E of Parabolic & Hyperbolic Orbits: Explained

[E92M3]

Homework Statement

Show the the total energy of a parabolic is zero.
Show that the energy of a hyperbolic orbit is positive.

Homework Equations

r=\frac{L^2}{GM\mu^2*(1+e cos\theta)}
v^2=GM \left (\frac{2}{r}-\frac{1}{a} \right )

The Attempt at a Solution

E=T+U=\frac{1}{2}\mu v^2 -\frac{GM\mu}{r}
=\frac{1}{2}\mu GM \left (\frac{2}{r}-\frac{1}{a} \right ) -\frac{GM\mu}{r}
= \frac{\mu GM}{r}-\frac{\mu GM}{2a} -\frac{GM\mu}{r}
=-\frac{\mu GM}{2a}
So somehow I have to show that a, the semi-major axis is infinity for a parabola and a is negative for a hyperbola. But what is "a" for a parabola and hyperbola? How can I define them?

 

[tiny-tim]

Hi E92M3!

So somehow I have to show that a, the semi-major axis is infinity for a parabola and a is negative for a hyperbola. But what is "a" for a parabola and hyperbola? How can I define them?

For an ellipse, the semi-major and semi-minor axes are a and b, with x²/a² + y²/b² = 1.

For a hyperbola, x²/a² - y²/b² = 1, and a is still the semi-major axis (and there is no semi-minor axis).

For a parabola, a is infinite (a bit obvious, since one focus is at infinity anyway )