Conservation Laws: Energy & Momentum

Green arrow: distance covered in days at the current velocity. View radius: AU. Time: 1 s on screen = .

What to notice 4 bullets
  • Press Play (or Step) on the Elliptical preset: $K$ and $U$ trade places while $\varepsilon = K + U$ stays fixed.
  • Escape is set by energy: $\varepsilon<0$ (bound), $\varepsilon=0$ (escape), $\varepsilon>0$ (unbound).
  • At the same $r_0$, $v_{\rm esc}=\sqrt{2}\,v_{\rm circ}$. The slider passes from elliptical at 1.41 to hyperbolic at 1.42; the Escape preset sets exactly $\sqrt{2}$.
  • At the same speed, a larger $|h|$ keeps the closest approach $r_p$ larger.
Model notes AU / yr / $M_\odot$
  • Teaching units: AU / yr / $M_{\odot}$ with $G=4\pi^2\,\mathrm{AU}^3/(\mathrm{yr}^2\,M_{\odot})$ and $\mu = GM$.
  • The body starts at $r_0$ on the $+x$ axis, moving at $v/v_{\rm circ}$ times the circular speed there, tilted from tangential (+ is outward).
  • The green arrow is the distance the body would cover in the stated number of days at its current velocity, drawn at the orbit's scale, unless the caption says it was shortened to fit.
  • Paths reaching beyond $6\,r_0$ are clipped to that window, bound or not, and the window stays between $1.5\,r_0$ and 50 AU; the caption gives how far a clipped bound orbit reaches.
  • With zero speed there is no angular momentum and no conic: the body falls straight in.