Wolverhampton Astronomical Society

Established 1951

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One evening in 1781, a professional musician named William Herschel became the first person to be credited with finding a new planet. For millennia, most people had accepted that the Solar System contained only five other worlds – Mercury, Venus, Mars, Jupiter and Saturn – besides our own. Herschel’s discovery of Uranus turned this perception on its head. In a stroke, he identified a planet four times bigger than Earth and more than doubled the known radius of the Sun’s empire to around two billion miles.

His work also vindicated one of the most peculiar ‘rules’ on record; a rule that, for more than half a century, was used as the basis for a number of calculations to predict where new, as-yet-unfound planets might reside. Known as ‘Bode’s Law’ or, more correctly, the ‘Titius-Bode Rule’, it has long since proven itself to be neither a law or a rule, but a quirky historical curiosity that still defies easy explanation.

Its seeds were sown over a decade before Uranus’ discovery, when German mathematician Johann Titius translated a book by Swiss naturalist Charles Bonnet. As part of efforts to demonstrate the ‘orderliness’ of God’s handiwork, Bonnet had used the Solar System as a supreme example and Titius took the opportunity to add a paragraph of his own: one which offered an intriguing progression in the distances of known planets from the Sun.

Starting with Mercury at zero, he assigned each world a number – three, six, 12, 24, 48 and 96 – which doubled that of its predecessor; he then added ‘four’ to each figure and divided their sums by ten. For example, Mercury lies 36 million miles or 0.387 astronomical units (AU) from our parent star. By following Titius’ rule, a distance of 0.4 AU emerged; in other words, an astonishingly close match with Mercury’s actual distance from the Sun.

Moreover, it continued to ‘work’ well, and surprisingly accurately, for Venus (‘three’), Earth (‘six’), Mars (‘12’), Jupiter (‘48’) and Saturn (‘96’), with just one glaring flaw: the place reserved for ‘24’ – somewhere around 260 million miles or 2.8 AU – was an empty region of space. No planet was known to occupy this ‘gap’ between the orbits of Mars and Jupiter and both Titius and German astronomer Johann Bode, who subsequently plagiarised the rule as his own, speculated that it could contain as-yet-undiscovered moons of those worlds.

A few years later, after Herschel’s detection of Uranus, Bode realised that if ‘his’ rule was taken a step further beyond Saturn, working the next number in the sequence (‘192’) would have yielded the distance of the new planet. Uranus lies 19.2 AU – almost two billion miles – from our parent star: the rule predicted it at a tantalisingly-close-to-target 19.6. Although based on an entirely arbitrary set of numbers, with an equally arbitrary figure added to each one, for some peculiar reason the rule seemed to work.

This prompted Bode and others to doubt that God had left the Mars-Jupiter gap empty and that there must be an unknown body circling the Sun at 2.8 AU. Several astronomers, including Franz von Zach and Johann Schröter in Germany, joined the search, but it was Sicilian priest and amateur skywatcher Giuseppe Piazzi who spotted a faint, star-like object in the constellation of Taurus on January 1st 1801. Later named Ceres to honour the ancient Roman corn goddess, in keeping with the Titius-Bode Rule’s prediction, it orbited at 2.77 AU.

Bode must have let out a cry of triumph when he learned of the discovery. However, his euphoria would not last for long. Barely a year later, in March 1802, German doctor and astronomer Wilhelm Olbers found another small world, which he called Pallas.This new discovery complicated the rule, however, because the new object occupied the same patch of sky as Ceres. Ironically, after two decades fruitlessly hunting one world in the Mars-Jupiter gap, the rule was now unsettled by two.

Furthermore, when Herschel later estimated the sizes of both objects, they turned out to be much smaller than even the tiniest planet. Rather than shooting the rule in the foot, he tried to ‘save’ it by speculating that Ceres and Pallas were not, in fact, ‘planets’, but a new breed of wanderers which he dubbed ‘asteroids’. Olbers, however, argued instead that they were fragments of a much bigger planet that had been torn apart, aeons ago, by a cataclysmic cometary impact.

He advocated scrutinising the Mars-Jupiter gap, particularly where Ceres’ and Pallas’ orbits came closest to intersecting, to find more pieces from this hypothetical world. Indeed, two more – Vesta and Juno – were found by 1807 but, like their predecessors, were just too small and insubstantial to have ever formed part of a large planet. Even today, with around 10,000 named asteroids in a Mars-Jupiter ‘belt’ around the Sun, their combined mass is barely a fraction of our Moon.

Still, the rule was used by French mathematician Urbain Leverrier and his British counterpart John Couch Adams, both of whom independently used it in the 1840s to find an eighth planet at the next number in the sequence (‘384’) at 39 AU. Neptune, however, when it was sighted and its distance plotted, turned out to reside at 30.06, producing a significant 23 percent error margin in the rule’s accuracy.

Admittedly, in 1930 with the detection of Pluto, Titius-Bode advocates argued that it orbited at a close-to-target 39.44 AU – although its overall, highly-elliptical path meant that this varies widely – but it was only halfway as far from the Sun as the rule predicted a ninth planet to be. The sequence, it seemed, ‘worked’ as long as one ignored Neptune and forced Pluto to fit the figures! This disobedience of the rule by both outermost worlds proved its downfall.

Today, although no satisfactory explanation has been offered for the peculiarities of the rule and why it proved so accurate in identifying each planet’s distance, precisely pinpointing Uranus and the asteroid belt, opponents argue that its arbitrary choice of numbers – why choose multiples of three and why add four? – render it neither a rule or a law of nature, but simply an intriguing coincidence. One question it does engender, however, is whether the regular spacing of planetary orbits offers a clue to the origin or evolution of the Sun’s empire.

Picture References:

Johann Elert Bode (1747-1826), whose interest in Titius’ rule prompted efforts in the early nineteenth century to find a ‘missing’ planet between Mars and Jupiter. Click on the image for a larger view.
Johann Daniel Titius (1729-1796), who translated Charles Bonnet’s work and demonstrated an intriguing mathematical progression of the planets’ distances from the Sun. Click on the image for a larger view.

Ben Evans is a schoolteacher with a lifelong interest in astronomy and space exploration. He has written for Astronomy Now, Spaceflight and Countdown magazines and is the author of NASA’s Voyager Missions (2003) and Space Shuttle Columbia (2005), both published by Springer-Praxis. He is also a regular contributor to the Midlands Spaceflight Society Magazine "CapCom".

The Wolverhampton Astronomical Society is affiliated or a member of the following organisations: 

British Astronomical Association, The Society for Popular Astronomy, the Federation of Astronomical Societies and the West Midlands Federation of Astronomical Societies.