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Taking the Earth's sidereal rotation period as $86164\mathrm{s}$, $\omega =2\pi /(86164\mathrm{s})$, and $G=6.6743\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ givesThe orbital radius is therefore about $6.61R$, or an altitude of about $5.61R$. Replacing the sidereal day by $86400\mathrm{s}$ changes this estimate by less than $0.2\%$.

The source's tangential arrows $a,c$ respectively point along and against the original motion; $b,d$ point radially outward and inward. Write $r_0$ for the initial orbital radius, $v_c=\sqrt{GM/r_0}$ for the circular orbital speed, and $\delta >0$ for the small added speed. The resulting trajectories follow the velocity kick on a circular Kepler orbit formulas. Here orbital energy includes kinetic energy and gravitational potential energy; angular momentum refers to the central object.

Neglecting the planets' masses compared with the Sun's mass, Newton's law of universal gravitation supplies the centripetal acceleration of each circular orbit:Thus Kepler's third law, calibrated by the Earth's orbital period of one year and orbital radius of one astronomical unit, givesBoth 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 periodUsing 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.

Coulomb centripetal balance and $\mu vr=n\hslash$ giveThe radius is the electron-nucleus separation. For positronium, $Z=1$ and $\mu =m_e/2$, so $r_n=2a_0{n}^2$ and $E_n=-\mathrm{Ry}/(2n^2)$. Each constituent's radius about the center of mass is half the separation; calling both that radius and $r_n$ the orbital radius without distinction would create another factor-two error.

Separate center-of-mass motion and replace $m$ by the reduced mass $\mu =mM/(m+M)$ in the relative-coordinate formulas. Then $r_n$ is the proton-electron separation, energy levels and the Rydberg constant scale by $\mu /m$, and transition wavelengths increase by $m/\mu$. The electron's own orbital radius about the center of mass is $M{r}_n/(M+m)$; the proton radius is $m{r}_n/(M+m)$. The relative speed retains $\alpha c/n$, while the electron's individual speed acquires $M/(M+m)$. Replacing the mass in the radius alone but leaving energies unchanged would be inconsistent.