= Solution
{c}
Neglecting the planets' <masses> compared with the <Sun>'s <mass>, <Newton's law of universal gravitation> supplies the <centripetal acceleration> of each <circular orbit>:
$$
\frac{GM_\odot}{r^2}=\frac{4\pi^2r}{T^2}.
$$
Thus <Kepler's third law>, calibrated by the <Earth>'s <orbital period> of one year and <orbital radius> of one <astronomical unit>, gives
$$
\frac{T}{1\,\mathrm{yr}}=\left(\frac{r}{1\,\mathrm{AU}}\right)^{3/2},\qquad
\boxed{T_J=5.2^{3/2}=11.86\,\mathrm{yr},\quad T_S=9.5^{3/2}=29.28\,\mathrm{yr}.}
$$
Both <orbits> are taken to have the same direction. Their relative <angular speed> is $2\pi(1/T_J-1/T_S)$, so the <Sun>-centred interval between successive <conjunctions> is the <synodic period>
$$
\boxed{T_{\rm syn}=\frac{1}{1/T_J-1/T_S}=19.93\,\mathrm{yr}.}
$$
Using the supplied rounded <orbital radii> explains the small difference from the commonly quoted approximately twenty-year interval.
From the <Earth>, the two lines of sight are displaced differently because <Jupiter> and <Saturn> are at different <distances>. The <Earth>'s annual <motion> along its <orbit> therefore shifts the dates of apparent <conjunctions> by amounts of order months. The broad recurrence remains about twenty years, but successive apparent <conjunctions> need not be equally spaced: <apparent retrograde motion> can produce a group of three <conjunctions> separated by months during one encounter. The <Earth>'s daily rotation adds a much smaller change of viewpoint. This is a geometrical comparison in the stipulated coplanar model; daylight and the horizon can prevent a particular <conjunction> from being visible at a chosen surface location.
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