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For a spherical planet, combine the hydrostatic pressure support equation with $dm/dr=4\pi r^2\rho$:Neglect surface pressure. Since $r(m)\le R$, the hydrostatic lower bound on planetary central pressure isFor a physically usual mass density decreasing outward, the mean interior density exceeds the global mean. Hence $r(m)\le R(m/M{)}^{1/3}$, giving the sharper minimum within this class,Uniform mass density attains the sharper bound; direct integration with $m(r)=M{r}^3/R^3$ gives $P_c=3G{M}^2/(8\pi R^4)$. Real planets are centrally concentrated through compression and dense cores, so their central pressure is higher.

Using $M_{\oplus }=5.97\times 10^{24}\mathrm{kg}$, $R_{\oplus }=6.37\times 10^6\mathrm{m}$, $M_J=1.90\times 10^{27}\mathrm{kg}$, and $R_J=7.15\times 10^7\mathrm{m}$, the universal bounds are approximately $5.75\times 10^{10}\mathrm{Pa}$ for Earth and $3.66\times 10^{11}\mathrm{Pa}$ for Jupiter. The uniform-density estimates are respectively $1.72\times 10^{11}\mathrm{Pa}$ and $1.10\times 10^{12}\mathrm{Pa}$, or $1.72$ and $11.0$ megabars. Either comparison gives a Jupiter minimum about 6.4 times the Earth minimum.

At $10\mathrm{au}$ from a four-solar-luminosity star, the incident flux is proportional to $4/10^2=1/25$, approximately the same as for Jupiter at $5\mathrm{au}$ around the Sun. Thus Jupiter supplies a possible old comparison at roughly the same mass and irradiation.

Assume the radius law holds from the young epoch to an old age of $4.5\mathrm{Gyr}$, that the irradiation history can be represented by the same fixed value, and that the present old radius is $R_J$. If $t$ is approximated by stellar age, thenIf $t$ instead starts at completion of formation, the model's formation delay of up to $3\mathrm{Myr}$ leaves the young planet with thermal age between $2$ and $5\mathrm{Myr}$. Taking the old thermal age as approximately $4500\mathrm{Myr}$ then gives $\alpha \simeq 0.090$–$0.102$.

Thus an exponent of order 0.1 is a reasonable conditional estimate. A single radius at one age cannot determine both the exponent and normalization of a power law; the old Jupiter comparison and the assumptions about formation time and irradiation are essential.

Let $L_J$ be Jupiter's current total emitted luminosity. Under the stipulated split, its intrinsic luminosity after removal of sunlight is approximately $L_J/2$. If the accessible energy reservoir down to $T_{\mathrm{cold}}$ is $E_J$, the constant-luminosity estimate gives $\tau \simeq E_J/(L_J/2)$.

Assume the more massive giant has approximately the same radius as Jupiter, a comparable structure factor, and an accessible cooling reservoir scaling as $G{M}^2/R$, as in the Kelvin-Helmholtz cooling time. Assume also that the final cold state contributes negligibly to that reservoir and that the giant radiates at the constant stipulated luminosity $L_J$. ThenThereforeThis is a characteristic scaling under the stated assumptions, not a detailed evolutionary age. A different mass-radius relation, a different accessible internal-energy fraction, or time-dependent luminosity changes the result; the mass and luminosity alone do not uniquely determine a cooling time.

Assume a hydrogen-helium perfect gas with mean particle mass $2.3m_H$, Jupiter gravity $g\simeq 24.8\mathrm{m}{\mathrm{s}}^{-2}$, and mixing over one atmospheric scale height. At $T=1000\mathrm{K}$,The eddy mixing time is ${\tau }_{\mathrm{mix}}\simeq H^2/K_{zz}$. A chemical quench level at one bar requires ${\tau }_{\mathrm{mix}}\lesssim {\tau }_{\mathrm{chem}}=10^5\mathrm{s}$, hence the vertical eddy diffusion coefficient must be of orderThe number depends quadratically on the assumed mixing length; using a fraction of $H$ reduces it accordingly.

Above the chemical quench level, neglect photochemistry, condensation, and molecular diffusion. The quenched atmospheric mixing ratio $f_A=n_A/n$ is approximately constant, while the number density isThus the abundance fraction is frozen, but the absolute number density falls with pressure and altitude. Examples are carbon monoxide–methane quenching, through $\mathrm{CO}+3{\mathrm{H}}_2\rightleftharpoons {\mathrm{CH}}_4+{\mathrm{H}}_2\mathrm{O}$, and nitrogen–ammonia quenching, through ${\mathrm{N}}_2+3{\mathrm{H}}_2\rightleftharpoons 2{\mathrm{NH}}_3$.

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.