Problem of Two Fixed Centres (Part III)
From the archive (2024). Corrected in October 2026; see the note at the end.
Summer research 2024: the three-body problem · Part 11 of 11
A weekly research log from my summer project at the University of Edinburgh, supervised by Prof. Jenni Smillie.
This post continues the problem of two fixed centres from part 4 and part 6.
New types of orbits
At the end of part 6 I wrote:
The moving body never crosses the saddle point of the potential between the two centres, so it stays around only one of the fixed masses. This disagrees with the theory: \(S\) should behave like \(\operatorname{sn}\), so \(\sigma\) should sweep through its whole range. […] A larger initial velocity may let it pass the saddle point.
That guess was right: the assumption of zero initial velocity was too restrictive. Carefully varying the initial velocity produces two new types of orbit, lemniscate and planetary.
To classify them, we need two constants of motion. With the centres at \((\pm d, 0)\) and masses \(m_1\) and \(m_2\),
\[\begin{aligned} G&=\frac{1}{2}\left(x p_y-y p_x\right)^2+\frac{1}{2} d^2 p_x^2 \\ &\quad+d x\Biggl(\frac{m_1}{\sqrt{(x+d)^2+y^2}} \\ &\qquad-\frac{m_2}{\sqrt{(x-d)^2+y^2}}\Biggr), \\ H&=\frac{1}{2}\left(p_x^2+p_y^2\right) \\ &\quad-\frac{m_1}{\sqrt{(x+d)^2+y^2}}-\frac{m_2}{\sqrt{(x-d)^2+y^2}} . \end{aligned}\]\(H\) is the Hamiltonian (the energy), and \(G\) is the additional constant of motion discovered by Euler, which makes the problem integrable.
The paper Syzygies in the Two Center Problem shows that the values of \(H\) and \(G\) determine the type of orbit: \(S\) (satellite, the type we found earlier), \(L\) (lemniscate) and \(P\) (planetary).
My simulations reproduce all three types:
With these orbits, \(S\) now covers its full range. However, it still looks rather irregular and does not yet clearly follow \(\operatorname{sn}\):
I also computed the parameter \(e\) from Integrable Systems in Celestial Mechanics, where \(e>1\) and \(e<1\) correspond to different types of orbit. The lemniscate orbit has a much larger value, around \(e \approx 2.5\); the satellite orbit has the smallest, around \(e \approx 0.98\); and the planetary orbit has \(e \approx 1.08\). Understanding this is the next step.
Future work
- Map out which initial conditions lead to each of \(S\), \(L\) and \(P\), in the hope of finding new types of orbit.
- Extend the analysis to unequal fixed masses.
- Check whether the mass of the test particle matters.
Progress
Progress has been faster than I expected. With three weeks of the project left, it is time to start drafting the final report and poster, which keeping this log up to date makes much easier. Making a genuinely original contribution in the remaining time is ambitious, but I will see how far I can get.
Credits
This post draws on Syzygies in the Two Center Problem and the book Integrable Systems in Celestial Mechanics by Diarmuid Ó’Mathúna.
Changes on 25 October 2026
What was wrong. Nothing in the results.
What changed. The set-up now gives the positions of the two fixed masses, and the eccentricities read off the plots are marked as approximate.