Equations for Specific Orbits

Energy determines whether an orbit is bound or not. Circles and ellipses are the only bound orbits; parabolae and hyperbolae are the only unbound ones. Note that $e = 1$ orbits (rectilinear or straight-line orbits) may be elliptical, parabolic, or hyperbolic.

$$E_B < 0 E_B \ge 0$$

Circle Ellipse Parabola Hyperbola

$$a = r a > 0 a \rightarrow \pm \infty a < 0$$ Semimajor Axis:

$$e = 0 0 < e \le 1 e = 1 e \ge 1$$ Eccentricity:

$$r = a r \ge a(1 - e) r \ge a(1 - e) r \ge a(1 - e)$$ Distance:

$$r \le a(1 + e) r \rightarrow \infty r \rightarrow \infty$$

$$v = \sqrt{\frac{GM}{a}} v \le \sqrt{\frac{GM(1 + e)}{a(1 - e)}} v = \sqrt{\frac{2GM}{r}} v \le \sqrt{\frac{GM(1 + e)}{a(1 - e)}}$$ Speed:

$$v \ge \sqrt{\frac{GM(1 - e)}{a(1 + e)}} v_\infty = 0 v_\infty = \sqrt{- \frac{GM}{a}}$$

($v_\infty = v$ in the limit $r \rightarrow \infty$, i.e., speed at ``infinity'')