The parabolic and hyperbolic branches on declared model data: $E=0$ with $e=1$ and periapsis $p\mathrm{/}2$, and $E=1.5\cdot {10}^{18}$J with $e=2$, periapsis $p\mathrm{/}3$ and angular domain $\mid \theta -{\theta }_0\mid <2\pi \mathrm{/}3$

Declare the same model data as the companion elliptic check (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 $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$): 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 with $m_1=m_2=2\times {10}^{24}\mathrm{k}\mathrm{g},\mu ={10}^{24}\mathrm{k}\mathrm{g},\kappa =G_{\mathrm{N}}{m}_1{m}_2={10}^{24}\mathrm{k}\mathrm{g}{\mathrm{m}}^3{\mathrm{s}}^{-2},J={10}^{27}\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1},p=\frac{J^2}{\mu \kappa }={10}^6\mathrm{m},$ where $G_{\mathrm{N}}=2.5\times {10}^{-25}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ is a declared datum of the computation and the couplings are identified with the inertial masses. Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as in the cited conic item. On this data the formulas 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 give:

1. Parabolic branch. The energy $E=0$ corresponds to eccentricity $e=1$; the conic is $\rho =p\mathrm{/}(1+\cos (\theta -{\theta }_0))$, whose denominator ranges over $(0,2]$, and the periapsis is ${\rho }_{\min }(0)=p\mathrm{/}2=5\times {10}^5\mathrm{m}$, with $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}({\rho }_{\min }(0))=0=E$.
2. Hyperbolic branch. With eccentricity $e=2$ the energy is $E=\frac{\mu {\kappa }^2}{2J^2}(e^2-1)=\frac{3}{2}\times {10}^{18}\mathrm{J},$ the periapsis is ${\rho }_{\min }(E)=p\mathrm{/}(1+e)={10}^6\mathrm{/}3\mathrm{m}$, with $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}\left({10}^6\mathrm{/}3\right)=E$, and the orbit is the branch $\rho =\frac{p}{1+2\cos (\theta -{\theta }_0)}$ evaluated where $1+2\cos (\theta -{\theta }_0)>0$, that is $\mid \theta -{\theta }_0\mid <2\pi \mathrm{/}3$ modulo $2\pi$.
3. Consistency of the classification. In both cases $e\ge 1$ and the corresponding focus-directrix conic loci 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 are unbounded; the barrier of clause 2 gives $\rho \ge {\rho }_{\min }(E)>0$ along the motion, so the collision is excluded; no claim is made that the motion traverses the entire locus.

• Conditional on the existence of the motions. As in the companion elliptic check, the computations verify the formulas for any motion with the stated conic data; they do not construct the motions or settle their asymptotics.

• Declared data only. The numbers are model data chosen for exact arithmetic; no measured value, observation or scattering measurement is used.