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Three-body dynamics in a (1+1)-dimensional relativistic self-gravitating system

J. J. Malecki and R. B. Mann
Phys. Rev. E 69, 066208 – Published 10 June 2004
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  • INTRODUCTION
  • HAMILTONIAN FORMULATION OF THE…
  • GENERAL PROPERTIES OF THE THREE-BODY…
  • METHODS FOR SOLVING THE EQUATIONS OF…
  • SOLUTION TO THE EQUATIONS OF MOTION
  • DISCUSSION
  • ACKNOWLEDGEMENTS
    • References

    Abstract

    The results of our study of the motion of a three particle, self-gravitating system in general relativistic lineal gravity is presented for an arbitrary ratio of the particle masses. We derive a canonical expression for the Hamiltonian of the system and discuss the numerical solution of the resulting equations of motion. This solution is compared to the corresponding nonrelativistic and post-Newtonian approximation solutions so that the dynamics of the fully relativistic system can be interpreted as a correction to the one-dimensional Newtonian self-gravitating system. We find that the structure of the phase space of each of these systems yields a large variety of interesting dynamics that can be divided into three distinct regions: annulus, pretzel, and chaotic; the first two being regions of quasiperiodicity while the latter is a region of chaos. By changing the relative masses of the three particles we find that the relative sizes of these three phase space regions changes, and that this deformation can be interpreted physically in terms of the gravitational interactions of the particles. Furthermore, we find that many of the interesting characteristics found in the case where all of the particles share the same mass also appear in our more general study. We find that there are additional regions of chaos in the unequal mass system which are not present in the equal mass case. We compare these results to those found in similar systems.

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    • Received 19 June 2003

    DOI:https://doi.org/10.1103/PhysRevE.69.066208

    ©2004 American Physical Society

    Authors & Affiliations

    J. J. Malecki* and R. B. Mann†

    • Department of Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

    • *Email address: jjmaleck@uwaterloo.ca
    • †Email address: mann@avatar.uwaterloo.ca

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    Issue

    Vol. 69, Iss. 6 — June 2004

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    Images

    • Figure 1
      Figure 1
      A cross section of the four potentials at V≈1.3Mtotc2 in the case that all three particles have the same mass. N—solid, pN—dashed, UpN—dotted, R—dash dotted. ρ and λ are dimensionless variables defined using the dimensionless positions ẑi of Eq. (38).Reuse & Permissions
    • Figure 2
      Figure 2
      Cross sections of the Newtonian potential at V≈1.3Mtotc2 for various mass ratios m1:m2:m3. Solid—1:1:1; dashed—1:1:4; dotted—4:4:1; dash dotted—1:4:8. Note that all discontinuities lie on one of the three bisectors (33, 34, 35) regardless of the mass ratio. ρ and λ are dimensionless variables as in Fig. 1.Reuse & Permissions
    • Figure 3
      Figure 3
      Critical values of the relativistic potential VR as a function of a given particle mass in units of Mtot (here set equal to 3). The maximum critical value occurs in the case when mj=Mtot∕2. The minimal value approaches the limit VR≃Mtot as mj→0 or Mtot.Reuse & Permissions
    • Figure 4
      Figure 4
      Cross sections of the relativistic potential at V≈1.3Mtotc2 for different ratios of the particle masses. The correspondence between line style and ratio is the same as in Fig. 2. The deformation of the potential due to changing the mass ratio is the same as in the Newtonian case. The ρ and λ are dimensionless variables as in Fig. 1.Reuse & Permissions
    • Figure 5
      Figure 5
      Cross sections of both the untransformed (left) and transformed post-Newtonian potentials at V≈1.3Mtotc2 for various mass ratios. The correspondence between ratio and line is the same as in Fig. 2. The ρ and λ are dimensionless variables as in Fig. 1.Reuse & Permissions
    • Figure 6
      Figure 6
      Examples of trajectories in the N system for different values of η and α. Each trajectory was run for 150 time steps. The small box indicates the starting position of the trajectory. Proceeding clockwise from the top left plot the symbol sequences are B¯, B¯, AB3A2B3¯, and B6AB9A¯.Reuse & Permissions
    • Figure 7
      Figure 7
      Examples of relativistic annulus trajectories for different values of η and α. Note the characteristic boxy shape at higher values of α. Each trajectory was run for 200 time steps. All trajectories have the symbol sequence B¯.Reuse & Permissions
    • Figure 8
      Figure 8
      Examples of relativistic pretzel trajectories for different values of η and α. Each trajectory was run for 200 time steps. The top right plot has symbol sequence A2B3¯.Reuse & Permissions
    • Figure 9
      Figure 9
      The relative position of each particle with respect to the center of mass is plotted as a function of time for various values of α in the R system. The particles 1, 2, and 3 have relative masses in the ratio 1:1:α. Solid line—particle 1, dotted line—particle 2, and dashed line—particle 3. Each plot uses the same initial values of (ρ,λ,pρ,pλ) but the total energy η+1 is fixed by the energy constraint (12). The top two plots display annulus motion (B¯) while the bottom two are classified as pretzel trajectories [(B6A)7B3¯ and A2B3(AB3)5¯, respectively].Reuse & Permissions
    • Figure 10
      Figure 10
      The relative positions of each particle with respect to the center of mass as a function of time in the N system. These plots were created using the same procedure as in Fig. 9 and follow the same conventions except that Eq. (20) is used to fix the value of the total energy η. The first is an annulus trajectory (B¯) while the remaining are pretzels [B9A¯, (B3A2)4B3A3¯, and (B3A5)2B3A4¯ from top to bottom].Reuse & Permissions
    • Figure 11
      Figure 11
      Relative motion of the particles with respect to the center of mass plotted as a function of time for the R system (top) and the N system (bottom). Both plots have mass ratios 1:1:100 or α=100. The lines are as defined in Fig. 9. The insets show the small perturbation in the motion of the large mass due to the crossing of the smaller masses.Reuse & Permissions
    • Figure 12
      Figure 12
      Relative motion of the particles for the case where the mass ratio is 1:1:0.01 or α=0.01 for both the R (top) and N (bottom) systems. The insets show the motion of the stable, two body subsystem made up of the two heavy particles, as well as the effect of encounters with the light particle.Reuse & Permissions
    • Figure 13
      Figure 13
      A Poincaré map of the Newtonian system when all of the particle masses are equal.Reuse & Permissions
    • Figure 14
      Figure 14
      A Poincaré map of the relativistic system when the masses of all of the particles are equal.Reuse & Permissions
    • Figure 15
      Figure 15
      Poincaré plots with α=0.1 for the Newtonian (left) and relativistic (right) systems. The insets on the right show the onset of chaos in the pretzel region.Reuse & Permissions
    • Figure 16
      Figure 16
      Poincaré plots with α=10 for the Newtonian (left) and relativistic (right) systems. The insets show additional regions of chaos in the pretzel region that are not present in the corresponding region on the equal mass Poincare section.Reuse & Permissions
    • Figure 17
      Figure 17
      Poincaré plots with a mass ratio of 1:5:10 for the Newtonian (left) and relativistic (right) systems. On the left, (a) marks the region of chaos separating annulus trajectories (inside) and predominantly pretzel trajectories (outside) while the densely filled area directly above and below (b) marks a new region of chaos amongst the pretzel trajectories. On the right, the densely filled regions marked by a (1) were created by a single trajectory separating the annulus and pretzel orbits while the chaotic regions marked by (2) were created by a trajectory within the pretzel region.Reuse & Permissions
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