Chapter 1: How Science Works

Picture of Alan Turing, age 16. He was the one who came up with the idea of a digital computer that then led to the technology we have today.

The mathematics that explains the natural world usually takes the form of equations. Examples include Newton's law of gravity, Maxwell's equations of electromagnetism, and Schrodinger's equation for describing the behavior of a quantum system. Sometimes equations do not have exact solutions so calculations are required. For example, Newton's law of gravity is only exact in the simple but artificial situation of two objects isolated in space. As soon as three or more objects are in the system, Newton's law becomes an approximation and the gravity between the objects must be calculated. In the nineteenth century, clever mechanical devices were invented to do calculations reliably and fairly rapidly, at least compared to doing calculations by hand on paper. The idea of a digital computer was first presented by Alan Turing in 1936, and after the Second World War the first modern computers were built. These early computers were the size of a living room, they used a lot of power, and they used relays and vacuum tubes since transistors had not yet been invented.

Computer simulation of the Earth's field in a period of normal polarity between reversals. The lines represent magnetic field lines, blue when the field points towards the center and yellow when away. The rotation axis of the Earth is centered and vertical. The dense clusters of lines are within the Earth's core.

When a computer is used to simulate some aspect of the universe, the simulation is only as good as the physics that is fed into the computer. In other words, a computer cannot take the place of theory, no matter how cleverly it is programmed. However, computation can be used to model or simulate very complex physical situations and in those cases it can lead to insights. For example, in the Solar System, Newton's law of gravity cannot be applied with perfect accuracy because there are eight planets and other small bodies orbiting the Sun and each one exerts gravity on all the others. When astronomers started to simulate the long-term behavior of the Solar System, they made some surprising discoveries. They found that small changes in the starting configuration accumulate and lead to large eventual differences in the orbits. This is sometimes called the "butterfly effect," after the idea of mathematician Edward Lorentz that the flapping of a butterfly's wings could affect the development of a tornado many miles away. Astronomers also found that the system could become chaotic, where some objects stopped being stable and repeatable in their orbits and changed their positions or were even ejected from the system entirely. Now that we know of many planetary systems beyond the Solar System, simulations are extremely important in understanding and predicting their properties.

An example of an N-body simulation: Freezeframe of a 2D Barnes-Hut simulation. 1000 bodies.

Computers allow problems to be attacked by brute force but programmers still use clever tricks to make the simulations better or run faster. For example, since gravity weakens with the inverse square of distance, in practice it's not necessary to calculate the force of every particle on every particle. It's a good approximation to only consider the nearest particles and estimate the net force from all the others. N-body simulations have been used to show how the universe went from an early hot and smooth state to a larger and cooler state filled with galaxies and clusters of galaxies. Simulations have also been used to show what happens when two galaxies interact or collide. So far we've only talked about Newtonian gravity, which is fine for simulating a solar system or a star cluster or a galaxy. But computers can also be "fed" the equations of general relativity so that the conditions near a black hole can be simulated. Or a simulation can have gas particles added since a diffuse gas behaves differently from compact objects like stars. The result is called a hydrodynamic simulation.