Astronomical Images : Problem 4: Measuring a comet's parallax

Johannes Regiomontanus

Astronomical Images

Johannes Regiomontanus died in 1475, leaving behind a printing press, instruments and a library containing printed books and manuscripts. Most of the library was bought by Bernhard Walther (1430-1504), the merchant-astronomer at Nuremberg and then ended up in the possession of Willibald Pirckheimer (1470-1530), the patrician friend of Albrecht Duerer. Pirckheimer sold on several of Regiomontanus's works to Johannes Schoener (1477-1547), who taught mathematics at the gymnasium in Nuremberg. Regiomontanus's work on comets, which was listed in his own printing advertisement, was first edited and published by Schoener in 1531 as Sixteen Problems on the Magnitude, Longitude and True Position of Comets. It was printed again, with several other works of Regiomontanus in 1544. Regiomontanus's fourth problem is to find the parallax of the comet when two observations are taken at the same altitude (and thus before and after the meridian), useful when the comet crosses the meridian during daytime. Correction to the figure: the intersection of the altitude circle through G and the horizon is L. ABCD: horizon circle DZB: semicircle of the meridian CKA: the comet's diurnal arc (carried by the primum mobile) [i.e. Regiomontanus suggests correction to observed values to take into account the comet's own motion] G: the true place of the comet at first observation (a.m.) M: the true place of the comet at second observation (p.m.) Z: zenith Two great quadrants ZL and ZP drawn through G and M O: apparent place of the comet at first observation (a.m.) N: apparent place of the comet at second observation (p.m.) OL = NP [because O and N have the same altitude, by assumption; note that O and N lie on ZL and ZP respectively, because the apparent and true positions due to parallax error are always in vertical alignment.] So GL = MP and GK = KM [by symmetry] H: celestial pole Draw great arcs HG and HM, which are equal by geometry The time between the two observations is measured from some fixed star, whence angle GHM is known; Hence angle GHK or angle GHZ is known, for each is equal to half angle GHM. Angle GZK is measured, which gives the azimuth; Hence we know angle GZH, because it is the complement of angle GZK. Since we know arc ZH is the complement of altitude of the pole H, the triangle GZH is determined, namely two angles GZH and GHZ, with arc HZ, whence both arcs HG and ZG are known. Arc ZO is known by observation, hence the remainder GO [= ZO ' ZG] is known, which is the parallax sought. This supposes negligible proper motion of the comet. We can also calculate the declination of the comet from the equator, ie. the complement of the arc HG, and since we know angle GHK, we know the true position of the comet on the ecliptic. Problems five, six and eight depend on this problem.