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Effective potential

  1. Mar 17, 2005 #1
    Hey everyone!

    I have an exam question, but I don't know how to approach it. The question is,

    "Discuss orbits of bodies in the Solar System using the effective potential method."

    I thought about every planet having a certain amount of kinetic and potential energy, showing how they balance out as a planet orbits the Sun. During lectures we havn't mentioned the effective potential, so hopefully someone here will enlighten me. I really hope so, and it would make me very grateful.

  2. jcsd
  3. Mar 17, 2005 #2

    Andrew Mason

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    The energy of an orbiting body is:

    [tex]E = KE + PE = \frac{1}{2}mv^2 + V(r) = constant[/tex]

    If you break the velocity into a radial and tangential component and use polar coordinates:

    [tex]KE = \frac{1}{2}m(\frac{dr}{dt})^2 + \frac{1}{2}m(rd\theta/dt)^2 = \frac{1}{2}m(\frac{dr}{dt})^2 + \frac{1}{2}mr^2\omega^2[/tex]

    Substituting angular momentum [itex]L = mr^2\omega[/itex]:

    [tex]KE = \frac{1}{2}m(\frac{dr}{dt})^2 + \frac{L^2}{2mr^2}[/tex]

    Since the force is central (no torque) L is constant. If you let:

    [tex]V_{eff}(r) = V(r) + \frac{L^2}{2mr^2}[/tex] you can write the energy equation as:

    [tex]E = \frac{1}{2}m(\frac{dr}{dt})^2+ V_{eff}(r) = constant[/tex]

    Then you can think of the variable energy in terms of the rate of change of the radius.

    Consider an oribit in which the V_eff = constant; where V_eff has a minimum and maximum; where V_eff has a minimum but no maximum.

    Last edited: Mar 17, 2005
  4. Mar 18, 2005 #3
    Thanks alot! And I mean a lot! :-)
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