Planets, planets everywhere

kw: speculation, planets

I read an article in which the author used a chart like this one to argue that many small extrasolar planets are yet to be found. While the conclusion is right, some of the reasoning is flawed.
This chart shows a frequency diagram of the 432 exoplanets so far catalogued, plotted by their line-of-sight masses (mass times the sine of inclination). The "1" bin includes all those with a mass of one Jupiter or less. The strong inflection in the shape of the distribution is said to indicate that the smaller masses are underrepresented. And they are, as we will see.

The appropriate analysis of wide-ranging data is not linear but either logarithmic (power law) or lognormal. The same data, collected into bins with a range of 2:1 in mass, show a more useful picture:
Here, "1" is again "one Jupiter", but the mass range is 0.71-1.41 rather than 0-1. The parenthesis points out that one Earth mass is 0.0031 Jupiter mass; one Mj is 318 Me. Here the sharp inflection to the left of 0.5 Mj also indicates that the mass distribution is artificially cut off at low masses. But now we have the ammunition needed to estimate just how much.

The only stellar system for which we have a reasonably complete catalog of planetary masses is our own. This chart of the distribution of planetary masses is based, not on Mj but on Me, the mass of Earth. Looking closely, we see four pairs:

• Mercury and Mars have very small masses, 0.055 and 0.11 Me.
• Venus and Earth are the second pair: 0.815 and 1 (of course).
• Uranus and Neptune are next, at 14.5 and 17.1 Me, respectively.
• Saturn and Jupiter are the heaviest: 95 and 318 Me.
  

These fit a lognormal distribution with a logmean (base 10) of 0.57 and a logSD of 1.36. The confidence interval about both these parameters is rather large, but I'll use them to produce a fit from which we might learn something useful.

Here a Lognormal distribution with our system's parameters and the exoplanet data are cast in the same coordinates, re-figured so that unit mass is one Earth. The lightest exoplanet on this list (which includes only those found by astrometric means, rather than eclipses) is just under 2 Me. Making the quite unfounded assumption that the larger exoplanets are fairly sampled, I fitted a distribution with the logmean and logSD of the Solar System. This chart tells us that if our own Solar System is at all representative, a complete census of the stars already searched would yield about 4,000 planets, with a statistical peak near 4 Me (3.7).

Is this reasonable? For it to be anywhere close to true, we must also assume that the preponderance of "hot Jupiters" so far found is also due to selection bias, because they are so much easier to find than any other kind. I sure look forward to the first data from the Terrestrial Planet Finder mission (canceled last week by the President, but sure to be reinstated by a future administration).