Quantizing comets: semiclassical methods in action

One key aspect of theoretical physics is that the (rather few!) basic equations are not tied to a narrow field of applications, but that insights from celestial mechanics are equally relevant for the dynamics of quantum objects. I have written before how to interpret the Coulomb Green’s function of hydrogen or Rydberg molecules in terms of Lamberts theorem of cometary orbit determination.

Potential landscape around Jupiter (rotating frame) showing the unstable saddle points (Lagrange points), where comets and asteroids can enter or escape to join Jupiter for a while.
Potential landscape around Jupiter (rotating frame) showing the unstable saddle points (Lagrange points), where comets and asteroids can enter or escape to join Jupiter for a while.

Equally interesting is the dynamics of small celestials bodies in the vicinity of a parent body (the “restricted three body problem“). Near the Lagrange points the attraction of the sun and the effective potential in a coordinate system moving with the parent body around the sun cancel and the small objects can be trapped.
The comet Shoemaker-Levy 9 (SL9) is a prime example: SL9 was captured by Jupiter, broke apart at a close approach, and finally the string of fragments collided with Jupiter in 1994.

What would have happened with SL9 if Jupiter was contracted to a point mass?

Since SL9 was once captured, it should also have been released again. Indeed, in 2014 SL9 would have left Jupiter, as shown in the numerically integrated JPL orbit. To illustrate and simplify the transient dynamics, I have assumed in a recent publication a circular orbit of Jupiter and that SL9, Jupiter, and the sun are located in a plane. In reality the changing distance from the sun can open and close the entry points and in conjunction with precise location of the comet an escape or trapping becomes feasible:

Dynamics around Jupiter (located at (0,0)) for two slightly different initial kinetic energies. The shaded area indicates the energetically allowed regions. On the right: after encircling Jupiter many times, the object escapes (or if you reverse time, becomes trapped).
Dynamics around Jupiter (located at (0,0)) for two slightly different initial kinetic energies. The shaded area indicates the energetically allowed regions. On the right: after encircling Jupiter many times, the object escapes (or if you reverse time, becomes trapped). (C) Tobias Kramer.

The Shoemaker-Levy 9 case has been studied extensively in the astronomical literature (see for instance the orbital analysis by Benner et al). So what new insights are there for quantum objects? The goal is not to claim that comets have to be treated as quantum mechanical objects, but to realize that exactly the same dynamics seen in celestial mechanics guides electrons in magnetic fields through wave guides. I refer you for the details to my article Transient capture of electrons in magnetic fields, or: comets in the restricted three-body problem, but want to close by showing the electronic eigenfunctions, which show a real quantum feature absent in the classical case: electrons can tunnel through the forbidden area and thus will always escape from the parent body:

Spectrum and gallery of eigenstates for the quantized version of the celestial dynamics around Lagrange points. The quantum case describes the motion of an electron in a magnetic field.
Spectrum and gallery of eigenstates for the quantized version of the celestial dynamics around Lagrange points. The quantum case describes the motion of an electron in a magnetic field. (C) Tobias Kramer.

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