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I'm doing a nbody simulation, and I'm interested in the formation of binary systems in the temporal evolution. I can identify them by eye, but I don't know an algoritmic criteria that can say me if two stars are a binary system or not.

Is there a deterministic criteria that can say me if two arbitrary stars in the simulation are a binary system?

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  • $\begingroup$ I think you want a discrete answer to a continuous question. $\endgroup$ – peterh - Reinstate Monica May 17 '16 at 13:55
  • $\begingroup$ It depends what you're trying to achieve and how rigorous you want to be. A simple method might simply be to identify pairs of stars that remain within some threshold distance for at least some threshold time interval. More complex approaches involve performing Fourier analysis on the coordinates to identify resonances shared amongst two or more coordinates (e.g.). $\endgroup$ – lemon May 17 '16 at 14:15
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This is really a difficult problem, but possibly not for the reason you imagine. The following naif criterion seems at first highly appropriate:

take any pair of stars, subtract the center-of-mass motion, compute the kinetic ($T$) and gravitational ($W$) energies, and check whether

$T + W < 0$.

If so, the pair is bound, otherwise it is not.

How can this criterion fail? I can think of three ways:

  1. Even in an uncrowded field, there may be three or more stars bound. It is well-known that multiple systems are unstable, so establishing which of the stars will remain bounded requires more analysis. Normally, the two heaviest stars are those that remain bounded, because the lighter ones gain energy at the expense of the heavy double, to escape the bound system.

  2. A supposedly unbound pair may later turn out to be bound after all, because of loss of angular momentum. It is well-know that the centrifugal barrier, under angular momentum conservation from the original cloud in which the stars form, is so high that nearly no stars should form. Since gravity conserves the overall amount of angular momentum, there must be non-gravitational forces at work that remove it from the collapsing gas. No one knows for sure what is responsible for these torques, even though everyone claims that they must be magnetic in origin.

    Thus it can happen that a given pair has, initially, so much angular momentum as to violate the above condition, but it can also happen that angular momentum removal is still occurring (at the time your simulations end) so that more binary pairs will form than is apparent from your simulations. Since this involves processes which are, as of now, largely speculative, no one can claim to know the exact time-evolution of these torques. This is a fundamental element of ignorance, on our part.

  3. Lastly, and least likely, in crowded fields, interactions between the (transient) pair and passers-by may unbind the binary, at the expense of the kinetic energy of the passer-by. In the limit of fully-developed stars, this has been studied especially in connection with the evolution of binaries in globular clusters (GCs, see Lyman Spitzer's Dynamical Evolution of Globular Clusters, especially Ch. 6). But you should keep in mind that in GCs two circumstances enhance the relevance of these interactions: the high stellar densities (higher than in Open Clusters), and the long time scales available, of the order of a Hubble time as compared to the few million years available before newly-formed stars walk away from their formation sites.

    Still, one circumstance makes many-body encounters more effective in Open Clusters, the large size of forming stars: when a binary encounters a passer-by, energy may be dumped from the enterloper's kinetic motion into tidal heating of either star in the binary, to be later dissipated thru radiative mechanisms. This tidal heating slows down the interloper, hence it makes the contact last longer, and the impulse transferred to the stars in the binary larger. In other words, it facilitates unbinding the binary.

Of the above, objections 1 and 3 are easily kept under control if your simulations do not involve unrealistically crowded fields (after all, Open Clusters are barely bound, and do no last long!). But objection 2 is more fundamental, and harder to keep under control. Keep in mind that, in Nature, 1 out of two stellar systems observed from Earth is a binary, so just about any simulation resulting in many fewer binaries than this is likely to be tainted by an improper accounting of electromagnetic torques.

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  • $\begingroup$ This is a really nice answer. I would suggest that all three issues are not really limitations however. Instead, they just mean that binaries are not static, i.e. they change over time. This is necessarily true. There is a fourth case however, which is more important. 4) two stars which are bound, may not be in a binary --- because they are either far apart, or are more bound to a different object. To resolve this, you can constrain your search to only nearby stars, and/or to make sure that a binary counterpart is the one which is star is most bound to. $\endgroup$ – DilithiumMatrix May 17 '16 at 16:24
  • $\begingroup$ @DilithiumMatrix are more bound to a different object is the same as my case with multiple stars: they are always unstable, when all is said and done. $\endgroup$ – MariusMatutiae May 17 '16 at 16:43
  • $\begingroup$ @DilithiumMatrix You are right about using only the nearest stars to check for a negative total energy, I should have stated that myself. My bad. $\endgroup$ – MariusMatutiae May 19 '16 at 6:31
  • $\begingroup$ @DilithiumMatrix I disagree about the transient nature of binarity: not wrong but we are interested on the statistics of binaries on the Main Sequence: we would like simulations to predict rate of binaries and distribution of major/minor axes at the ZAMS, because this is what is observable (you may include the path down the Hayashi track, but it lasts little compared to MS and it is not expected to make any difference). Basically, these statistics remain unaltered until one star fills its Roche lobe, then orbital parameters are changed. I repeat, I want theoretical predictions for the ZAMS. $\endgroup$ – MariusMatutiae May 19 '16 at 6:39
  • $\begingroup$ ... Okay... maybe you are interested in binaries on the MS, in field stars. maybe someone (perhaps the OP) is interested in stars in globular clusters, or near the galactic center, where interactions are important. For the general problem, you need to take my point into account. That's it. It's a minor addition. $\endgroup$ – DilithiumMatrix May 19 '16 at 17:52

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