# Energy of orbits

1. Jan 28, 2010

### E92M3

1. The problem statement, all variables and given/known data
Show the the total energy of a parabolic is zero.
Show that the energy of a hyperbolic orbit is positive.

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

3. 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?

2. Jan 28, 2010

### tiny-tim

Hi E92M3!
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 )