Almost Perfect: The Titius-Bode Law

It’s not perfect, and it only works for 7 of the 8 planets, but it’s still great for getting an approximation for how far apart to hang the glow-in-the-dark planets from your ceiling.

The distance from the Sun to Mercury is approximately 4 tenths of an AU (Astronomical Unit — the average distance from the Sun to the Earth), and the average distance from the Sun to Venus, Earth, Mars, the asteroid belt, Jupiter, Saturn, and Uranus is approximately [4+(3·2^N)] tenths of an AU, where N=0,1,2,3,4,5, and 6 respectively. Isn’t that a neat equation? It was first observed by David Gregory in 1702 (in Latin; 1715 in English) in his book The Elements of Astronomy and is therefore named after Johann Titius (who published a German translation of a 1724 book by Christian Wolff that contained the same description) and Johann Bode (who read Titius’s translation and put it as a footnote in his own textbook). And here’s where I’m wondering if I can get my name added to the Law by virtue of just mentioning it here.

Here’s a table showing the prediction and actual average distances of these planets/asteroids from the Sun:

	Titius-Bode	Actual Distance
Mercury	0.4 AU	0.39 AU
Venus	0.7 AU	0.72 AU
Earth	1.0 AU	1.00 AU
Mars	1.6 AU	1.52 AU
Aseroid Belt (Ceres)	2.8 AU	2.77 AU
Jupiter	5.2 AU	5.20 AU
Saturn	10.0 AU	9.54 AU
Uranus	19.6 AU	19.19 AU
Neptune	38.8 AU	30.07 AU

Except when the Gregory-Wolff-Titius-Bode Law was around, no one knew about the Asteroid Belt, Uranus, or Neptune. Titius wrote “From Mars there follows a space of 4+24=28 such parts [tenths of an AU], but so far no planet was sighted there. But should the Lord Architect have left that space empty? Not at all. Let us therefore assume that this space without doubt belongs to the still undiscovered satellites of Mars, let us also add that perhaps Jupiter still has around itself some smaller ones which have not been sighted yet by any telescope.” Bode didn’t think it was a moon in that spot, but agreed that the Lord Architect wouldn’t have left it blank.

This Law got a lot more attention in 1781 when Uranus was discovered and it fit the pattern! Woo hoo! So then Court Astronomer Baron Franz Xaver von Zach of Gotha (whose name is fun to say out loud) suggested that astronomers work together to find the missing planet between Mars and Jupiter, and so they did, declaring themselves to be the “celestial police” in search of a missing planet. (Seriously. See here.) But it was a mathematician astronomer monk who actually found it: Guiseppe Piazzi of Palermo beat them to the punch and discovered the missing planet Ceres in very early 1801. He lost it after a brief time, but then Carl Friedrich Gauss himself did the math (80 variables in 3-space!) to find it again the following year, and everyone was happy. At least until William Hershel invented asteroids in 1802 and said Ceres was one. Piazzi wasn’t thrilled with this and Ceres and several of its soon-discovered asteroid friends (Juno and Pallas) got to be planets for several decades, before people decided Hershel had been right all along and they were just asteroids. [Incidentally, Ceres was promoted back to (dwarf) planethood in 2006 along with Pluto and Eris (the planet formerly known as Xena)].

Getting back to Titius-Bode, their Law was all the rage after these discoveries, but it fell out of favor when Ceres was declared to be just one of many asteroids between Mars and Jupiter and when Neptune was discovered in 1846, since Neptune doesn’t really fit the law. And Pluto and Eris really don’t match: according to Titius-Bode, the next planet should be at 77.2 AU, which is between Pluto (average 39.48 AU) and Eris (average 97 AU), though both Pluto and Eris come as close as Neptune at times.

So what’s the end result? Nothing in a sense — the formula doesn’t work, and even though there’s some idea that planets may naturally fall into this kind of pattern [where the gaps between planets grow by powers of 2], there aren’t enough planetary systems to do more than make rough conjectures. But at the same time, the Titius-Bode Law gives rough estimates that are remarkably close most of the time, and so worth a note.

Thanks to Hiroyo for sharing this equation with me! The photos of the Sun and Ceres and the data (except for Ceres) come from NASA, and most of the story comes from the Dawn Website (The Dawn spacecraft is on its way to visit the asteroid belt.) who got it from Michael Hoskin’s article here. You can also see the orbits of the planets in a 3d applet here.

Tags: bode, ceres, planet, titius