Stellar Flybys and Long-term Stability

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).

Motivation

Many studies have investigated how stellar flybys affect the architecture and evolution of planetary systems. In many of these numerical studies, the systems evolved for hundreds of millions of years after the flyby event. However, many planetary systems, including the solar system, have expected lifetimes of billions of years.

We wanted to know

1. How little could a stellar flyby change the solar system as it is today and measurably increase instability?
2. How likely is a stellar flyby of this magnitude to occur?
3. What are the dominant dynamical mechanisms that lead to instability?

Methods

We ran 2,880 N-body simulations of the solar system using REBOUND to investigate the stability of the solar system in the presence of stellar flybys. For the sun and eight planets, we had one stellar flyby for each simulation. The perturbation strengths from the flybys ranged from numerical noise to unity. Each simulation ran for 4.8 billion years following the flyby event or until an escape or collision event. We used REBOUNDx to add a gr_potential to incorporate the effects of general relativity on the precession rate of Mercury. We also used the WHCKL integrator which is the Wisdom-Holman integrator ¹ with symplectic correctors and the lazy implementation of the kernel method. ² For initial conditions of the solar system we used NASA JPL Horizons data at the J2000 epoch. ³ Finally, we used a timestep of dt = 8.062 days which gives about 28 days of walltime to simulate one system for 4.8 Gyrs.

1. Main Results

We found that the solar system is in a robustly stable region of parameter space. Without any external influences, there is about a 1% chance of instability before the death of the Sun ~5 billion years from now. ^{4 5} We divided our ensemble of simulations into a control group (N = 960) and an experimental group (N = 1920). After 4.8 Gyrs, the control group had 4 instabilities (~0.42%) and the experimental group had 26 instabilities (~1.35%). However, when divided over perturbation strengths, the experimental group shows a significant difference in the fraction of instability for relative changes larger than 0.1%, see Figure 1. A change to the semi-major axis of Neptune of just 0.1% increases the chance of instability by more than 10 times.

2. Expected Flyby Rates

How often do we expect stars to flyby the solar system?

On some level stars pass by the Solar System all the time, but they are often very far away (many light years away) and have practically no influence on the system's current state. Recent observational surveys like GAIA have provided more accurate data for the closest stellar flybys to the Solar System. Two notable stars are HD 7977 (~1.2 M _⦿), which may have passed within 3,000 AU (0.0457 light years) of the Sun some 2.5 million years ago, and Gliese 710 (~0.6 M _⦿), which is expected to pass within about 10,000 AU (0.1696 light years) of the Sun in about 1.3 million years from now. ⁶

The stellar flyby rate is given by Γ = nσv, where n is the stellar density, σ is the cross-section, and v is the velocity dispersion of the stars in the region. ⁷ If we make some assumptions about the distribution of stellar masses (0.01 M _⦿ ≤ m _★ ≤ 10 M _⦿) in the region around the solar system (local neighbourhood), using GAIA data we can determine that the stellar density is 0.14 pc^-3^. Additionally, if we assume a Maxwellian distribution for the velocity of the stars in the local neighbourhood, then GAIA can give us a typical values of 20-60 km/s. ⁸ Finally, we need to assume some reasonable size for the interaction cross-section. Since the influence of a stellar flyby on the semi-major axes of the planets drops off exponentially, ^{9 10} for numerical precision reasons, we define a reasonable sphere of influence to be 10,000 AU. Thus, accounting for the effects of gravitational focusing, ¹¹ , we expect stellar flybys in the local neighbourhood to occur every 97.6 Myrs, on average.

How likely is a critical flyby?

The left panel in Figure 2 shows the average time until a flyby of a particular magnitude will occur for various stellar environments based on analytical and numerical estimates. The panel on the right shows the expected time until a flyby from a average mass star as a function of the pericentre of the flyby star (scaled by the semi-major axis of Neptune). By tracing between the left and right panels, the relationship between pericentre and perturbation strength for an average mass star can be determined.

We can also run N-body simulations to analyze the frequency of encounters that perturb a system by a particular amount. The analytical estimates in our work are for adiabatic flybys, where the relative velocity between the stars is small compared to the orbital velocity of the planets. Thus, there are some limitations in the strong interaction regime where N-body simulations perform better. We define a critical flyby to be one that changes the system by more than 0.1%. We expect a critical flyby in the local neighbourhood to occur once every 100 Gyrs. For an average mass star (~0.3 M _⦿) to cause this, it would need to have a closest approach of about 150 AU. Both of HD 7977 and Gliese 710 will have no appreciable effect to the evolution of the Solar System. For comparison with the numbers in our paper, the closer of the two stars, HD 7977, may have changed the semi-major axis of Neptune by about 8.2e-11 AU, or about 12 meters. Again, we found that critical changes to the orbit of Neptune needed to be on the order of 0.03 AU or 4.5 billion meters in order to have any impact.

3. Secular Resonances

What does secular mean?

In celestial mechanics secular comes from the classical Latin saeculum meaning "occurring once in an age," or meaning "span of time". ¹² Secular dynamics describe the long-term orbital evolution of planets around a central star. This is done by averaging over the orbital phases, allowing an investigation of the orientation of an orbit without worrying about the particular location of the planet around the star. One way to think about it is to consider smearing the mass of the planet into a ring. For a perfectly circular orbit, the ring will have a uniform mass. For a more eccentric orbit, the mass will be more heavily concentrated where the planet would spend most of its time---the apocentre. The rings then apply torques on each other influencing their shape (a, e) and orientation (i, Ω, ϖ)

What are secular resonances?

Under the influence of planet-planet interactions, the shape and orientation of planetary orbits evolve. For example, in the Newtonian two-body problem, a planet will orbit a star in a perfect ellipse. However, with an additional exterior planet, the argument of the pericentre will precess around the star. The cumulative effects of planet-planet interactions on Mercury cause Mercury's pericentre to precess at a rate of ~5.32"/yr (general relativity adds an additional ~0.43"/yr, read more). ¹³

One type of secular resonances is when the pericentre precession rate of two planets are the same. For the solar system, secular resonances are the dominant mechanism for causing instability. Because of how light Mercury is compared to Jupiter, when there is a Mercury-Jupiter pericentre resonance, the eccentricity of Mercury can be increased substantially. This effect is similar to a force-driven oscillator where the amplitude of oscillation increases dramatically when the driver closely matches the natural frequency.

The g₁-g₅ secular resonance encapsulates the pericentre resonance between Mercury and Jupiter. With Laplace-Lagrange secular theory, one can construct a matrix that captures the dominant secular interactions between the planets. ¹⁴ The eigenvaules of this matrix are the principle secular frequencies of the system. With some further linearization, these modes are sometimes associated with particular planets in the system (g₁ with Mercury, g₂ with Venus, etc.). However, these principle frequencies aren't exclusive to specific planets because they are the result of the linearization of a coupled, non-linear system. While there are other reasons why the frequencies are associated with specific planets, when g₁ and g₅ are commensurate, the eccentricity of Mercury increases dramatically.

How are secular resonances and stellar flybys related?

Due to the coupled nature of the solar system, changes to one planet are propagated over time throughout the system. Thus, while perturbations from a stellar flyby will most strongly affect Neptune, the alterations to Neptune's orbit will ripple to the other planets in the solar system, see Figure 3. These changes alter the natural principle frequencies of the system by an amount proportional to the magnitude of the perturbation.

If the changes to the system are large enough, secular resonances could occur on relatively short timescales. This is especially true for planets which are already close to resonance like Mercury and Jupiter, but more especially Earth and Mars. Figure 4 shows how artificial alterations to the orbit of Neptune directly affect the secular structure of the solar system and its dynamical stability.

Conclusions

• A change of 0.1% to the semi-major axis of Neptune's orbit can increase the chance of solar system instability by more than 10 times.

• We don't expect a flyby would critically alter the solar system within the next 100 Gyrs.

• Due to chaos, there is no one-to-one map between changes to the secular frequencies and dynamical instability, but we see a strong correlation.