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For more than two centuries it has been known that the distances of
planets from the Sun follow a simple and mysterious law. Now, at long last,
two French scientists have come up with a plausible explanation.

The best way to see the law is to write down the sequence 0, 3, 6, 12
and so on, where each number is obtained by doubling its predecessor. Next,
add 4 to each number, and divide the result by 10. Now get out any astronomy
textbook and look up the distances of the planets from the Sun in astronomical
units, the Earth-Sun distance being defined as 1. The distances are virtually
identical to the terms in the number sequence for all but the outermost
planets.

This curious ‘coincidence’ has divided astronomers into two camps ever
since its publication in 1766 by the Prussian astronomer Johann Daniel Titius.
Some think it really is just a coincidence, while others think it reflects
an undiscovered feature of the Solar System.

For a mere coincidence, the ‘law’ has an astonishing track record. Shortly
after Titius’s announcement, his compatriot Johann Bode suggested that the
lack of a known planet at 2.8 astronomical units, the fifth term in the
series, might make a search for the ‘missing planet’ worthwhile. Bode’s
idea got a major boost in 1781 with the discovery of Uranus, the first planet
to be found since antiquity. According to the Titius-Bode law, Uranus should
lie at 19.6 astronomical units; it was found 19.2 astronomical units from
the Sun.

By 1796, Bode had persuaded a group of astronomers to hunt for the missing
planet between Mars and Jupiter. Five years later it turned up: the ‘minor
planet’ Ceres, 1000 kilometres across and orbiting at 2.8 astronomical units
from the Sun – just where the Titius-Bode law predicted.

Some years later, astronomers found that the newly discovered Uranus
was mysteriously wandering off course, prompting some to wonder if yet another
missing planet was pulling it off course, this time orbiting at the vacant
eighth Titius-Bode position of 38.8 astronomical units. In 1842, the young
Cambridge mathematician John Couch Adams based his first stab at solving
the mystery on the idea of a planet sitting at this eighth position. So
did Urbain Le Verrier, the distinguished French astronomer who in 1846 pipped
Adams and a lacklustre British team in the race to find the missing planet,
now called Neptune.

Ironically, it was this apparent triumph for the Titius-Bode law that
led to its current uncertain status among astronomers. The orbit of Neptune
turned out to have a radius of 30.1 astronomical units, which is much lower
than the law predicts. The final straw came in 1930, when Pluto was discovered
at about 40 astronomical units – barely more than half the 77.2 predicted
by the ‘law’.

So, were all the earlier successes of the Titius-Bode law merely flukes?
Today, the relationship is widely seen as no more than a numerological curiosity.
Sceptics point to research showing that even random numbers subject to plausible
constraints can be persuaded to obey such a law (Nature, vol 242, p 318).

But now two French researchers have found evidence that the Titius-Bode
law may have some deep significance after all. According to Francois Graner
of the Ecole Normale Superieure in Paris, and Berengere Dubrulle of the
Observatoire Midi Pyrenees in Toulouse, such a ‘law’ is a natural consequence
of certain symmetry properties, some of which are almost certain to feature
in any planetary system (Astronomy and Astrophysics, vol 282, p 262 and
p 269).

Models explaining how planets form typically feature a cloud of material
under the influence of the Sun’s gravity. Graner and Dubrulle point out
that such models have two symmetries. The first is so-called rotational
invariance – no matter how the cloud and its contents are turned, they always
look the same.

The second symmetry is more subtle. Known as scale invariance, it means
that the cloud and its contents look the same on all length scales. A familiar
example of scale invariance is the so-called ‘fractal’ coastline of a country:
no matter how closely you examine a coastline, it always looks jagged. Graner
and Dubrulle point out that scale invariance is a property of many concepts
thought to be important in planetary formation: collapsing dust clouds,
inverse-square force fields, turbulence. But they then show that in any
mathematical model that combines scale and rotational invariance the points
at which a physical parameter, such as density, reaches a maximum or minimum
will always follow a simple relationship. Specifically, for a disc-like
model of the Solar System, the location of the nth such point will be proportional
to some number K raised to the power of n.

The Titius-Bode law is just such a relationship: its sequence of planetary
distances is based on doubling each successive term – that is, setting K=2.
In fact, a much better fit can be obtained by setting K=1.7, raising it
to the first power for Mercury, the second for Venus and so on, and multiplying
each result by 0.23.

Does this mean that Bode, Adams and the others were right to put their
faith in the law after all? The answer is that they might have been – but
it isn’t certain. ‘If the early planetary system had both rotational and
scale invariance, then you can expect to find a type of Titius-Bode law
in the planetary orbits,’ says Graner. ‘However, as well as the physical
laws, the conditions within the system, such as temperature and chemical
distribution, must also be scale invariant – and we don’t know if they were.’

So belief in the validity of the Titius-Bode law ultimately boils down
to a belief in the scale invariance of physical conditions in the early
Solar System. Graner admits that even he and and Dubrulle do not agree on
this. ‘Berengere still thinks the law may have something in it, but the
debate is not settled – and it may never be’.