Interactions between a planetary system and an FFP: A fuzzy approach

For many years, astronomers and cosmologists thought planets were always bound to stars, doomed to journey in one parabolic orbit until the star that held it in its place died. However, recent far-ranging forays into regions of space lying much beyond the solar system and observations of vast sweeps of space by farsighted telescopes have revealed a clutch of exoplanets that don't orbit any star. Instead, they wander the universe as nomadic bodies, owing no other body any allegiance.

Because of their non-aligned nature, interesting things happen when they get too close to a proper planetary system with a star at the centre. The free-floating planet (FFP) exerts its gravitational force on the system and distorts its gravitational force distribution. For the sake of simplicity, let's assume that the system consists of one sun-like star and one Jupiter-like planet going round it. As the FFP gets close enough, the force that holds the planet (BP, for binary planet) around the star will experience a pull because the newcomer will have a gravitational field that pulls the BP away from the star.

There are four possible outcomes when this encounter happens, and the probability of one outcome happening over another is determining by a set of two parameters. The first is d, the impact parameter, and the second is φ, which essentially is the position of the BP in its journey around the star. For reasons better left untold, the value of d is less than zero if the FFP is on one side of an appropriately chosen reference line, usually one that runs through the centre of the star and bifurcates the planetary system into two symmetrical mechanical systems. When the FFP is on the other side, d is positive.



The four possible outcomes are:

1. Fly-by - The newcomer just flies by, leaving the planetary system untouched and unperturbed


2. Exchange - The newcomer swaps its position with the BP, becoming the new journeyman and liberating the BP into nomadhood


3. Resonance - The newcomer joins the planetary system and becomes the second journeyman


4. Disruption - The newcomer distorts the incumbent gravitational field enough to break up the system and leave all three wandering

(The likelihood of each of these outcomes is more precisely determined by the energy of the system at its centre-of-gravity. However, the ergonomics of the system is dependent absolutely o the masses of the star, the BP and the FFP, the approach-velocity of the FFP, r₀, the distance between the BP and its star - all of which are fixed constants - and φ and d - both of which are the only controllable variables.)

After this point, it becomes incredibly difficult to assess the probability of each outcome because of the immense variability in the BP's position relative to the newcomer. So, three smart physicists at the University of Thessaloniki, with access to what I can only imagine to be a bare-minimum of computing resources, decided to tackle the problem statistically.

In statistical mechanics, there is a lack of predictability: there are no formulae that correlate the behaviour of millions of small events to the entire system. Instead, there may be a few well-established formulae that provide correlations between other aspects of the system and its state. And when comes to statistical mechanical modelling, these few formulae are used to simulate the system in different states and look for patterns. Then, with the help of such patterns, a conclusion can be gradually drawn about the influence of small-scale patterns on large-scale ones.

In this case, the physicists of Thessaloniki programmed the initial conditions of the system into a computer: velocity of the incoming FFP, φ, d, masses of the FFP, BP and the star, and r₀. Then, they brought the newcomer close enough to the planetary system to affect its mass distribution 350,000 different times, each cycle corresponding to a uniquely chosen value of φ and d. In doing this, they gained a statistical understanding of the system that solidified an "estimated" understanding of it into a perfectly understood one.

Each time they brought the planet closer to the system, the value of φ they picked had to lie between 0 and 360 (for obvious reasons). The value of d, on the other hand, had to lie between 7r₀ and -7r₀ because of the assumption that the BP would be Jupiter-like, and in being Jovian like that, that's the value of d at which disturbances would begin to be felt. Such conditions are called initial conditions and play a very important role in fuzzy logic, which is the encompassing branch of logic at work in this example.

[caption id="attachment_21228" align="aligncenter" width="450" caption="The gap between crisp logic and fuzzy logic is the same as between precision and significance: sometimes, all you need to know is what's going to make all the difference, and fuzzy logic is a good way to get that information."][/caption]

Fuzzy logic is the set of mathematical tools that work with the fact that given a set of ground rules and an initial state, the final state can be deciphered to a heartening degree of accuracy, a degree dependent only on the accuracy of the ground rules. This gives us a basis for exploring the nature of satellite concepts to those currently understood: by using what we already know as initial conditions and an initial state, we get a fuzzy idea of the nature of the new concept. Fuzzy logic is particularly powerful when studying very large systems wherein the interference of chance is great.

The abstract of the original paper can be found here.