“The past year has given to us the new [minor] planet Astraea; it has done more – it has given us the probable prospect of another. We see it as Columbus saw America from the far shores of Spain. Its movements have been felt, trembling along the far-reaching line of our analysis with a certainty hardly inferior to ocular demonstration.”

— Sir John Herschel addressing the British Association of the Advancement of Science on Sept. 15, 1846, two weeks prior to the discovery of Neptune.

Yesterday, Ryan Montgomery gave his presentation at the AbSciCon meeting in Washington DC, and laid forth our provocative hypothesis. We think that Earth-mass planets are common in the habitable zones of the lowest-mass red dwarf stars, and we think that these planets can potentially be detected by targeted photometric searches of the nearest known low-mass stars. The closest stars on this list are accessible to transitsearch.org observers, and we are advocating that the search begin immediately.

Our calculations use John Chambers’ Mercury integrator to follow the last evolutionary stages of a planetesimal swarm in the protoplanetary disk of a young low-mass red dwarf star. The underlying physical picture in the simulations is that the star and disk are of order one million years old. The initial stages of planet formation are assumed to already have been completed. Grains of solid material have stuck together to build larger and larger objects in the disk. Most of the gas that was originally in the disk has either accreted onto the star, or has been photoevaporated by high-energy photons from the star itself and the neighboring stars in the birth aggregate.

We’ve completed three sets of calculations, and our computers are currently working on a large number of additional runs. In the first set (containing sixty individual simulations) we assume that two Neptune-like giant planet cores have already managed to form beyond the protostellar ice line, where the temperature is lower than 150K, and where planets can grow more quickly because of the availability of ices. We also assume that the innermost Neptune-mass core has been able to migrate a small ways inward to a distance of ~0.2 AU from the central star. This situation was chosen so as to be in analogy with the known Neptune-mass planets orbiting the red dwarfs GL 436 and GL 581 (see yesterday’s post). In a second set of sixty simulations, we didn’t include the giant planet cores. In our simulations, the Neptune-mass cores assume a role similar to that which Jupiter and Saturn are believed to have had during the formation phases of the terrestrial planets in our own solar system.

In each of the 120 simulations that comprise the first two sets, we distribute 1000 planetesimals in initially circular orbits in the region between 0.04 AU and 0.12 AU surrounding the eventual stellar habitable zone for the 0.12 solar mass star. Each planetesimal contains 0.003 Earth masses (about a quarter of a lunar mass). The swarm of planetesimals is then allowed to evolve under its own self-gravity, the gravity of the star, and the gravity of the ice-giant cores (if they are present). Planetesimals that collide with each other are assumed to conserve total angular momentum in the collision, while merging into a larger composite body. Some planetesimals collide with the ice giants or with the star, or are thrown out of the system. In a typical simulation (shown below) the swarm rapidly works itself down over a period of a few thousand years into a system of several terrestrial mass planets. Earth-mass planets in the habitable zone of the star are a very common outcome of the simulations.

In a third set (of thirty) simulations, we lowered the masses of the planetesimals to 0.0003 Earth-masses, that is, a factor of ten times lower. The results of these simulations were the formation of Mars-sized or smaller bodies in the stellar habitable zone.

The results have a simple interpretation. The final stages of terrestrial planet formation in the protoplanetary disks of red dwarf stars appears to be an efficient process. If one starts with an adequately high effective surface density of solid material in the disk, then one frequently gets Earth-mass planets in the habitable zone. If one starts with a lower surface density, then one gets final sets of terrestrial planets that (on average) have proportionally lower masses, i.e., no deal.

We believe that the key issue, then, is: what is the appropriate surface density to use?

If one makes reasonable extrapolations from the minimum-mass solar nebula that formed our own solar system, or if one extrapolates from the dust disks which are observed around young stars in the solar neighborhood (see the photo below of the disk orbiting AU Microscopium), then one should adopt a low surface density. This was the approach taken by Sean Raymond in his talk (which followed Ryan at AbSciCon). Sean’s results agreed quite well with our low-surface density simulations, namely, Mars-sized or smaller planets in the habitable zones of red dwarfs.

Submillimeter observations of dust masses in young stellar systems measure the amount of mass in dust, and are not directly sensitive to the amount of mass in large, planetesimal-sized bodies. Furthermore, such measurements give the dust mass at large distances (say greater than 1 astronomical unit at least) from the star, and hence do not give information about the mass of solids present in the innermost region of the disk.

Our preferred high surface density scenario is based on the “Minimum Mass Nebulae” for the inner regions of GJ 876 (0.32 solar mass), and Jupiter (0.001 solar mass). These are the two objects closest in mass to our hypothetical 0.12 solar mass star whose “terrestrial planet” systems we can measure.

In the case of Jupiter, the moon Io has a mass of 8.93e+25 grams, an orbital radius of 0.0028 AU, and an orbital period of 1.8 days. This implies a solid surface density of approximately 12,000 grams per square centimeter at the 1.8 day orbital radius in the proto-Jovian nebula.

In the case of GJ 876, planet “d” (which you can characterize from the actual Keck and Lick radial velocity data using the Systemic Console) has a mass of 4.5e+28 grams (7.5 Earth masses), an orbital radius of 0.02 AU, and an orbital period of 1.94 days. If we assume that GJ 876 d fed off material reaching out to a radius of 0.075 AU, then this implies a solid surface density of 11,000 grams per square centimeter at the 2.0 day orbital radius in GJ 876’s protoplanetary nebula. This is remarkably close to the value for Io. That is, the “rule of thumb” from these two systems suggests an effective surface density of solid material of ~10,000 grams per square centimeter at a 2-day orbital period.

The similarity between the solid surface densities obtained by grinding up Io and GJ 876d suggests that we also adopt a solid surface density of 11,000 grams per square centimeter at the 2-day orbital radius for our 0.12 solar mass star (0.015 AU). Using a reasonable r^-3/2 falloff in surface density as we move away from the star, this suggests a fiducial density of 2000 grams per square centimeter at a habitable-zone radius of 0.045 AU, which is the value that we use in our preferred (sets 1 and 2) simulations.

Once we’ve run a particular simulation, we choose a random angle from which the system is to be viewed. We then generate photometry that is typical of what high-end amateur observers such as Ron Bissinger or Tonny Vanmunster are capable of regularly achieving. For instance, here’s an example of Ron Bissinger’s observation of HD 149026b shortly after it was discovered.

We then “observe” the system by creating a simulated photometric time-series over a period of several hours, during the intervals in which a transit might possibly occur.

Our simulations imply about a 1.0% a-priori probability that a 0.12 solar mass red dwarf has a detectable, habitable planet. That means that most of the simulated systems, when observed at a random viewing angle, don’t show any transits:

With the omnipotence afforded by the simulation output files, we know that some of the simulations were not that far away from having a transiting planet:

Whereas some were closer still:

In this case, a tiny planet produces a grazing transit that is completely unobservable with 0.4% differential photometry:

And then, finally, gloriously:

That’s how I think we’ll get our first look at a truly habitable world orbiting an alien star.

Finally, back to the passage from John Herschel that starts this post off with an egregious bang. At first glance, it looks like a totally outrageous bit of self-serving grandstanding. Moreover, the quote itself is well-known to the extent that a reasonable person might justifiably press charges of second or even first degree cliche. On second glance, however, it actually seems rather appropriate.

Columbus thought he was headed for the East Indies, and he was justifying his expedition on an assumed distance from the Canary Islands to Japan of only 4444 km (as opposed to the true distance of 19,600 km). He had no conception whatever of America while he was still on the “far shores of Spain.”

Furthermore, the prediction of the existence of Neptune by Urbain Jean Joseph LeVerrier, was based on the large perturbations to the orbit of Uranus which occured from ~1810-1840, and which only occur once per Uranus-Neptune conjunction. The large derangement of Uranus’ orbit allowed LeVerrier to compute predicted ephemerides for the location of Neptune that were accurate enough for it to be quickly discovered by Johann Galle and Heinrich d’Arrest on the night of Sept. 23, 1846. LeVerrier was lucky, however. Even though he assumed an incorrect distance for Neptune of 36.15 AU, based on Bode’s spurious “law”, his method — which was essentially a laborious hand-cranked version of what goes on beneath the hood of the Systemic Console — was able to compensate for this incorrect assumption by invoking a mass for Neptune that was too large (2.9 times too large, in fact), and an eccentricity, e=0.11, that was also too large. Neptune’s actual orbit is currently nearly circular, with e=0.00884. As a result, LeVerrier’s orbital predictions of the location of Neptune in the skies of 1846 were close enough to allow it to be found, even though his predicted planet had an orbital period of 217 years, in comparison to Neptune’s actual period of only 166 years.

This point is often glossed over in the astronomical lore, and LeVerrier (with Adams invariably in tow) is lionized a bit too assiduously as a hero of the scientific method. In fact, luck, in the form of the fact that Uranus and Neptune happened to be close to conjunction, played a major, if not leading role. At the end of the day, we expect the same situation to hold true for those habitable planets transiting nearby low-mass red dwarf stars.