# Hierarchical Triple

Bodies · mechanical systems

```
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](https://www.wavelace.com/app#p=84) · [This page](https://www.wavelace.com/presets/hierarchical-triple)

### 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.

### Read more

- [Star system](https://en.wikipedia.org/wiki/Star_system)
- [Three-body problem](https://en.wikipedia.org/wiki/Three-body_problem)
- [Two-body problem](https://en.wikipedia.org/wiki/Two-body_problem)
- [Circular orbit](https://en.wikipedia.org/wiki/Circular_orbit)