Tycho seems to have built up his catalog in steps. Dreyer describes a sequential process where ever larger sets of standard stars were used by Tycho to assess the positions of new stars for the catalog.
"On the basis of the nine standard stars and twelve additional stars near the zodiac, Tycho Brahe built up his star catalogue. Of a star to be determined, the declination was measured directly by the armillæ or a meridian quadrant, and the distance from a known star was measured with a sextant. This furnished, as before, a spherical triangle, with the three sides known, from which the angle at the pole or the difference of right ascension could be computed. Generally the star was connected with two known stars, one preceding and one following it, which gave two results for the right ascension as a control."
This indicates a hierarchical approach in which the number of standard stars were slowly increased, with the accuracy of new additions depending on the accuracy of the prior set of standards. Note that Tycho realized that more than one standard star could be used to determine the position of a new star, but rarely used more than two standards in the process.
This means that any errors in a prior set of standards would propagate their errors into any larger set that used them as reference.
If this was Tycho's method, then every new expansion of the standard star set must have slightly larger errors compared to the prior (smaller) set of standards.
In order to test this thesis myself, I downloaded a well-studied catalog of Tycho's from a paper by Verbunt.
This paper points to a catalog known as "KeplerE", which contains Tycho's original positions alongside errors from modern positions transformed back to Tycho's epoch. Because each version of Tycho's catalog has different column entries, it can be difficult to compare two catalogs. The KeplerE catalog did not have star names, only an index into the list of stars in each constellation. Before I could study the position errors, I needed to identify Tycho's standard stars in the KeplerE catalog.
Because modern positions are in the celestial ra/dec system, and Tycho's are in the ecliptic lat/long system (epoch 1601), I needed to take my list of standards, fetch their ra/decs, and transform to ecliptic_1601 coordinates. Such a transformation involves a precession of ~400 years, followed by a rotation into the ecliptic system. I could not find an internet tool that could do both so I contacted Dr. Patrick Wallace who modified one of his tools (COCO) into a form that could do my required transformation. Thank you Dr. Wallace!
With the transformed coordinates of Tycho's 100 standards, I was able to identify his stars in the KeplerE catalog. I could then extract Tycho's errors for each star. With Dr. Rosa's identification of the modern names of Tycho's 100, I could then group his standard stars into successively larger sets.
Here are the 4 standard stars first mentioned by Tycho in his attempt to show that his technique spanned correctly the celestial sphere in right ascension. (I call these 4 Group 1)
Gem,beta
Ari,alpha
Aql,alpha
Vir,alpha
He then added five stars (Group 2, below) to the initial 4 to create a list of 9 standards.
Oph,delta
Peg,alpha
Tau,alpha
Gem,mu
Leo,alpha
He then added 12 more stars (Group 3, below) to form a list of 21 standards.
Cap,gamma
Sgr,omicron
Ari,beta
Leo,gamma
Sco,alpha
Gem,gamma
Psc,alpha
Cnc,gamma
Lib,beta
Vir,gamma
Tau,epsilon
Aqr,beta
Finally he added 79 more stars (Group 4, below) to form his primary set of 100 standard stars.
Cap,alpha2
And,beta
Ori,lambda
Ori,delta
Lib,alpha
Cyg,epsilon
UMa,zeta
Oph,beta
Lep,beta
Oph,alpha
Ori,epsilon
Ser,alpha
Aqr,epsilon
Oph,eta
Ari,gamma2
Cep,beta
Oph,zeta
Cet,iota
Cyg,gamma
Sgr,pi
Cas,gamma
Cyg,delta
Lyr,alpha
Aqr,alpha
Cet,alpha
Peg,gamma
Her,beta
Gem,alpha
PsA,alpha
Cet,zeta
Cet,beta
Leo,beta
UMa,delta
Aql,zeta
Vir,delta
Peg,epsilon
Vir,epsilon
Sco,beta
UMa,gamma
Ori,beta
Leo,delta
UMi,akpha
Psc,gamma
Cyg,alpha
Cas,beta
Hya,alpha
Leo,zeta
And,alpha
CrB,alpha
Cas,alpha
Cyg,beta1
UMa,eta
Ori,alpha
Leo,eta
Ori,zeta
UMa,epsilon
CMi,alpha
Tau,eta
Aur,alpha
Tau,beta
CMa,alpha
Tau,gamma
UMa,alpha
Peg,beta
Cap,delta
Cap,beta
Psc,alpha
Tau,zeta
Her,alpha
And,gamma
Pup,rho
UMa,beta
Cas,delta
Per,beta
Ori,gamma
Dra,gamma
Boo,alpha
These stars would then be used to determine the positions of the
remaining (~600) stars in the catalog.(Group 5)
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