Zenith star reductions

The zenith stars were those stars in Tycho's catalog that had declinations that were within three degrees of my latitude. Such stars passed within three degrees of my zenith when they crossed the meridian, and my mount precluded accurate altitude measurements near the zenith, as detailed on an earlier page.

The measurements of meridian altitudes allowed the calculation of a star's declination, as shown on an earlier page. But Tycho himself, as pointed out by Dr. Rosa, offered an alternate way to measure the declination of a star by using two stars nearby that were already well measured, as shown on this earlier page.

With these tools, I generated 55 sets of three stars for further measurement. Here is a snapshot of the first few triplets:

The zenith star name is in the first column, followed by its 2020 ra/dec coordinates. You can see that the decs all lie between 36 and 42 degrees. The two primary stars used to triangulate the zenith star are then given. Note that the zenith star betaPer is included in 8 different triplets in order to pin down its declination accurately because it is used also as a primary star sometimes, as seen in line 6 and 7. More than 900 separation measurements were made to supply the data necessary to solve these 55 stars.

After the separation measurements of the three stars in the triplet were made, a script generated the rest of the geometry and solved for the zenith star's correct meridian altitude. The 'cxx' below refers to column numbers in the spreadsheet.

Zenith star formulae using sides (s1,s2,s3,s4,s5 in the diagram below) to calculate theta (1,2,3) and s6

c26=90-(c19-50.98) 's1 primary star#1 co-declination 90-dec, dec=alt-50.98

c27=90-(c22-50.98) 's2 primary star#2 co-declination 90-dec, dec=alt-50.98

c28=c16 's3 primary#1 to primary#2 separation

c29=c10 's4 zenith star to primary#1 separation

c30=c13 's5 zenith star to primary#2 separation

c32=invcos( (cos(c27)-cos(c26)*cos(c28)) / (sin(c26)*sin(c28)) ) 'theta1

c33=invcos( (cos(c30)-cos(c28)*cos(c29)) / (sin(c28)*sin(c29)) ) 'theta2

c34=c32-c33 'theta3=theta1-theta2

c36=invcos( cos(c26)*cos(c29) + sin(c26)*sin(c29)*cos(c34) ) 's6 zenith star codec

c38=90-c36+50.98 'zenith star meridian altitude

It is humbling to remember that Tycho had to do all these computations by hand to 6 significant figures, with no computers, calculators, or slide rules. Sine and cosines were interpolated from large tables of chords painstakingly calculated. None of the worksheets or blackboards survive.

The connection between Tycho's diagram and my script is shown here:

After all the observations and calculations were done, these newly minted meridian altitudes had no specific time associated with them. They had been calculated from measurements taken on different nights. Because the exact time of a meridian altitude was not important to the catalog, I assigned all these synthetic meridian altitudes a standard date of MJD 700.00 and entered the 'method' as 'S-S-S' indicating that a side-side-side spherical triangle solution was used to determine this value. An example of this is shown in the last two lines of this meridian altitude history snippet. The calculated meridian altitudes are near 90 degrees.

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