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Chaos in the vicinity of a singularity in the Three-Body Problem: The equilateral triangle experiment in the zero angular momentum limit
by Hugo D. Parischewsky, Gustavo Ceballos, Alessandro A. Trani, Nathan W. C. Leigh
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Alessandro Trani |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202207_00010v1 (pdf) |
| Date accepted: | 2022-10-17 |
| Date submitted: | 2022-07-07 14:52 |
| Submitted by: | Trani, Alessandro |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We present numerical simulations of the gravitational three-body problem, in which three particles lie at rest close to the vertices of an equilateral triangle. In the unperturbed problem, the three particles fall towards the center of mass of the system to form a three-body collision, or singularity, where the particles overlap in space and time. By perturbing the initial positions of the particles, we are able to study chaos in the vicinity of the singularity. Here we cover both the singular region close to the unperturbed configuration and the binary-single scattering regime when one side of the triangle is very short compared to the other two. We make phase space plots to study the regular and ergodic subsets of our simulations and compare them with the outcomes expected from the statistical escape theory of the three-body problem. We further provide fits to the ergodic subset to characterize the properties of the left-over binaries. We identify the discrepancy between the statistical theory and the simulations in the regular subset of interactions, which only exhibits weak chaos. As we decrease the scale of the perturbations in the initial positions, the phase space becomes entirely dominated by regular interactions, according to our metric for chaos. Finally, we show the effect of general relativity corrections by simulating the same scenario with the inclusion of post-Newtonian corrections to the equations of motion.
Author comments upon resubmission
List of changes
Please see the attached PDF.
Published as SciPost Phys. Core 6, 016 (2023)
Anonymous on 2022-10-05 [id 2879]
The authors have provided satisfactory replies to all the comments raised previously. They have also made meaningful changes in line with the comments. The paper is now acceptable for publication.
Anonymous on 2022-09-13 [id 2809]
The authors have provided clear and satisfactory responses for all the points raised previously. It would be great if the authors could address following minor points before publication:
Page 2: It would be more accurate to say that “More than a century passed”, instead of “Many centuries passed”.
Page 14: Authors point out that that regular regions appear at approximately regular intervals. Is there a simple explanation for this behavior? Is it possible to derive an analytical expression for the location of these regular regions?
I wanted to make sure I understood Figure 6 correctly. In the panel g, it looks like all the particles are grey, meaning that these systems ended in mergers. It is my understanding that these systems belong to binary-single regime. Since, you are simulating 1 solar mass objects for 10^5 years, it looks like the binary components need to be very close (a1~ 10^-3 AU) for them to merge within the integration time. Are all binary separations in panel g this close?
In the Section 5.2 where the authors describe post-newtonian results, it should also be noted that while newtonian results are scale invariant, post-newtonian results are not. More specifically, the results from these simulations cannot be directly applied to other systems with different of masses.
Page14, Section 5.3.1: Typo: The top panel of Figure 7 does not show binary energies. It would be great if authors could comment and describe each panel of the figure. Also, It is not clear if the bottom row is necessary as the theoretical predictions can be plot on similar plots in the first two rows.
It looks like Figure 9 and Section 5.3.4 would work better in Section 5.1 where authors describe the results of simulations.
Section 6.1.2: It is not clear how authors can conclude that the binaries formed from 3-body interactions in isotropic star clusters would rapidly coalesce. Since post-newtonian results are not scale invariant, results from the simulations done in this work may not always be applicable. For instance, many low mass stars can take billions of years to merge even at smaller separations.
Anonymous on 2022-10-03 [id 2869]
(in reply to Anonymous Comment on 2022-09-13 [id 2809])We thank the referee for the throughout review. We implemented the revisions, and improved the paper. In the attachment is our reply to each of the raised points. Due to the manuscript's size, its file cannot be uploaded here, therefore we provide the following link:
http://gofile.me/6fCm8/u8HUevDnx
Kind regards,
On behalf of the authors,
A.A.T.
Attachment:
Equilater_triangle_ref_reply.pdf