Kepler Statistical Analysis Suggests Earthlike Planets Extremely Rare



by John Rehling



San Francisco CA (SPX) Mar 15, 2012



Overall, we see that our solar system is qualitatively typical in placing larger planets farther out than smaller planets. However, it is quantitatively atypical: While Kepler shows us the happy result that there are almost certainly several planets for every star, it shows us that our solar system is distributed freakishly outwards, in comparison to more typical planetary systems.

Last week, the Kepler science team released its list of candidate planets based on the data collected during the mission's first sixteen months. The last comparable publication summarized the mission's first four months of data.

The update thus yields a much larger number of candidate planets. Significantly, the longer time window means pushing back the veil of two biases that make Kepler's data set, at any point in time, relatively incomplete:

Bias favoring Close-in Orbits

A geometric bias means that Kepler's data is always more complete for the regions closest to their respective stars. The probability of an alignment that produces transit is linearly proportional to the distance from the host star. This also manifests as an added temporal factor as planets farther away revolve more slowly and are thus less likely to transit in a given time window.

Reflecting this bias, over 90% of observed candidates have shorter periods around their stars than Mercury has around the Sun. Many of these planets will therefore be extremely hot.

Bias favoring Larger Planets

Because the light-dip caused by a planet's transit can be small in comparison with the star's intrinsic variability, there is a profound bias in favor of larger planets. In many cases, a large planet will be detected whereas a smaller planet in the same context would not have been detected.

Reflecting this bias, most observed candidates are significantly larger than Earth, making many of them Jovian-class gas giants, devoid of any solid surface where life could flourish.

In many respects, the most exciting and critical result of Kepler so far is not the list of actual planets (whose existence must be verified after Kepler has identified their candidacy) but the statistics comprising a universal census of planet frequencies, qualified by radius and distance from the primary.

It has been noted that the rate of candidates which are actually planets is quite high, so for the purposes of this article, candidates are taken as direct evidence of universal abundance. By placing observed candidates into bins according to planetary radius and orbital period, we have raw data to classify into various bins.

By modeling the two forms of bias, we can calculate a numerical de-bias factor for each bin. Where the positive observations are significant in number, we can calculate the universal abundance of such planets. Where there have been few or no observations, we can use the de-bias factor to infer probabilistically a ceiling on the number of such worlds.

A similar approach by Wesley Traub used the data from the earlier four-month data release to calculate absolute frequencies and furthermore to extrapolate trends in planet radius and orbital period to project that about 34% of stars host an earth-sized planet in the habitable zone, a happy speculation for the future goal of finding truly earthlike planets as possible abodes for life.

This was dependent upon an approach that took patterns seen with bins of no more than 50-day orbital periods and extrapolated them out to the much longer orbital periods of the habitable zone.

With the latest release, we come much closer to making such projections accurate, by filling bins that the 4-month release left empty and requiring us to extrapolate only a couple of bins over from the bins where we have hard data. 16 months of observation is categorically insufficient to detect any precisely earthlike planet, because the ground rule that only those earth-sized candidates with three transits observed means that a minimum of 24 months of observation will be required.

However, as new data comes in, the barriers enforced by the geometric bias are pushed outward, and as more candidates are reported, more terrestrial worlds like the Earth, rather than giants like Jupiter, are revealed.

This release shows two favorable trends. As a statement from the Kepler team puts it, "With each new catalog release a clear progression toward smaller planets at longer orbital periods is emerging. This suggests that Earth-size planets in the habitable zone are forthcoming if, indeed, such planets are abundant."

However, the fine details of this latest release favor more pessmistic projections. Although the release shows, in comparison to the previous release, one favorable trend regarding planet size and another favorable trend regarding planet distance from the star, these trends are, unfortunately, unfolding in an either-or respect: We see more Earth sized planets which are very close to their stars, and therefore likely very hot; and, separately, we see more giant planets which are located farther out from their stars.

This table shows the estimated frequency of planets per star in each bin. The planet sizes are presented in terms of Earth radius, Re, while the orbital periods are in days (d). The bins are populated with frequency estimates based on the de-bias factor, times the observed candidate count, where that count was 4 or greater. In addition, more tentative numbers appear in parentheses in certain bins, based on a smaller n or the n that would have resulted from a single discovery. In each row, the maximum value appears in bold.

This shows that for each terrestrial planet size catgeory, we have observed the frequency max out at a very short period between 4 and 16 days, then exhibit decreasing frequency for longer periods.

For three of the terrestrial size categories, we have data on at least two bins to the right of the peak, and in all of these cases, we see the frequency drop with longer periods. The median factor relating a bin's frequency to that of the bin to its left is 0.72. Using that factor to extrapolate to the right, the following figure shows how the four size categories of terrestrial planet drop in frequency as we move from the peak orbital period out into the habitable zone.

The darker color represents bins with hard data: Lighter bins involve the extrapolation only. For the smallest size category, there is meager positive data; however, the zero-valued results in longer-period bins is meaningful, and makes it very likely that this category plunges in frequency from the peak at periods of only a few days.

For the bin corresponding precisely to the Earth, the projected frequency is 0.7%, a far cry from Traub's projection of 34%, owing in part to differences in the size of the bin: If we consider the 3x3 collection of bins that surround Earth's bin, the current projection rises to 7.8%. By including also the smaller Mars-sized planets, to 9.0%.

This is still less than a third of Traub's projection, and the reason is primarily a single assumption in his work: "To be clear, these estimates are based on projecting the total of all planets around all bright stars in the database, then simply applying the terrestrial fraction for short periods to the longer [Habitable Zone] periods."

This is, unfortunately, too optimistic in two ways: One, only the larger planets which particularly dominated the four-month release continue to increase in frequency in longer-period bins. Secondly, the "Super Earth" and Neptune-sized categories of planet peak in frequency at orbital periods roughly corresponding to the upper end of the four-month release's window.

These planets, though no Super Earths exist in our solar system, are the most frequently observed in Kepler data, and their dominance suggested that the trend of longer periods continued overall. However, with the sixteen-month release we see that Super Earths and even Neptunes peak in frequency at short periods.

Only larger giant planets like Jupiter continue to increase in abundance at the limits of the sixteen-month release. The following chart presents the projected frequencies for broader size bins, with terrestrial-class planets - including all smaller planets with a size of 0.4 Re to 1.4 Re. MidSized (super-Earth) planets includes all planets from 1.4Re to 4.0 Re, and all larger planets are grouped as Jovian. As we see, the peak frequency moves farther out for larger planet sizes.

Overall, we see that our solar system is qualitatively typical in placing larger planets farther out than smaller planets. However, it is quantitatively atypical: While Kepler shows us the happy result that there are almost certainly several planets for every star, it shows us that our solar system is distributed freakishly outwards, in comparison to more typical planetary systems.

The data also indicates that as Kepler's mission continues, it may not find precise Earth analogues, although this will depend in part upon luck: The de- bias factor for the Earth's bin is 5800, meaning that out of the 156,000 stars being monitored, we are effectively searching only 27 for perfect Earth analogues: If they are less abundant than 3%, we may very likely find none. On the other hand, if Kepler's mission is extended to allow several years more of data collection, the noise-based de-bias factor may be relaxed by finding candidates on the basis of multiple noisy events, as opposed to the three or so transits possible in a three-year mission.

Possibly the worst ramification of these results doesn't pertain to Kepler per se: The Kepler candidates are mostly located quite far from us, making possible follow-up science with spectroscopy and imaging extremely hard. We would be better able to make observations of earth-like planets located closer to us, at distances of tens of light years instead of hundreds. But if the abundance of earthlike planets is only a few percent, there will be comparatively fewer of these worlds in our neighborhood.

Any future effort to find and examine earthl-ike planets in our corner of the galaxy will be limited by the frequency of such planets, and this result serves to dim prospects somewhat, or to require considerably larger and more expensive telescopes than would be needed if the optimistic projections proved to hold true.

In a memorable scene of Carl Sagan's Cosmos, Sagan considers the Drake Equation and estimates that perhaps 1/4 of all stars have planets and that each such system might have two planets suitable for the development of life. That estimate was recorded before any planet outside our solar system had been discovered.

With the current Kepler release, we find that the number of earthlike planets per star is likely to be considerably lower than the 0.5 implied by Sagan's estimate. Of course, the picture is considerably complicated: We may find habitable worlds, whether earth-like or Europa-like, orbiting giant planets. We may find earth-like worlds harbored as Trojans of giant planets in the habitable zone.

The frequency as a function of orbital period may have a second peak or flatten and fail to drop with still longer periods. Comfortable temperatures may also be found at planets close in to dim red dwarf stars, although that may likely result in the world being tidally locked, and therefore subject to one daylight hemisphere and one in eternal night. And if nothing else, the pessmistic characteristic of these results suggest that to find earth-like worlds elsewhere, we should prepare to look hard - and quite possibily very hard for decades if not centuries.

John Rehling is a graduate of computer science programs at Dartmouth College and Indiana University. He has performed research and engineering work in artificial intelligence and natural language processing at NASA, Carnegie Mellon University, and for several companies in Silicon Valley. He is currently employed as a research scientist at Reputation.com.