<p>In September 1777 Ruđer Bošković observed and measured the sunspot 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 Bošković. We recalculated Bošković’s original example using his measurements of sunspot positions from 1777 and the equations developed here, confirming his results from 1785. Bošković’s methodology of arithmetic means determines <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2497_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>i</mi> </math></EquationSource> <EquationSource Format="TEX">$i$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2497_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> <EquationSource Format="TEX">$\Omega $</EquationSource> </InlineEquation>, and the sidereal period <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2497_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mo>′</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$T'$</EquationSource> </InlineEquation> separately, while the planar trigonometric solution determines <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2497_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>i</mi> </math></EquationSource> <EquationSource Format="TEX">$i$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2497_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> <EquationSource Format="TEX">$\Omega $</EquationSource> </InlineEquation> together. His spherical trigonometric solution calculates <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2497_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>i</mi> </math></EquationSource> <EquationSource Format="TEX">$i$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2497_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> <EquationSource Format="TEX">$\Omega $</EquationSource> </InlineEquation>, and the sidereal period <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11207_2025_2497_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mo>′</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$T'$</EquationSource> </InlineEquation> in a single procedure.</p>

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

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发表日期：

2025年7月4日

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Mirko Husak,
Roman Brajša,
Dragan Špoljarić,
Davor Krajnović,
Domagoj Ruždjak,
Ivica Skokić,
Dragan Roša,
Damir Hržina

摘要

In September 1777 Ruđer Bošković observed and measured the sunspot 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 Bošković. We recalculated Bošković’s original example using his measurements of sunspot positions from 1777 and the equations developed here, confirming his results from 1785. Bošković’s methodology of arithmetic means determines $i$ $i$ , $Ω$ $\Omega $ , and the sidereal period $T^{'}$ $T'$ separately, while the planar trigonometric solution determines $i$ $i$ and $Ω$ $\Omega $ together. His spherical trigonometric solution calculates $i$ $i$ , $Ω$ $\Omega $ , and the sidereal period $T^{'}$ $T'$ in a single procedure.