Bošković's Spherical Trigonometric Solution for Determining the Axis and Rate of Solar Rotation by Observing Sunspots in 1777

Abstract

In September 1777 Ruđer BoÅ¡ković observed and measured the sun-spot positions to determine the solar rotation elements. In 1785, among other methods, he described a trigonometric spherical solution for the determination of the position of the axis and rate of the solar rotation using three sunspot positions, but without equations. For the first time, we derive the equations that are applicable to modern computers for calculating the solar rotation elements, as they were described by BoÅ¡ković. We recalculated BoÅ¡ković's original example using his measurements of sunspot positions from 1777 and the equations developed here, confirming his results from 1785. BoÅ¡ković's methodology of arithmetic means determines $i$, $\hat{I}\text{c◯}$, and sidereal period $T^{\prime }$ separately, while the planar trigonometric solution determines $i$ and $\hat{I}\text{c◯}$ together. His spherical trigonometric solution calculates $i$, $\hat{I}\text{c◯}$, and the sidereal period $T^{\prime }$ in a single procedure. Keywords: Ruđer BoÅ¡ković, Sunspots, Solar rotation, Spherical trigonometry

The article is the submitted one to the publisher upon the Licence to Publish by Springer Nature. Since its submission to acceptance, section 4.4. Accuracy Versus Precision of the Measurements on the Apparent Solar Disk, is added, the old section 4.4 became 4.5, and some references are added. Editor-in-Chief copy-edited the article. Published article is https://doi.org/10.1007/s11207-025-02497-x