-06.txt
The list of 379 primary pairs was then duplicated, and in this copy the first and second stars (and their modern position and measured altitude data) in the pair were swapped. This copy was then concatenated with the first list to form a master list of all primary pairs taken in both directions. This doubling of the data was to account for the dual nature of the separation measurement. Each primary pair was now listed twice, with each star getting a chance to be positioned first, as the reference star in the pair.
For each separation pair a mdra had already been calculated, so, for all the pairs of this concatenated list, the mdras values needed to be signed correctly to reflect the direction in which the separation was to be applied. I apologize for the obscurity of this preparation, but it took me quite a while, and many iterations, to find the right way to honor the bi-directionality of the separation measures. The separation values, which were originally all positive, were now both positive and negative to reflect the direction of the separation.
Here were the successive steps to generate the ra separations. The altitudes would be converted to decs in a later step to finalize the catalog.
The start of the ra-linking process began with entering the starter star set provisional ra values. Remember that at this point the absolute ra values were not available, so everything was relative to the first star in the starter set. That first star (alphaAri) had a provisional ra of zero, and the rest of the stars in the starter set were assigned provisional ras relative to the first star. From now on, the words 'provisional ra' or 'prov' will always appear where normally 'ra' would appear.
In the starter set 'E' the star deltaVir had a provisional ra of 162.09 degrees. The list was sorted by the modern (Ycat) ra values of the first star in the pair. This sorting was indicated by a '^' in the header title label (Ycat_ra1^). The 2020 ra of alphaAri was 32.08 degrees, so the Ycat values would differ from the provisional ra values by 32.08 degrees. Ycat stands for the Yale Bright Star Catalog.
The provisional ra of deltaVir was entered in column15 four times (right blue outline) because there were four other star separation measurements made relative to deltaVir; betaLeo, alphaBoo, deltaLeo, and gammaBoo. The delta ra (mdra, col10) was added to the prov ra of deltaVir (col15) to get the first estimates of the provisional ras of the four linked stars. Those values were entered in col16 under the heading 'prov_ra2.1' indicating that the values were provisional ras of the second star in the pair in the first iteration.
Of these four stars, I arbitrarily selected deltaLeo, outlined in green above, as the star to follow for the next steps. It was a proxy for any star whose provisional ra has been estimated from a starter star. The first estimate of its provisional ra is from one linkage only and has the value 136.709 degrees.
I then sorted, below, on the just-determined provisional ra2.1 (col16), to bring together the first estimates on deltaLeo. The four stars that have links to deltaLeo were outlined in blue along with their starter set provisional ra, also in blue.
Below, the four deltaLeo estimates (green outline) were averaged together to get a new, hopefully better, estimate of 136.72 degrees, and was entered four times into col17, the averaged provisional ra of the second star in the pair after the first iteration.
Next, the list was sorted, below, on the ra of the second star (Ycat_ra2) to find all possible linkages to deltaLeo, even those not in the starter set. I found that deltaLeo is now linked to 14 other primary stars (blue outline). The initial starter set linkages were outlined in green in col15. The empty cells in this column indicate new stars that were linked to deltaLeo. The average deltaLeo provisional ra (136.72) was filled manually into col17, the average provisional ra of the second star in the pair in this first iteration.
Now, below, came the first back-propagated step. The second star in the pair (deltaLeo) now became the reference point for the first stars in the pairs linked to it. The mdras of the first stars in the pairs was subtracted from the constant averaged value of deltaLeo to get provisional ras (green outline) for the first stars in the pair in this iteration (prov_ra1.1). Note that the four starter stars, listed in col13, had new provisional ra estimates. These starter star's provisional ras had been modified by the averaging and back-propagation. The remaining ten stars were still on their first estimate. Of course, this same process was happening simultaneously, to all the other starter stars and all the other primary star pairs on the spreadsheet.
The previous step had been sorted by Ycat_ra2 to gather the deltaLeo's in the second star position. I now sorted, below, by Ycat_ra1, the ras of the first star in a pair, to bring together all the deltaLeo pairs where it is the first star in the pair. The col18 (prov_ra1.1) now contained all the estimates of the provisional ra of deltaLeo from all 14 of its linked stars. These 14 values were combined to eliminate outliers and to generate a new average estimate. The new provisional ra estimate was 136.70 degrees, and it was entered into col19 as the new best estimate of the provisional ra of deltaLeo.
In the above example, deltaLeo served as a reference star for the second star in the pair. Now, with all the new averaged estimates for the provisional ras of the first stars in a pair, I sorted, below, on the Ycat_ra2 column to gather all the deltaLeo-referenced average provisional ra values (col19). I then added in the delta ras (mdra col10) to get back-propagated estimates of deltaLeo in a second iteration.
These second iteration values for the provisional ra of deltaLeo were averaged together (minus outliers) to get, below, the final estimate of the deltaLeo's provisional ra, 136.71 degrees, (purple outline), and the 1-sigma value of the estimate in degrees.
So, deltaLeo's estimated provisional ra started at 136.709 degrees, then went to 136.72, and then 136.70, before finally settling on 136.71, almost exactly where it started.
This list was then sorted on the final averages (avg_prov_ra2.2) and produced a list of 99 primary stars and their provisional ras, the basis for the final catalog step. This list of averaged provisional positions of the primaries can be found here. The ra2_err column was a sanity check, generated by adding the known ra offset of alphaAri (32.08) to the provisional ras, then subtracting the catalog ra. The header of the 'dec2' column is sometimes interpreted by the spreadsheet program as a date. A sample of that file of primary provisional positions:
To show that this iterative process improves the results, I show the errors in the starter set 'E' before and after the iteration process.
