Model A (Attractive)

Consider a unified model universe of particles where the only force acting between them is Gravity.

Newton's laws of motion apply at slow speeds, while Special Relativity enforces a global limit at the speed-of-light.

Assuming low speeds, Newton's law of gravity states:

F = - G mg1 mg2  ⁄  r2

For the sake of simplicity, we'll assume that G = 1 from here on.

Newton's Second Law of Motion states:

F = mi a

In standard physics, all masses are constrained to be positive. This leads to a universe where gravity is always attractive, which was bascially fine before 1998.

We can further simplify things by assuming that all particles in our model have the same mass m = 1. This means that all particles are essentially the same and we can give them the same label "A".

1-Body Problem

If we consider a single particle (or "body") of type A, then it either sits still or moves in a straight line according to Newton's first law of motion.

From a philosophical point of view, if there's only one particle then the concept of relative motion is somewhat redundant because there is nothing else to measure against. We can however consider the Gravitational Field to give an idea of what the effect would be on other particles, if there were any.

In particular, we can make a plot of the Gravitational Potential versus distance. Using the convention that higher potentials are plotted in lighter shades and lower potentials in darker ones, we get:

A (No Lines)

This shows us what the environment would be like around a single planet or star, for example, where the attractive force of gravity increases as you get closer.

We can make things a little easier to understand by plotting lines of equal potential as follows:

A (With Lines)

We can think of the lines as a bit like a "marble run" to give us an idea of how other particles would behave if they were introduced into the system. This leads us neatly onto the next section...

2-Body Problem

We can analyse what happens when 2 particles of type A attract each other due to the force of gravity. This is known as the 2-body problem.

They behave in a familiar way, either colliding, orbiting around each other according to Kepler's laws of planetary motion, or flying past each other if their relative speed is high enough.

The shape of each orbit is one of:

  • Straight line
  • Circle
  • Ellipse
  • Parabola
  • Hyperbola

depending on their initial positions and velocity:

Newtonian Orbits

Note that because the mass of each particle is the same, they each move in a similar way due to the gravitational attraction from the other. Another way of expressing this is via Newton's third law of motion which states that "for every action, there is an equal and opposite reaction".

Again we can make a plot of the gravitational potential to get an idea of what their combined gravitational effect would be on other particles:

AA

This shows us what the environment would be like around a binary star, for example.

Note that although there is a theoretical point mid-way between the 2 particles where the forces balance and other particles could be located, in practice this arrangement is only semi-stable.

3-Body Problem

Things get more involved as soon as introduce a 3rd particle. It turns out that it is not possible to solve this problem analytically and so we must resort to numerical methods (i.e. computer simulation) to find out what happens in the general case.

Nevertheless, we can get a flavour of what goes on if we start with just 2 of the particles and consider the environment around them, as experienced by the 3rd particle.

With 2 particles we can assume that they will be rotating around each other in the general case. If we measure things relative to the rotating frame then there is is also a rotational potential as well as the gravitational potential from the other 2 particles:

AA (Rotating)

Effectively, a 3rd particle experiences the rotating frame as a repulsion away from the centre point, at the same time as still being gravitationally attracted to the other 2 particles. This is commonly known as Centrifugal Force.

This tells us that if it's sufficiently far away from the first 2 then the 3rd particle might orbit around them as if they're a single combined A particle with twice the mass, located at the centre point.

On the other hand, if the 3rd particle approaches either of the first 2 or flies between them near the centre, then the resulting collision or a near miss can send them in all sorts of directions, disrupting whatever was going on beforehand.

Lagrange Points

There is a further bit of investigation we can do with the 3-body problem, which is to consider the case when the masses of the first 2 particles are different and the mass of the 3rd particle is neglible by comparison. This is known as the Restricted 3-Body Problem.

For example, if we use the relative masses of the Sun and Earth for the first 2 particles we get something like:

AAA (Rotating)

Interestingly, the repulsion due to the rotating frame dominates the attraction due to the Sun in all cases except a very close fly-by, which explains why comet orbits are very elongated.

The attraction due to the Earth dominates close to the Earth (an effect known as the Hill Sphere).

We notice that there are 5 points on the plot (L1 to L5) where the gravitational and rotational potentials balance. These are known as the Lagrange Points.

It turns out that L1, L2 and L3 are unstable, but can be useful places to position spacecraft with a minor amount of stability correction.

On the other hand, L4 and L5 (known as the Trojan Points) are stable, which means that naturally-occuring bodies such as asteroids may be found there.

Near-misses can result in bodies orbiting around L4 or L5 in what are known as Tadpole Orbits.

Slightly further out we find that bodies can orbit around L3, L4 and L5 in what are known as Horseshoe Orbits.

n-Body Problem

We can introduce yet more particles of type A, all of which attract each other gravitationally.

Some restricted cases may be analysed such as the Solar System or Moons of Jupiter.

In the general case there is little that can be done to study n-body systems analytically, but they are amenable to computer simulation, using readily-available software such as Grav Sim.

There are many examples of star systems that are essentially "stable", give-or-take the occasional close-encounter which results in the ejection of a star at high speed.

These systems can be found at a wide variety of scales:

SystemNumber of BodiesExampleImage
Open Clusters10-1k starsPleidesPleides
Globular Clusters10k-1m starsOmega CentauriOmega Centauri
Dwarf Galaxies10m-100m starsSmall Magellanic CloudSmall Magellanic Cloud
Galaxies1b-100b starsAndromedaAndromeda
Galaxy Clusters10-10k galaxiesVirgo ClusterVirgo Cluster
Superclusters10-1k clustersComa SuperclusterComa Supercluster

At the Galaxy scale and above we find that there seemingly isn't enough matter there to hold everything together in purely gravitational terms. This is the problem known as Dark Matter.

Furthermore, at the scale of the entire universe, we find that gravity seemingly isn't attractive at all, but rather is repulsive. This is the problem known as Dark Energy.

Computer Simulation

All of the above can be simulated on a home computer using standard physics. A good place to start to understand the calculations can be found here, courtesy of Professors Piet Hut and Jun Makino.

You can even download some PC software from Grav-Sim, courtesy of the author. Included are some ready-made models of globular clusters, which you can use as a starting point.

For example, an artificially-generated cluster with 10,000 stars appears as follows:

Model 10000

Summary

Although we can get a long way with Model A and always-attractive gravity, we find that there are cases where it breaks down. In particular, it cannot model any situation where the bodies are repelled by each other, as appears to be happening at the largest scales in the universe.

Conversely, if we attempt to use it as a unified model at the smallest scales then it's a non-starter because Electrons - the very first sub-atomic particles discovered by J.J. Thomson in 1897 - repel each other.

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