Hierarchical Triple
gravity m=1 x=-0.85 y=0 vx=0 vy=0.9577 m=1 x=-1.15 y=0 vx=0 vy=-1.6243 m=1 x=2 y=0 vx=0 vy=0.6667
Open in the app The dials and keys named below are the app's.
The setup
Three equal masses, but not on equal terms. Two of them sit at x = −0.85 and x = −1.15, a mere 0.3 apart, while the third waits out at x = 2. That is a close pair with a distant companion, the arrangement most triple star systems settle into.
Two clocks
The speeds come from two circular-orbit sums stacked on top of each other. For the pair, each body circles their common centre at √(1/(2 · 0.3)) ≈ 1.291, a turn every 2π√(0.3³/2) ≈ 0.73 seconds. Now treat the pair as one mass of 2 at x = −1. The outer separation is then 3, giving a relative speed of √(3/3) = 1.
The third body and the pair split that speed two thirds to one third about the total centre of mass at the origin: 0.6667 for the loner, −0.3333 for the pair. Add the two contributions and the pair's lines read vy=0.9577 and vy=-1.6243. The outer turn takes 2π√(3³/3) ≈ 18.85 seconds, so the inner clock runs some twenty-six times faster. In the plot the pair's turns come out near 0.74, a shade long because the third body is pulling on it too.
Why the hierarchy holds
A ratio of periods that large is what keeps a triple together. The outer body feels the pair almost as a single mass and never gets between them. Over three hundred seconds the pair's separation stays between 0.300 and 0.304, and the outer distance holds at 3.00. Bring the two clocks closer together and that ceases to be true.
Try
- Top in the deck: the small circle of the pair riding around the wide one.
- Bring the companion in, x=0.8 on the third line. Its speed no longer suits that distance, so the outer orbit turns eccentric and the pair is squeezed, its separation swinging between 0.13 and 0.37.
- Widen the pair, x=-1.4 on the second line. The hierarchy fails, the pair is torn apart and the three scatter.
- Drop Speed below 1 to follow the fast pair one turn at a time.