What to notice 3 bullets

• Escape is set by energy: $\varepsilon<0$ (bound), $\varepsilon=0$ (escape), $\varepsilon>0$ (unbound).
• Angular momentum controls how close you get: larger $|h|$ prevents deep plunges.
• At the same $r_0$, $v_{\rm esc}=\sqrt{2}\,v_{\rm circ}$.

Model notes AU / yr / $M_\odot$

• Teaching units: AU / yr / $M_{\odot}$ with $G=4\pi^2\,\mathrm{AU}^3/(\mathrm{yr}^2\,M_{\odot})$.
• We start at $r_0$ on the +x axis and set $v$ using $v/v_{\rm circ}$ and a direction offset from tangential.
• For open orbits, the plotted path is clipped to a finite radius window for visibility.