Tycho's reductions

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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