An elliptic gravitational relative orbit checked on declared model data: $p={10}^6$m, $e=3\mathrm{/}5$, $a=25\cdot {10}^6\mathrm{/}16$m, $b=5\cdot {10}^6\mathrm{/}4$m, $E=-3.2\cdot {10}^{17}$J and $T=125\pi \cdot {10}^9\mathrm{/}32$s, with $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$

Declare the gravitational two-label model of Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (i) with time window $I_0=\mathbb{R}$ and the following model data (decimal numbers declared for this computation; $G_{\mathrm{N}}=2.5\times {10}^{-25}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ is a declared datum of the example, not the measured value of clause (iii), and the couplings are identified with the inertial masses as in clause (ii)): $m_1=m_2=2\times {10}^{24}\mathrm{k}\mathrm{g},M=4\times {10}^{24}\mathrm{k}\mathrm{g},\mu =\frac{m_1{m}_2}{M}={10}^{24}\mathrm{k}\mathrm{g},\kappa =G_{\mathrm{N}}{m}_1{m}_2={10}^{24}\mathrm{k}\mathrm{g}{\mathrm{m}}^3{\mathrm{s}}^{-2}\mathrm{.}$ Suppose a motion $x$ of this model has a relative orbit with angular momentum magnitude $J={10}^{27}\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, conic parameter $p=\frac{J^2}{\mu \kappa }={10}^6\mathrm{m},$ and eccentricity $e=\frac{3}{5}$ (the conic data of The inverse-square gravitational relative trajectory has the Kepler polar form: $\rho =p\mathrm{/}(1+e\cos (\theta -{\theta }_0))$, $E=(\mu {\kappa }^2\mathrm{/}2J^2)(e^2-1)$, elliptic completeness and barycentric scaled ellipses clause 1). Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as in the cited conic and period items. Then the cited items predict, as exact consequences of these data:

1. the energy $E={\displaystyle \frac{\mu {\kappa }^2}{2J^2}}(e^2-1)=-{\displaystyle \frac{8}{25}}\times {10}^{18}\mathrm{J}=-3.2\times {10}^{17}\mathrm{J}$, and $E<0$ with $e<1$, consistent with the elliptic branch;
2. the semiaxes $a={\displaystyle \frac{p}{1-e^2}}={\displaystyle \frac{25}{16}}\times {10}^6\mathrm{m}$ and $b=a\sqrt{1-e^2}={\displaystyle \frac{5}{4}}\times {10}^6\mathrm{m}$, with $a=-\kappa \mathrm{/}(2E)$;
3. the relative period $T=2\pi \sqrt{{\displaystyle \frac{\mu a^3}{\kappa }}}=2\pi {\displaystyle \frac{\mu ab}{J}}={\displaystyle \frac{125\pi }{32}}\times {10}^9\mathrm{s}$, so $T^2={\displaystyle \frac{4{\pi }^2\mu a^3}{\kappa }}$;
4. the third-law form $T^2={\displaystyle \frac{4{\pi }^2{a}^3}{G_{\mathrm{N}}M}}$ of Kepler's third law for the gravitational two-label model: the relative period is $T=2\pi \mu ab\mathrm{/}J=2\pi \sqrt{\mu \mathrm{/}\kappa }{a}^{3\mathrm{/}2}$, so $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$ under the mass identification, and both labels are periodic relative to the centre of mass clause 2 under the declared identification;
5. the areal rate $J\mathrm{/}(2\mu )=500{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and the identity $\pi ab\mathrm{/}T=J\mathrm{/}(2\mu )$;
6. both labels trace ellipses with the centre of mass at a focus, the same eccentricity $e$, and semimajor axes $a_1=a_2=\frac{1}{2}a={\displaystyle \frac{25}{32}}\times {10}^6\mathrm{m}$ (equal masses), each periodic about the centre of mass with period $T$.

• Conditional on the existence of the motion. The check verifies the numerical consequences of the cited items for any motion with these conic data; it does not construct the motion, and the existence of the elliptic branch is not part of the computation.

• Declared data only. The numbers $m_1,m_2,J,p,e$ and the declared $G_{\mathrm{N}}$ are model data chosen for exact arithmetic; no measured constant is used, and no real system is modelled.