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I performed an N-body simulation of a binary system ($m_1 = 0.3 M_\odot$, $m_2 = 1.3 M_\odot$, $e = 0$, $T_\text{orb} = 10 \,\text{yr}$) on a circular orbit around the center of a Plummer's distribution of particles at equilibrium. The simulation, over multiple revolution orbital periods, shows that the COM of the binary, is inspiraling towards the center of the distribution. Meanwhile, the rotational period of the internal motion of the binary decreases with time, namely, the binary hardens. The binary also seems to get more and more eccentric during the simulation. Now, I should interpret the result from a physical point of view. I expect the inspiral of the center of mass to be linked to dynamical friction, but what about the internal motion of the binary? I was thinking about tidal force due to a gradient in gravitational potential, but I would expect that to soften the binary instead of hardening it. Here is the result of the simulation.

Vincent Thacker
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Physicyclist
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1 Answers1

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The simulation, over multiple revolution orbital periods, shows that the COM of the binary, is inspiraling towards the center of the distribution.

As you suggest, this can be attributed to dynamical friction (or equivalently mass segregation). The binary system is much more massive than other particles in your system, so it has a lot more orbital energy, i.e., it is dynamically hotter. It will tend to transfer that heat to other particles. Thus it loses energy and sinks to the center.

Meanwhile, the rotational period of the internal motion of the binary decreases with time, namely the binary hardens.

This effect can be attributed to Heggie's law. If your binary has an internal orbital speed much faster than the velocities of the field particles, or if the binary particles are much more massive, then the binary is said to be hard, and it will continue to harden over time due to interactions with the field particles. Essentially this is because the binary is internally hotter than its surroundings, so heat will transfer from the binary orbit to the surroundings. That transfer heats the binary further over time, due to its negative heat capacity. An approximate criterion for hardness might be written as $$\mu v_\mathrm{orb}^2 \gtrsim \bar m \sigma^2,$$ where $\mu=m_1 m_2/(m_1+m_2)$ is the reduced mass of the binary, $v_\mathrm{orb}\simeq\sqrt{G(m_1+m_2)/r}$ is the binary orbit speed (where $r$ could be e.g. the binary semi-major axis), $\bar m$ is the mean mass of field particles, and $\sigma$ is the velocity dispersion of field particles. If this criterion is satisfied, then we expect the binary orbit to shrink over time as the binary hardens.

The binary also seems to get more and more eccentric during the simulation.

This is likely due to a combination of the following two effects:

  1. Eccentricity thermalization. Interactions with other particles tend to drive binaries towards a thermal eccentricity distribution, $$f(e)\propto e.$$ Basically this corresponds to all angles being randomized. Although this process is random, it would tend to drive an initially circular binary orbit away from circular.

  2. Secular evolution due to tidal forces. Basically, the tidal force from the broader system means that the gravitational potential for the internal binary orbit is not spherically symmetric, so the angular momentum of the binary orbit is not conserved. You can derive detailed equations of motion for the eccentricity (see for example arXiv:1902.01344). One feature that emerges is that a binary system won't secularly evolve away from an exactly circular orbit ($e=0$). However, if there is a small perturbation away from $e=0$ (e.g. due to eccentricity thermalization), then that will be amplified by the secular evolution.

Sten
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