Sunday, April 4, 2010

Bode's Law Doesn't Bode Well for Us

Both photometry and spectroscopy have been used to discover the more than 400 known exoplanets. The photometric light curve of a distant star with a transiting planet, when combined with detailed spectroscopic data, yields the planet radius, mass, and semimajor orbital distance, as well as the mass, radius, and effective temperature of the host star. The holy grail of this impressive scientific endeavor is, of course, to find Earth-like planets (possibly inhabited with Earth-like beings). However, the current odds of achieving such a find are not looking favorable because the trend indicates that Bode's law is being broken by the gaseous exo-giants.

Relative sizes of our planets (distances not to scale)
array of pyramidal quantum dots

The standard model has it that our planets formed, under the influence of gravity, out of an accreting disk of interstellar dust, with the bulk of the mass at the center of this disk. The Sun, at the center of our solar system, does account for more than 99% of all the matter. Jupiter, on the other hand, contains more than twice the matter of the other planets combined and is located in the middle of our planetary sequence. The sequence of semimajor-axis distances (A) from the Sun is given by the semi-empirical relationship known as Bode's law:

A  =  0.4 + 0.3 (2k),(1)

where k = -∞, 0, 1, 2, ..., 8 and A is expressed in AU. The following table shows a numerical comparison with measured planetary distances.

Since Bode's law is a power law, it is appropriate to compare it with the data on a log-scale plot. In Bode's day only six planets (P = 1, 2, 3, 4, 6, 7) were known.

The discovery of Uranus (P = 8) in 1781 was seen to be in agreement with eqn. (1). The location of the then unknown planet Ceres (P = 5) was predicted by Bode and discovered in 1801. Today, it's a member of the asteroid belt. The fit at P = 1 using k = -∞ is a hack and clearly the outermost planets do not fit; although Pluto has since been officially demoted from planet status by the IAU.

Without getting too exercised over whether eqn. (1) is really a law or a not, let's take it as read that it (or a similar power law) has some statistical meaning in the context of our Solar system, which might ultimately be explained by a more sophisticated planetary formation model. Whether such power laws hold up for exosolar systems remains to be seen. Although Bode's law doesn't predict the mass distributions (only the relative orbital distances), ultimately the planetary masses are related to their distances via Newton's law of gravitation once the planets proper have formed. Getting there from the accreting dust cloud, however, is a gravitational maelstrom.

What has been observed photometrically and spectroscopically about most of the exogiants is that they orbit very close to their respective exostars—much closer than our own gas giants and often with much higher eccentricities. Using the current measurement techniques, the semimajor-axis is so small that it makes detection of terrestrial-like exoplanets elusive, if not completely excluded. Indeed, if Jupiter were to move closer to the Sun, it would destroy all the inner planets and, along with it, the possibility of us contemplating this exoplanet enigma. Since such Jovian instability appears unlikely to happen any time soon, we can continue our investigations in comfort—but alone.