Deep impact

If half of the earth is blown away by the impact of a comet, what happens to the orbit of the moon?

During impact

During impact

After impact

After impact

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Details and Assumptions:

  • The mass of the earth \(M\) is simply halved by the impact, without the fragments interacting with the moon of mass \(m.\)
  • Before the impact, the moon's orbit is a perfect circle with radius \(r_0.\)
  • \(M \gg m,\) so the moon can hardly affect the motion of the earth.
  • Both the energy \(E = \frac{1}{2} m v^2 + V(r)\) and the angular momentum \(L = m r^2 \dot \phi \) of the moon are preserved, where \(r\) is an arbitrary distance between the earth and moon after the impact.
  • The gravitational potential energy of the earth and moon system reads \[V(r) = - G \frac{M m}{r}, \] where \(G\) is the gravitational constant.
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