Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 13.djvu/48

Formules du mouvement de Jupiter.

Soient

${\begin{aligned}n^{\text{ıv}}t+\varepsilon ^{\text{ıv}}=&\quad 4^{\circ }{,}17858+t.33^{\circ }{,}721121,\\n^{\text{v}}\ t+\varepsilon ^{\text{v}}\ =&257^{\circ }{,}07259+t.13{,}579357,\\n^{\text{vı}}t+\varepsilon ^{\text{vı}}=&353^{\circ }{,}96753+t.\ \ 4^{\circ }{,}760710.\end{aligned}}$

Ces trois quantités sont les longitudes moyennes de Jupiter, de Saturne et d’Uranus, comptées de l’équinoxe fixe de 1750.

Soient encore

${\begin{aligned}\omega ^{\text{ıv}}=&\ \ 11^{\circ }{,}47940+t.\ \ 20''{,}527446+t^{2}.0''{,}00061759,\\\omega ^{\text{v}}\ =&\ \ 97^{\circ }{,}96370+t.\ \ 59''{,}736418+t^{2}.0''{,}00049627,\\\theta ^{\text{ıv}}\,=&108^{\circ }{,}82267+t.105''{,}832375,\\\theta ^{\text{v}}\ \,=&123^{\circ }{,}88341+t.\ \ 94''{,}677500,\end{aligned}}$

précession annuelle des équinoxes $=154''{,}63\,;$

$\varphi ^{\text{ıv}}=n^{\text{ıv}}t+\varepsilon ^{\text{ıv}}+\left(3739''{,}05-t.0''{,}11224+t^{2}.0''{,}0001078\right)$

${\begin{aligned}\times \sin &\left(5n^{\text{v}}t-2n^{\text{ıv}}t+5\varepsilon ^{\text{v}}-2\varepsilon ^{\text{ıv}}+5^{\circ }{,}0093\right.\\&\ \ \ \quad \qquad \qquad -\left.t.242''{,}25+t^{2}.0''{,}03789\right)\end{aligned}}$

${\begin{aligned}-40''{,}86\sin 2&\left(5n^{\text{v}}t-2n^{\text{ıv}}t+5\varepsilon ^{\text{v}}-2\varepsilon ^{\text{ıv}}+5^{\circ }{,}0093\right.\\&\ \ \ \quad \qquad \qquad -\left.t.242''{,}25+t^{2}.0''{,}03789\right).\end{aligned}}$

$\varphi ^{\text{v}}=n^{\text{v}}t+\varepsilon ^{\text{v}}-\left(9111''{,}406-t.0''{,}2738+t^{2}.0''{,}0002534\right)$

${\begin{aligned}\times \sin &\left(5n^{\text{v}}t-2n^{\text{ıv}}t+5\varepsilon ^{\text{v}}-2\varepsilon ^{\text{ıv}}+5^{\circ }{,}0510\right.\\&\ \ \ \quad \qquad \qquad -\left.t.237''{,}84+t^{2}.0''{,}03635\right)\end{aligned}}$

${\begin{aligned}&+(94''{,}72-t.0''{,}0053)\sin 2\left(5n^{\text{v}}t-2n^{\text{ıv}}t+5\varepsilon ^{\text{v}}-2\varepsilon ^{\text{ıv}}+5^{\circ }{,}0510\right.\\&\qquad \qquad \qquad \qquad -\left.t.237''{,}84+t^{2}.0''{,}03635\right).\\&+95''{,}76\sin \left(3n^{\text{vı}}t-n^{\text{v}}t+3\varepsilon ^{\text{vı}}-\varepsilon ^{\text{v}}-95^{\circ }{,}0779\right).\end{aligned}}$

La longitude vraie $\mathrm {V} ^{\text{ıv}}$ de Jupiter dans son orbite, et comptée de l’équinoxe moyen, sera

${\begin{aligned}\mathrm {V} ^{\text{ıv}}=\varphi ^{\text{ıv}}&+\ t.154''{,}63+(61208''{,}23+t.1''{,}9446)\sin \left(\varphi ^{\text{ıv}}-\omega ^{\text{ıv}}\right)\\&+(1838''{,}54+t.0''{,}1168)\sin 2\left(\varphi ^{\text{ıv}}-\omega ^{\text{ıv}}\right)\\&+(\quad 76''{,}57+t.0''{,}0072)\sin 3\left(\varphi ^{\text{ıv}}-\omega ^{\text{ıv}}\right)\\&+(\quad \ \ 3''{,}65+t.0''{,}0005)\sin 4\left(\varphi ^{\text{ıv}}-\omega ^{\text{ıv}}\right)\\&+\ \ \quad \qquad \qquad \qquad 0''{,}19\sin 5\left(\varphi ^{\text{ıv}}-\omega ^{\text{ıv}}\right)\end{aligned}}$