Three-body problem in modified dynamics

Abstract

General properties of the three-body problem in a model of modified dynamics are investigated. It is shown that the three-body problem in this model shares some characters with the similar problem in Newtonian dynamics. Moreover, the planar restricted three-body problem is solved analytically for this type of extended gravity and it is proved that under certain conditions, which generally happen at galactic and extragalactic scales, the orbits around $L_{4}$ and $L_{5}$ Lagrange points are stable. Furthermore, a code is provided to compare the behavior of orbits in the restricted three-body problem under Newtonian and modified dynamics. Orbit integrations based on this code show contrasting orbital behavior under the two dynamics and specially exhibit in a qualitative way that the rate of ejections is smaller in the modified dynamics compared to Newtonian gravity. These results could help us to search for observational signatures of extended gravities.

Appendix A: Supplementary figures

See Figs. 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27.

Fig. 13

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=7~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm 0.1x_{0},~y_{0} \pm 0.1y_{0})$ and this particle rotates initially in the opposite direction of $m_{1}$ and $m_{2}$

Fig. 14

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=7~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm 0.5x_{0},~y_{0} \pm 0.5y_{0})$ and this particle rotates initially in the opposite direction of $m_{1}$ and $m_{2}$

Fig. 15

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=7~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm x_{0},~y_{0} \pm y_{0})$ and this particle rotates initially in the opposite direction of $m_{1}$ and $m_{2}$

Fig. 16

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=3~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm 0.1x_{0},~y_{0} \pm 0.1y_{0})$ and this particle rotates initially in the same direction of $m_{1}$ and $m_{2}$

Fig. 17

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=3~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm 0.5x_{0},~y_{0} \pm 0.5y_{0})$ and this particle rotates initially in the same direction of $m_{1}$ and $m_{2}$

Fig. 18

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=3~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm x_{0},~y_{0} \pm y_{0})$ and this particle rotates initially in the same direction of $m_{1}$ and $m_{2}$

Fig. 19

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=3~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm 0.1x_{0},~y_{0} \pm 0.1y_{0})$ and this particle rotates initially in the opposite direction of $m_{1}$ and $m_{2}$

Fig. 20

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=3~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm 0.5x_{0},~y_{0} \pm 0.5y_{0})$ and this particle rotates initially in the opposite direction of $m_{1}$ and $m_{2}$

Fig. 21

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=3~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm x_{0},~y_{0} \pm y_{0})$ and this particle rotates initially in the opposite direction of $m_{1}$ and $m_{2}$

Fig. 22

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=1~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm 0.1x_{0},~y_{0} \pm 0.1y_{0})$ and this particle rotates initially in the same direction of $m_{1}$ and $m_{2}$

Fig. 23

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=1~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm 0.5x_{0},~y_{0} \pm 0.5y_{0})$ and this particle rotates initially in the same direction of $m_{1}$ and $m_{2}$

Fig. 24

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=1~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm x_{0},~y_{0} \pm y_{0})$ and this particle rotates initially in the same direction of $m_{1}$ and $m_{2}$

Fig. 25

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=1~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm 0.1x_{0},~y_{0} \pm 0.1y_{0})$ and this particle rotates initially in the opposite direction of $m_{1}$ and $m_{2}$

Fig. 26

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=1~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm 0.5x_{0},~y_{0} \pm 0.5y_{0})$ and this particle rotates initially in the opposite direction of $m_{1}$ and $m_{2}$

Fig. 27

The evolution of a test particle around two massive objects of $m_{1}=10~ \mathcal {M}$ and $m_{2}=1~ \mathcal {M}$ with the distance of $d=20~ \mathcal {D}$. The evolution time is $3000~\mathcal {T}$, the initial position of $m_{3}$ is chosen randomly to be within $(x_{0}\pm x_{0},~y_{0} \pm y_{0})$ and this particle rotates initially in the opposite direction of $m_{1}$ and $m_{2}$