Conservation Laws: Energy & Momentum

• Press Play (or Step) on the Elliptical preset: $K$ and $U$ trade places while $\varepsilon = K + U$ stays fixed.
• Switch the stage from Observatory to Potential and press Play (or Step) on Elliptical: the gold line from the energy level down to $U_{\rm eff}(r)$ is $\tfrac{1}{2}v_r^2$, and it shrinks to zero exactly where the orbit turns around, at $r_p$ and $r_a$.
• 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.

• 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.

Split the speed into radial and tangential parts, $v^2 = v_r^2 + v_t^2$. Angular momentum per unit mass is conserved, $h = r\,v_t$, so $v_t = h/r$ and the specific energy becomes

$\displaystyle \varepsilon = \tfrac{1}{2}v_r^2 + U_{\rm eff}(r), \qquad U_{\rm eff}(r) = -\frac{\mu}{r} + \frac{h^2}{2r^2}.$

• The tangential kinetic energy, $h^2/(2r^2)$, depends only on $r$ once $h$ is fixed, so it acts like a potential: a centrifugal barrier that keeps the body away from $r = 0$.
• $\tfrac{1}{2}v_r^2$ cannot be negative, so the body can only be where $U_{\rm eff}(r) \le \varepsilon$. It turns around ($v_r = 0$) where $U_{\rm eff}(r) = \varepsilon$: at $r_p$, and for a bound orbit at $r_a$.
• The curve is lowest at $r_c = h^2/\mu$, where $U_{\rm eff} = -\mu^2/(2h^2)$. An orbit with exactly that energy is circular.
• With $\varepsilon \ge 0$ there is no outer turning point: the body comes in once and leaves.
• In the plot, the gold line from the energy level down to the curve has length $\tfrac{1}{2}v_r^2$. It shrinks to zero at each turning point.