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

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Adopt the two-label gravitational 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): inertial masses $m_1,m_2>0$, $M=m_1+m_2$, reduced mass $\mu =m_1{m}_2\mathrm{/}M$, gravitational couplings $m_1^g,m_2^g>0$, $\kappa =G_{\mathrm{N}}{m}_1^g{m}_2^g>0$, and the central pair potential $u(\rho )=-\kappa \mathrm{/}\rho$ on $J_0=(0,\mathrm{\infty })$, and model time window $I_0$. Let $x:I\to Q$ be a motion on a time interval $I$ with nonempty interior, with relative coordinate $r$, attached angular momentum $j=\mu r\times r^{\mathrm{\prime }}$ and energy $E=\frac{1}{2}\mu \parallel r^{\mathrm{\prime }}{\parallel }_2^2-\kappa \mathrm{/}\mid r\mid$ (Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential), and suppose $j=J{e}_3$, $J>0$, for an oriented orthonormal triad $(e_1,e_2,e_3)$. Let $(\rho ,\theta )$ be polar data for $r$ and put $p:=\frac{J^2}{\mu \kappa },U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}(\rho ):=-\frac{\kappa }{\rho }+\frac{J^2}{2\mu {\rho }^2},{\rho }_{\ast }:=\frac{J^2}{\mu \kappa }=p\mathrm{.}$ Then:

In the orbital plane $\mathrm{\Pi }:=\mathrm{span}(e_1,e_2)$, for a point $F\in \mathrm{\Pi }$, a line $D\subseteq \mathrm{\Pi }$ with $F\mathrm{\notin }D$, and $e>0$, define the focus-directrix conic of eccentricity $e$ to be $C(F,D,e):=\{y\in \mathrm{\Pi }:\parallel y-F{\parallel }_2=e{d}_{\mathrm{\Pi }}(y,D)\},d_{\mathrm{\Pi }}(y,D):={\inf }_{z\in D}\parallel y-z{\parallel }_2\mathrm{.}$ Here $F$ is a focus, $D$ is its directrix, and $e$ is the distance ratio in the definition. The types $0<e<1$, $e=1$, and $e>1$ are called ellipse, parabola, and hyperbola, respectively. The case $e=0$ is the circle of radius $p$ centered at $F$, the circular ellipse with coincident foci at its centre. In this item, an unbounded conic is a conic locus that is unbounded, not an open subset of $\mathrm{\Pi }$; the periapsis distance is the minimum distance on the conic to its focus, and any point attaining it is a periapsis.

1. (Conic and eccentricity.) There are $e\ge 0$ and ${\theta }_0\in \mathbb{R}$ with $\rho =\frac{p}{1+e\cos (\theta -{\theta }_0)}$ on the interior of $I$, and the energy satisfies $E={\displaystyle \frac{\mu {\kappa }^2}{2J^2}}(e^2-1)$; consequently $E<0⟺e<1$, $E=0⟺e=1$ and $E>0⟺e>1$. The relative trajectory lies on a focus-directrix conic with the force centre as focus when $e>0$: its directrix is $D=\{y\in \mathrm{\Pi }:⟨y,n_0⟩=p\mathrm{/}e\}$, where $n_0:=\cos {\theta }_0{e}_1+\sin {\theta }_0{e}_2$. When $e=0$, the relative trajectory lies on the circle of radius $p$ centered at the force centre.
2. (Centrifugal barrier.) ${\rho }_{\ast }$ is the unique critical point of $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}$ in $(0,\mathrm{\infty })$ and the unique minimum, with $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}({\rho }_{\ast })=-\frac{\mu {\kappa }^2}{2J^2};$ setting ${\rho }_{\min }(E):=\frac{J^2}{\mu \kappa +\sqrt{{\mu }^2{\kappa }^2+2\mu E{J}^2}}>0,$ one has $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}({\rho }_{\min }(E))=E$ and $\rho (t)\ge {\rho }_{\min }(E)$ for every $t\in I$. In particular the relative motion never reaches the collision $r=0$, and ${\rho }_{\min }(E)=p\mathrm{/}(1+e)$ is the periapsis distance of the conic of clause 1.
3. (Elliptic case.) If $E<0$, then $e<1$ and: (a) the relative trajectory extends uniquely to a central-force trajectory for $(\mu ,u)$ defined on all of $\mathbb{R}$. Together with the unique affine extension of $X$, this defines global coordinate curves ${\bar{x}}_1=\bar{X}+(m_2\mathrm{/}M)\bar{r}$, ${\bar{x}}_2=\bar{X}-(m_1\mathrm{/}M)\bar{r}$ satisfying the same time-independent force equations; their restriction to $I_0$ is the unique extension of the model motion $x$. They are a model motion on all of $\mathbb{R}$ when $I_0=\mathbb{R}$. The relative motion is periodic, with $\theta (t+T)=\theta (t)+2\pi$ for one full angular revolution; (b) $r$ is an ellipse with focus at the force centre, semiaxes $a=\frac{p}{1-e^2}=-\frac{\kappa }{2E},b=a\sqrt{1-e^2},$ and $p\mathrm{/}(1+e)\le \rho \le p\mathrm{/}(1-e)$; (c) each label's trajectory relative to the centre of mass is the conic of clause 1 scaled by $m_j\mathrm{/}M$: with $p_1:=\frac{m_2}{M}p$, $p_2:=\frac{m_1}{M}p$ one has $\mid x_1-X\mid =\frac{p_1}{1+e\cos (\theta -{\theta }_0)},\mid x_2-X\mid =\frac{p_2}{1+e\cos (\theta -{\theta }_0)},$ so both labels trace ellipses with the centre of mass at a focus, the same eccentricity $e$ and semimajor axes $a_1=\frac{m_2}{M}a$, $a_2=\frac{m_1}{M}a$.
4. (Parabolic and hyperbolic cases on the domain of the motion.) If $E=0$, then $e=1$ and $\rho =p\mathrm{/}(1+\cos (\theta -{\theta }_0))$ with periapsis ${\rho }_{\min }(0)=p\mathrm{/}2$. If $E>0$, then $e>1$ and the formula of clause 1 holds at every $t$ with $1+e\cos (\theta -{\theta }_0)>0$, with periapsis ${\rho }_{\min }(E)=p\mathrm{/}(1+e)$. In both cases $e\ge 1$ and the corresponding conic locus is unbounded; the extent of the trajectory, its asymptotes and its scattering interpretation are not claimed here.

• Which items use the elliptic hypotheses. The completeness, closure and periodicity statements of clause 3(a) and the bounds of clause 3(b) hold only for $E<0$; the conic formula of clause 1 and the barrier of clause 2 hold for every nonzero angular momentum and every energy reached by the trajectory.

• The force centre and barycentre. In the reduced coordinate $r$ the focus is $r=0$, the force centre in relative-coordinate space. Clause 3(c) separately shows that each scaled coordinate $x_i-X$ has its focus at the centre of mass $X$.

• What is deliberately absent. No claim is made that the parabolic and hyperbolic branches have straight asymptotes, a deflection angle or a scattering cross-section on this page; those belong to the scattering development. No observation, measurement or empirical comparison appears here.