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Add more finding options to KerrGeoFindResonance#59

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TVTien83 wants to merge 7 commits intoBlackHolePerturbationToolkit:masterfrom
TVTien83:resonance
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Add more finding options to KerrGeoFindResonance#59
TVTien83 wants to merge 7 commits intoBlackHolePerturbationToolkit:masterfrom
TVTien83:resonance

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I added more options for finding resonance orbit to the existing KerrGeoFindResonance:

  • Find r- $\phi$ resonance and $\phi$ - $\theta$ resonance; with given ratio, a and two of (p, e, x).
  • Find triple resonance with given ratio and (a, e).
  • If there is no triple resonance, there will be a message for range of possible r-integer, $\phi$ -interger and $\theta$ -integer. To get a triple resonance from the wrong one, you can change one of 3 integers by picking one value in the corresponding interval from the message and keep the other 2.

Quick guide: The syntax is similar to the existing KerrGeoFindResonance, I just modify and add things to this:

  • Same as in r- $\theta$, for r - $\phi$ and $\phi$ - $\theta$, set the $\theta$ - integer and r-interger to zero, respectively
  • For triple-resonance, only give (a, e) and set of non-zero integer.
  • For retrograde orbits, remember to use negative integers for $\theta$ and $\phi$ input; for example:
    KerrGeoFindResonance[<|"a"->0.9, "e"->0.5, "x"->-0.6|>, {1, 0, -2}].
    This helps keeping the solution unique in some KerrGeoFindResonance options.

Caution:

  • The correction result might be too close the separatrix, which makes the frequency ratio diverges; the resonant equation is unsolvable. When it happens, I set the result to KerrGeoSeparatrix[a,e,x]+10^(-5). Try to choose set of integers not far from each other, since they tend to coincide as $p$ goes to infinity.

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