The German astronomer Wilhelm Bode ("bo-deh") published the law (without giving credit) in 1768, , and it is therefore often referred to as the "Titius-Bode law," or sometimes just "Bode's law" (a short name helps!). One exception was Mercury, where nothing needed to be added to 0.4. The other exception, more notable, was that the formula worked only if allowance was made for a missing planet between Mars and Jupiter. Here is the tabulation, using values from the section on "Kepler and his laws":
||0.4 + 0.3(2n)
The search was on, spurred on by the discovery of Uranus at 19.2 AU, where the Titius-Bode law gave 19.6. And a planet was duly found at the appropriate distance, on the first day of the 19th century, 1 January 1801 (There was no year zero, so the first century was 1-100, 2nd 101-200 and so forth). Discovered by the Italian astronomer Giuseppe Piazzi, it was named Ceres, after the goddess of wheat (hence the word cereal) and its mean distance from the Sun was 2.987 AU--a tolerable fit to the law.
The only problem was, it was surprisingly small, with a radius just shy of 500 km.
Another small planet--a little closer to the Sun--was discovered in 1807 and was named Vesta. But the floodgates of asteroid discovery opened wide once photographic cameras teamed up with telescopes with stable equatorial mountings, driven by clockwork. Since the telescope automatically tracked the rotation of the Earth, stars registered as dots: any image stretched into a short line indicated an object moving relative to the distant stars--hence, either an asteroid or a comet. Many thousands were discovered, catalogued and named, and since the discoverer had the privilege of naming, a large variety of names found their places in the sky.
Why didn't all these lumps form one single large planet? Perhaps because consolidation was disrupted by the big bully of their neighborhood, Jupiter. Many asteroids congregate at the L4 and L5 Lagrangian points of Jupiter and are known as "Trojans," with names related to Homer's account of the Trojan war. And while most asteroids orbit between Mars and Jupiter, some also reach other parts of the solar system.
Among those, special interest is attached to "Earth-crossing" asteroids whose orbit approaches the Sun closer that Earth--such as "Eros" pictured here, an image from NASA's "NEAR" mission taken 15 December 2000. Earth-crossing asteroids may collide with Earth, a worrisome possibility. The craters of the Moon and on other bodies suggest that large collisions were frequent in the early days of the solar system, when many small objects orbited the sun (before most consolidated into a few bigger ones), but even now collisions cannot be ruled out.
Such a collision could be a catastrophe to us, since even a 1-km asteroid packs the energy of many billions of tons of TNT. The flash as it hits the atmosphere can ignite forests over nearly half the Earth, their smoke could later blot out sunlight and cause plants to wither, while the wave from an ocean impact could flood large areas. The extinction of the dinosaurs is attributed to such a collision, its evidence detected by Walter Alvarez as a thin layer of iridium-rich deposit extending though Italian limestone. A huge circular structure--part of it beneath the Caribbean sea, part under the Yucatan peninsula--has been ascribed to that impact.
Big impacts are rare. Smaller ones occur on scales of thousands of years, larger ones on scales of tens of thousands, still larger, millions of years apart--but all are hard to predict and almost impossible to prevent (no matter what Hollywood has suggested). On 15 February 2013, a 12,000-ton meteorite exploded at altitude of about 30 km near the city Chelyabinsk in eastern Russia: the energy release was estimated as that of 500,000 ton TNT, but damage was moderate and no one died.
Meteor Crater in Arizona was left by a bigger impact which struck the ground, and a bigger one still carved the Manicougan crater in Canada. Automatic telescopes now keep watch and catalogue all possible impactors. Still, if one heads our way, unless we know decades beforehand, it is doubtful humanity can divert it in time.
The "Dawn" mission.
NASA launched in 2007 the "Dawn" mission, which flew past the large asteroid Vesta and then entered an orbit around the largest one, Ceres. Among its distinguishing feature was the use of ion propulsion creating a jet of xenon ions, electrically accelerated using energy from photovoltaic "solar" cells. The thrust obtained is very small, but acting for months on end it can modify the spacecraft's velocity by an appreciative amount, more effectively than chemical fuel.