<p><em>Such a three-body system is in principle expected to show a chaotic behavior sooner or later.</em> <strong>Nope</strong>. <em>Hierchical</em> multiple systems (such as this), where the semi-major axes differ by a factor ten or larger may well be stable for ever (never become chaotic), in particular if the eccentricities are low and if the most massive object is in a tight binary. </p>

<p>An unstable three-particle system will eventually result in (typically) the two most massive objects in a tight binary and the third particle ejected (unbound). The time scale for this to happen is of the order of several (10-100) dynamical times and is indeed a highly chaotic process.</p>

<p>The concept of the Lyapunov time scale is not too useful here. One issue is that as soon as one object is ejected (unbound), the system is no longer bounded, when the concept of Lyapunov becomes problematic. Another issue is that the Lyapunov time is defined in the limit of infinite time and does not necessary reflect the behaviour of the system over any finite time.</p>

<p>Finally, to <strong>answer your question</strong>. I think there is no stringent way. What one can do is to integrate numerically many realisations of the system, each equally in agreement with the data (and their uncertainties). Then one can see whether there are any stable configurations and how frequent they occur. Given that the system didn't form yesterday, it appears likely that it is indeed stable.</p>

CC BY-SA 3.0
2014-01-11T08:44:34.850Z
Thanks for this very interesting answer! Do you have some pointers to further readings about the methods used to analyze multiple hierarchical systems for example? Cheers
Dilaton
CC BY-SA 3.0
2014-01-10T20:46:53.307Z
2014-01-10T21:46:17.103Z
2014-01-10T21:46:17.103Z
Walter
<p>As for example explained at the beginning of <a href="http://motls.blogspot.de/2014/01/trinary-star-wont-disprove-equivalence.html?m=1" rel="nofollow noreferrer">this</a> blog post, the trinary system consists of a millisecond pulsar ($1.438$ times the mass of the sun) orbited by two white dwarfs. One of the white dwarfs ( $0.198$ solar masses) is very close to the pulsar and has an orbit period of $1.6$ d, whereas the other one ($0.410$ solar masses) is farther away and and needs about a year ($327$ d) to orbit the central pulsar.</p>

<p>Such a three-body system is in principle expected to show a chaotic behavior sooner or later, which means that collisions between these three celestial bodies can be expected and a finit life time of the system can be assumed.</p>

<p>Making some in my opinion way too hand-waving arguments, the blog post further explains that the collisions can not be expected too soon however, by taking into account that the distant white dwarf "sees" the inner white dwarf and the pulsar as a single central body and the relative motion of the inner white dwarf around the pulsar is rather stable and eliptic too. </p>

<p>Thinking about such multiple star systems as chaotic dynamical systems, another approach to estimate the liftime could be to make use of some chaos-theoretic methods which could for example involve the <a href="http://en.wikipedia.org/wiki/Lyapunov_exponent" rel="nofollow noreferrer">Lyapunov Exponent</a> of the system, such that a large exponent would mean that the collisions happen soon and the stellar system has a rather short life time, whereas the converse would be true if the Lyapunov exponent is small (which is what I would expect for the system in my question).</p>

<p>So in short my question is: how can the liftime of a multiple star system be calculated in a not just hand-waving way?</p>

<p><a href="https://astronomy.stackexchange.com/q/58/92">This</a> question is interestingly related to my issue, but it does not yet answer it...</p>

CC BY-SA 3.0
2014-01-10T18:10:57.950Z
One place to look at is Kozai mechanism, which describes the effect of a third body on the parameters of the binary. It can potentially make the WD and NS collide with each other.
Alexey Bobrick
<stellar-dynamics><astrophysics>
How can the life time of a multiple star system, such as for example the trinary system PSR J0337+1715 be derived?
Dilaton
# How can the life time of a multiple star system, such as for example the trinary system PSR J0337+1715 be derived?

(Asked by: username_0 on 2014-01-10)

As for example explained at the beginning of [this](http://motls.blogspot.de/2014/01/trinary-star-wont-disprove-equivalence.html?m=1) blog post, the trinary system consists of a millisecond pulsar ($1.438$ times the mass of the sun) orbited by two white dwarfs. One of the white dwarfs ( $0.198$ solar masses) is very close to the pulsar and has an orbit period of $1.6$ d, whereas the other one ($0.410$ solar masses) is farther away and and needs about a year ($327$ d) to orbit the central pulsar.

Such a three-body system is in principle expected to show a chaotic behavior sooner or later, which means that collisions between these three celestial bodies can be expected and a finit life time of the system can be assumed.

Making some in my opinion way too hand-waving arguments, the blog post further explains that the collisions can not be expected too soon however, by taking into account that the distant white dwarf "sees" the inner white dwarf and the pulsar as a single central body and the relative motion of the inner white dwarf around the pulsar is rather stable and eliptic too.

Thinking about such multiple star systems as chaotic dynamical systems, another approach to estimate the liftime could be to make use of some chaos-theoretic methods which could for example involve the [Lyapunov Exponent](http://en.wikipedia.org/wiki/Lyapunov_exponent) of the system, such that a large exponent would mean that the collisions happen soon and the stellar system has a rather short life time, whereas the converse would be true if the Lyapunov exponent is small (which is what I would expect for the system in my question).

So in short my question is: how can the liftime of a multiple star system be calculated in a not just hand-waving way?

[This](https://astronomy.stackexchange.com/q/58/92) question is interestingly related to my issue, but it does not yet answer it...

**Tags:** stellar-dynamics, astrophysics

> **Comments:**
> **username_1**: One place to look at is Kozai mechanism, which describes the effect of a third body on the parameters of the binary. It can potentially make the WD and NS collide with each other.  

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## 1 answers

### Answer by username_2 on 2014-01-10. Score: 6 | Accepted answer

*Such a three-body system is in principle expected to show a chaotic behavior sooner or later.* **Nope**. *Hierchical* multiple systems (such as this), where the semi-major axes differ by a factor ten or larger may well be stable for ever (never become chaotic), in particular if the eccentricities are low and if the most massive object is in a tight binary.

An unstable three-particle system will eventually result in (typically) the two most massive objects in a tight binary and the third particle ejected (unbound). The time scale for this to happen is of the order of several (10-100) dynamical times and is indeed a highly chaotic process.

The concept of the Lyapunov time scale is not too useful here. One issue is that as soon as one object is ejected (unbound), the system is no longer bounded, when the concept of Lyapunov becomes problematic. Another issue is that the Lyapunov time is defined in the limit of infinite time and does not necessary reflect the behaviour of the system over any finite time.

Finally, to **answer your question**. I think there is no stringent way. What one can do is to integrate numerically many realisations of the system, each equally in agreement with the data (and their uncertainties). Then one can see whether there are any stable configurations and how frequent they occur. Given that the system didn't form yesterday, it appears likely that it is indeed stable.

> **Comments:**
> **username_0**: Thanks for this very interesting answer! Do you have some pointers to further readings about the methods used to analyze multiple hierarchical systems for example? Cheers  

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