The Prevalence of Earth-Size Planets Around Sun-Like Stars

The primary objective of NASA’s Kepler mission is to detect Earth-size planets orbiting Sun-like stars in Earth-like orbits. While the ongoing analysis of the huge amount of data from Kepler’s primary mission has uncovered thousands of planets to date including a number of potentially habitable planets orbiting dim red dwarfs, only one confirmed extrasolar planet in an Earth-like orbit around a Sun-like star has be found as of this date that is close to the Earth in size – Kepler 452b. But with a radius about 1.6 times that of the Earth (or R_E), it is probably a bit more likely to be a volatile-rich mini-Neptune than a rocky planet like the Earth (see “Habitable Planet Reality Check: Kepler 452b”).

Diagram showing the major components of NASA’s Kepler spacecraft. (NASA/Kepler Mission/Ball Aerospace)

While this may be disappointing, it should be remembered that the analysis of Kepler’s huge data set is continuing and that a number of potentially more Earth-like planet candidates are currently being examined (see “Earth Twins on the Horizon?”). In the mean time, the statistical analysis of more easily detected Earth-size planets in shorter period orbits around Sun-like stars found by Kepler has revealed some interesting insights that may apply to more Earth-like planets.

New Work

Erik Petigura (University of California – Berkeley) recently submitted his doctoral thesis on the prevalence of Earth-size planets orbiting Sun-like stars based on his analysis of the Kepler data set. This thesis is the culmination of a series of earlier peer-reviewed works including a 2013 paper where he collaborated with Andrew Howard (University of Hawaii at Manoa) and famed extrasolar planet hunter, Geoff Marcy (University of California – Berkeley). In this earlier paper, Petigura et al. estimated that 5.7% (+1.7%/-2.2%) of Sun-like stars are orbited by “Earth analogs” – planets with radii between 1 and 2 R_E orbiting Sun-like stars with periods ranging from 200 to 400 days (see “The Abundance of Earth Analogs”). This definition would include not only a habitable world like the Earth, but decidedly nonhabitable planets like Venus as well as worlds with radii greater than about 1.5 R_E which are now widely considered more likely to be mini-Neptunes than Earth-like rocky planets (see “Habitable Planet Reality Check: Terrestrial Planet Size Limit”).

In his thesis, Petigura summarizes his work analyzing Kepler’s huge photometric data set using a new algorithm called TERRA (Transiting Exoearth Robust Reduction Algorithm) specifically tailored to detect smaller planets. Also described were the efforts made to validate the finds as well as estimate the completeness of the survey results. A statistical analysis of all the planets he found orbiting solar-type G and K dwarf stars with surface temperatures in the 4100 to 6100 K range is presented with emphasis on those with periods in the 5 to 50 day range. The results from a second analysis of a larger number of Sun-like stars provide estimates of the number of Earth-size planets in longer period orbits allowing an extrapolation into the habitable zone of Sun-like stars.

A histogram showing the distribution of the sizes of Kepler finds for planets with periods between 5 and 50 days. The gray color indicates planets actually found while the red indicates the estimated number of planets missed by the analysis. Click on image to enlarge.(Petigura)

In Petigura’s first round of analysis, planets were confidently detected orbiting a total of 119 Sun-like stars out of the 12,000 photometrically least noisy type G and K dwarfs observed by Kepler during its primary mission with orbital periods ranging from 5 to 50 days. This group of extrasolar planets, found by analyzing the first three years of Kepler’s data, had radii ranging from 0.54 to 5.60 R_E. Based on Petigura’s analysis, the survey for planets with radii less than 1 R_E has a completeness of less than 50%. Excluding these smallest finds, Petigura determined that 41.7% (+6.8%/-5.9%) of Sun-like stars have planets with radii greater than 1 R_E in orbits with periods ranging from 5 to 50 days. This confirms the view that our Solar System is somewhat unusual in not having planets in short-period orbits (see “How Typical Is Our Solar System?”).

While the increase in the planet population with decreasing radius was found to follow a power law, there is definitely evidence that a plateau is reached around 2.8 R_E and that the population seems to remain flat out at least to the 1 R_E limit of the survey . This observation confirms what has been found earlier by other investigators analyzing Kepler’s discoveries. This finding suggests that there may be distinctly different planet formation processes at work either side of this threshold. For planets in the 1.0 to 1.4 R_E range (which would include worlds that are most likely to be rocky like the terrestrial planets of our Solar System) with periods of 5 to 50 days, Petigura finds an occurrence rate of 7.8% (+1.3%/-2.1%).

Out to the Habitable Zone

In order to get a sufficient number of extrasolar planets to analyze statistically in more distant orbits where the probability of producing a transit is less, Petigura had to expand his survey to include less ideal target stars. In this second part of his analysis, which builds on the work he had published earlier with Howard and Marcy, Petigura again chose Kepler targets with surface temperatures in the 4100 to 6100 K range but now included all targets with Kepler band magnitude values, K_p, between 10 and 15. He then used TERRA to analyze Kepler’s photometric data for this expanded set of 42,257 stars to look for planets with orbital periods in the 0.5 to 400-day range. After a painstaking process of weeding out various types of false positive signals, Petigura found a total of 603 objects of interest. Subsequent ground-based observations of 274 of these helped to reduce the uncertainties in the target stars’ characteristics resulting in an improvement in the accuracy of the derived planetary properties.

This plot shows the planets found in the analysis of Kepler data (indicated as red dots) as a function of orbital period and planet size. The color coding gives an indication of the completeness of the survey. Click on image to enlarge. (Petigura)

After taking into account the completeness of the survey results, Petigura once again found that the planet population increases with decreasing size for extrasolar planets with orbital periods in the 5 to 100-day range. With only four extrasolar planets found with radii smaller than 2 R_E and periods greater than 100 days, the survey results were too incomplete for reliable statistical analysis of extrasolar planets in orbits with longer periods. Taking into account the uncertainties in the survey, he found that the size distribution appears to reach a plateau at about 2 R_E broadly consistent with what he found in the higher quality sample of planets with short-period orbits. Petigura found that 26±3% of Sun-like stars have roughly Earth-size planets with radii between 1 and 2 R_E and orbital periods in the 5 to 100-day range. About 12% of Sun-like stars have planets in the 1.0 to 1.4 R_E range (and therefore are most likely to be predominantly rocky in composition) while only 1.6±0.4% of Sun-like stars have Jupiter-size planets in orbits with this period range.

Looking at the distribution of planets as a function of the logarithm of orbital period, Petigura found that the population increases with orbital period up to about 25 days. The population’s occurrence rate then appears to remain constant for longer orbital periods out to at least 100 days. These results are consistent with those found by other investigators with smaller sized samples. Petigura also analyzed his survey results as a function of stellar flux which ranged from 0.5 to 700 times that of the Earth. Correcting for the completeness of his survey, Petigura found that 11±4% of Sun-like stars have planets in the 1 to 2 R_E size range with stellar fluxes 1 to 4 times that of the Earth. This range of stellar fluxes would encompass the Earth and Venus in our Solar System.

These histograms show the distribution of the sizes (A on left) and orbital periods of Kepler finds for planets with periods between 5 and 100 days. The gray color indicates planets actually found while the red indicates the estimated number of planets missed by the analysis. Click on image to enlarge. (Petigura)

Because of the increasingly lower probability that extrasolar planets in longer period orbits will produce observable transits, Petigura had to do some modest extrapolation of his statistical results to estimate their occurrence rates. The certainty of this extrapolation is enhanced somewhat by his observation that the size distribution for planets in the 1 to 2 R_E range appears constant at orbital periods greater than 25 days and that the occurrence rate of planets appears to be constant per interval of the log of orbital period. In this way, Petigura reproduces the estimate of his earlier published work with Howard and Marcy that 5.7% (+1.7%/-2.2%) of Sun-like stars are orbited by planets with radii between 1 and 2 R_E with a periods ranging from 200 to 400 days – what they defined as “Earth analogs” in this earlier work.

As was pointed out earlier, this definition for “Earth analogs” is not the equivalent of that for potentially habitable planets. Petigura does push his extrapolations a bit further to estimate the number of Earth-size planets in the habitable zones of Sun-like stars which stretches well beyond the limits sampled by the Kepler mission. Assuming the statistical characteristics of the planet population holds at these greater distances, Petigura estimates that about 22% of Sun-like stars would have Earth-size planets in the 1 to 2 R_E range inside a “simple” definition of the habitable zone. This “simple” definition includes all planets with stellar flux values in the range of 0.25 to 4 times that of the Earth and largely encompasses the more optimistic definitions for the habitable zone. This result implies that the nearest Earth-size planet orbiting inside this very optimistic definition of the habitable zone of a Sun-like star is less than 12 light years away.

Petigura also scaled his results to estimate the occurrence rate of Earth-size planets for a variety of other definitions of the habitable zone. One of the more conservative definitions of the habitable zone, based on detailed atmospheric radiative transfer and geophysical modelling, is by Kopparapu et al. which itself builds on the earlier pioneering work of James Kasting (Penn State) – a definition I have made no secret over the past couple of decades that I prefer. Based on this more conservative definition, the habitable zone of a Sun-like star encompasses stellar flux values in approximately the range of 0.35 to 1.02 times that of the Earth. Petigura estimates that 8.6% of Sun-like stars have planets in the 1 to 2 R_E size range orbiting inside this definition of the habitable zone.

One has to be careful about this value because even it is still not the equivalent of the occurrence rate of potentially habitable planets around Sun-like stars. Recent analyses of Kepler finds has allowed the mass-radius relationship of planets to be characterized for the first time in the 1 to 4 R_E size range (i.e. sizes larger than the rocky planet Earth and smaller the volatile-rich planet Neptune in our Solar System). These analyses, although still in their earliest stages, strongly suggest that planets larger than about 1.5 R_E are increasingly likely to be mini-Neptunes with little prospects of being habitable. Smaller planets are much more likely to be rocky in composition like the Earth (see “Habitable Planet Reality Check: Terrestrial Planet Size Limit” and “A Mass-Radius Relationship for ‘Sub-Neptunes’”). Assuming a constant size distribution for planets in the 1 to 2 R_E range like that found by Petigura, I estimate that about 5% of Sun-like stars have planets in the 1.0 to 1.5 R_E size range that are most likely to be rocky planets.

But even this is not the occurrence rate of all potentially habitable planets orbiting Sun-like stars. Not counted are planets smaller than the Earth which Petigura rightfully excluded from his analysis because Kepler is ill-suited to detect them orbiting Sun-like stars. Planets with masses as small at 0.25 times that of the Earth (equivalent to a radius of 0.63 R_E, assuming an Earth-like composition) or even smaller could be potentially habitable. Pushing this extrapolation into this admittedly uncharted territory of sub-Earth size extrasolar planets, I estimate that possibly 11% of Sun-like stars could have potentially habitable planets. This is on the order of about a third of the value of 29% (+25%/-12%) found for red dwarf stars by Dressing and Charbonneau based on their recently published analysis of Kepler finds but still roughly consistent considering the large uncertainties involved (see “The Occurrence of Potentially Habitable Planets Around Red Dwarf Stars“). My estimated 11% occurrence rate implies that the closest truly habitable Earth-like planet orbiting a Sun-like star may be less than a still relatively modest 15 light years away.

Conclusion

This analysis of Kepler data by Erik Petigura has provided empirical data that is now allowing us to get a hint of the occurrence rate of Earth-size rocky planets inside the habitable zones of Sun-like stars – planets that could be considered true Earth-twins. Based on extrapolations of Petigura’s survey results, I estimate that maybe 11% of Sun-like stars could have potentially habitable planets in the 0.63 to 1.5 R_E radius range. Of course whether or not these worlds are actually habitable in the Earth-like sense (i.e. they have surface conditions that can support liquid water allowing for the possibility of life) depends on a host of other critical factors which are currently beyond our means of measuring as well as possibly other considerations we have yet to even recognize.

Artist’s depiction of NASA’s Transiting Exoplanet Survey Satellite (TESS) that promises to find even more Earth-size planets orbiting Sun-like stars after its scheduled launch in 2017. (NASA/GSFC)

It should be remembered however, that there is much uncertainty in this result since it involves various assumptions and extrapolations that, in some cases, reach far beyond what was actually detected in Petigura’s survey results. That being said, Petigura’s work should be considered preliminary since it based on an analysis of only three years of data from Kepler’s four-year primary mission. Future analyses, which will benefit from the techniques developed for TERRA and other analysis algorithms, will be able to use much more data and should detect many more Earth-size planets in and near the habitable zone. Kepler’s ongoing “K2” extended mission also promises to provide much more data from a larger sample of stars of extrasolar planets in short-period orbits. Follow on missions such as NASA’s TESS (Transiting Exoplanet Survey Satellite) and ESA’s CHEOPS (CHaracterizing ExOPlanets Satellite), both scheduled for launch in 2017, will only add more finds to better characterize the population of extrasolar planets. Astronomers’ estimates for the occurrence rate of potentially habitable planets will only improve in accuracy in the years to come.

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Related Reading

“The Occurrence of Potentially Habitable Planets Around Red Dwarf Stars”, Drew Ex Machina, January 12, 2015 [Post]

“The Abundance of Earth Analogs”, Drew Ex Machina, June 25, 2014 [Post]

“Habitable Planet Reality Check: Terrestrial Planet Size Limit”, Drew Ex Machina, July 24, 2014 [Post]

“A Mass-Radius Relationship for ‘Sub-Neptunes'”, Centauri Dreams, May 22, 2015 [Post]

“The Abundance of Venus Analogs”, Drew Ex Machina, September 15, 2014 [Post]

General References

Courtney D. Dressing and David Charbonneau, “The Occurrence of Potentially Habitable Planets Orbiting M Dwarfs Estimated from the Full Kepler Dataset and an Empirical Measurement of the Detection Sensitivity”, The Astrophysical Journal, Vol. 807, No. 1, Article id. 45, July 2015

R. K. Kopparapu et al., “Habitable zones around main-sequence stars: new estimates”, The Astrophysical Journal, Vol. 765, No. 2, Article ID. 131, March 10, 2013

Erik A. Petigura, Andrew W. Howard and Geoffrey W. Marcy, “Prevalence of Earth-size planets orbiting Sun-like stars”, Proceedings of the National Academy of Sciences of the United States, Vol. 110, No. 48, pp. 19273-19278, November 26, 2013

Erik Ardeshir Petigura, “Prevalence of Earth-size Planets Orbiting Sun-like Stars”, arXiv 1510.03902v1 (PhD dissertation), October 13, 2015 [Preprint]