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Homework Help: Energy of orbits

  1. Jan 28, 2010 #1
    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
    [tex]r=\frac{L^2}{GM\mu^2*(1+e cos\theta)}[/tex]
    [tex]v^2=GM \left (\frac{2}{r}-\frac{1}{a} \right )[/tex]

    3. The attempt at a solution
    [tex]E=T+U=\frac{1}{2}\mu v^2 -\frac{GM\mu}{r}[/tex]
    [tex]=\frac{1}{2}\mu GM \left (\frac{2}{r}-\frac{1}{a} \right ) -\frac{GM\mu}{r}[/tex]
    [tex]= \frac{\mu GM}{r}-\frac{\mu GM}{2a} -\frac{GM\mu}{r}[/tex]
    [tex]=-\frac{\mu GM}{2a} [/tex]
    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. jcsd
  3. Jan 28, 2010 #2


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    Science Advisor
    Homework Helper

    Hi E92M3! :smile:
    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 :wink:)
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