Sensitive Threesome

When ancient astronomers looked up to the skies, they found comfort in knowing that the stars and the planets would remain unchanging; they were anchored foundations that represented stability in the universe… but not for long.

The technological advancement of telescopes in the 1700s allowed for the gathering of extremely accurate astronomical data about the positions and movement of bodies in the sky. Physicists such as Newton and Kepler wanted to be able to describe the mechanics of the solar system in a succinct way, which led to them considering the n-body problem. The solutions to this problem would define the position and velocities of the bodies in that system over a certain amount of time.

Historically, this problem has been used by astronomers when trying to predict the motion of the Earth and Moon around the Sun, making up a three-body problem. Inherently, the three-body and above problems are chaotic, so we can’t use initial conditions that are precise enough to be able to fully describe the motions. This idea can be easily demonstrated using a simulation I found online which models the movement of a planet in a binary star system. By changing the initial mass of one of the stars by an extremely miniscule amount, the resulting path differs by a surprisingly significant amount. In Figure 1 the star has a relative of mass of 1, in Figure 2 a mass of 1.1, and in Figure 3 a mass of 1.00001. Try it out for yourself if you don’t believe me!

orbit1
Figure 1
orbit2
Figure 2
orbit3
Figure 3

But before considering the difficult three-body problem, it seems a sensible idea to start by looking at the closely-related two-body problem which may offer light on a suitable approach to the problem. Our two-body problem can be further simplified by considering one of the bodies as stationary, and only examining how the other body is affected by gravitational forces. In 1687, Newton used this method to solve the two-body problem; he found that the motion of two bodies which are bound together by gravity will always follow an ellipse, with its centre at the shared centre of gravity of the two bodies, known as the barycenter. Less than a century beforehand, Kepler published three laws of planetary motion based on extremely accurate data provided to him by Tycho Brahe, the first of which accounted for the elliptical orbit of planets around the Sun. Newton’s calculations had derived and confirmed the predictions made by Kepler, which demonstrates consistency between observation and theory.

More recently, the ideas behind solving the two-body problem have been applied when looking at a hydrogen atom, which involves a pair of a proton and an electron. Even though quantum mechanics underpins the nature of these very tiny particles, we can learn a lot about the electron shell structure by considering them as a two-body ‘orbit’ system, since the laws of electrostatics obey an inverse square law just like gravitation. Understanding electron configurations in atoms and the so-called ‘motion’ of subatomic particles is important in modern science. It allows scientists to describe chemical properties of compounds and the chemical bonds which bind them together.

Unlike the closely-related two-body problem, it has been proven that analytical solutions to the three-body problem cannot be found. Without going into too much detail, it’s unsolvability comes from the fact that there are more unknowns than conserved quantities throughout the motion, like momentum and energy. However, the fact that we aren’t able to find exact solutions hasn’t posed too much of a problem. Even though there are many bodies in the solar system, everything roughly behaves as if it were in a two-body system, because the Sun is the only significant gravitational influence in close proximity. Using approximations gives us about 99% accuracy, and from obtaining solutions we have discovered the existence of equilibrium points known as Lagrangian points. A peculiar characteristic of these points is that orbits can form around them, known as “halo” orbits, even though they are just points in empty space. Scientists now use these orbits to place satellites and telescopes. For example, the James Webb telescope is planned to occupy one of these regions when it is launched in 2018.

So we’ve witnessed chaos inherent in the motion of bodies, but where else might we find chaos in the Solar System? I think we’ll find out next week… but that’s just a prediction.

Harvey

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