ThatsMaths

An astrodynamical miracle is happening in the sky above. Our ability to launch a space-probe from the revolving Earth to reach a moving target billions of kilometres away almost ten years later with pinpoint accuracy is truly astounding. “New Horizons” promises to enhance our knowledge of the solar system and it may help us to understand our own planet too.

[See this week’s That’s Maths column (TM071): search for “thatsmaths” at irishtimes.com].

Until recently, Pluto was counted as the ninth planet. In 2006 the International Astronomical Union reclassified it as a dwarf planet. We knew that Pluto had a moon and, like the Earth and its moon, Pluto and Charon revolved around their common centre of mass in an orderly and predictable manner.

The Hubble Space Telescope images have shown up four tiny moons in addition to Charon, which have been named Styx, Nix, Kerberos and Hydra. Their orbits around the two larger bodies show signs of chaotic behaviour. They follow roughly circular paths but wobble and tumble erratically as they pass close to Pluto or Charon.

Pluto, about six billion km from Earth has not previously been observed close-up. But on 14th July – next Tuesday week – the New Horizons probe, launched by NASA in 2006, will pass close to the dwarf planet. It seems likely that more Plutonic moons will be discovered. After passing Pluto, New Horizons will follow the earlier Voyager probes into deep space, never to return to Earth.

The regular elliptic motion of the planets was observed by Johannes Kepler and explained in terms of the inverse square attraction of gravity by Newton. Laplace, sometimes styled the Newton of France, expressed the idea that for an intelligent being having complete knowledge of the present state of the universe, “nothing would be uncertain and the future, just as the past, would be present before its eyes.” According to Laplace, if the precise position and momentum of every particle in the universe is known, its past history and future motion follow from the laws of classical mechanics.

Developments in thermodynamics in the nineteenth century showed how there is an arrow of time: physical processes are inherently irreversible and the past history cannot be deduced from present conditions. Then Henri Poincaré showed that, even for a system as basic as three bodies orbiting each other, chaotic behaviour is found and the future motion cannot be predicted with certainty. Of the intricate solutions he found, he wrote: “One will be struck by the complexity of this picture, which I will not even attempt to draw.”

Poincaré’s ideas on chaotic motion go some way towards explaining why long-range weather forecasting is problematic. Tiny errors in the initial state grow rapidly, ultimately spoiling the forecast. Likewise, for chaotic planetary motions, uncertainties in the current position and movement make it impossible to predict the future motion with confidence. If the Earth’s orbit were chaotic, the lengths of days and years would vary, the progress of the seasons would be erratic and the climate would oscillate wildly. Probably, intelligent life could not have evolved in such conditions.

Thousands of “exoplanets” orbiting stars other than the Sun have now been found. Some of these are revolving around binary stars and their dynamics are correspondingly more complex. Since Pluto and Charon act like a binary system, the observations of the New Horizons probe may help us to understand such complex dynamics more fully, and to estimate the conditions on these exoplanets.