= Tangential launch trajectories across the escape threshold
{title2=$r=\tfrac{h^2/\mu}{1+e\cos f}$}
The same initial distance can lead to strikingly different <orbits>. Let $s=v_0/\sqrt{\mu/r_0}\geq1$ and launch tangentially in one <orbital plane>. Then <periapsis> is the launch point, <specific angular momentum> is $h=r_0v_0$, and
$$
\frac{\varepsilon}{\mu/r_0}=\frac{s^2}{2}-1,\qquad e=s^2-1,\qquad \frac{r}{r_0}=\frac{s^2}{1+(s^2-1)\cos f},
$$
where $f$ is <true anomaly>. These follow by combining the initial <specific orbital energy> with the <orbital eccentricity> relation and the <Kepler orbit> equation. With $\mu=8\,\mathrm{m^3/s^2}$ and $r_0=2\,\mathrm m$, the initial <speeds> $2$, $\sqrt6$, $2\sqrt2$, and $2\sqrt3\,\mathrm{m/s}$ give $e=0,1/2,1,2$: a <circular orbit>, an <elliptic orbit> with <apoapsis distance> $6\,\mathrm m$, a <parabolic trajectory>, and a <hyperbolic Kepler orbit>. The two open paths have asymptotic <speeds> zero and $2\,\mathrm{m/s}$ respectively. Their different directions at large distance are part of the <Kepler orbit>, not an extra scalar escape threshold.
\Image[/escape-threshold-tangential-kepler-trajectories.png]
{title=Tangential launches give circular, elliptic, parabolic, and hyperbolic <Kepler orbits>}
{description=All panels use equal spatial scales, a fixed attracting centre at the origin, a launch point $(r_0,0)$, and an initially upward <velocity>. The gray body's radius is stipulated as $R=r_0/2$, so all displayed paths remain exterior. Colored solid curves follow motion after launch; dashed curves on the open <orbits> are their incoming extensions, not an additional part of the launch. Open curves continue beyond the viewing window. The normalized launch <speeds> and <specific orbital energies> are computed from the stated isolated model.}
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