The Transition from Rocky to Non-Rocky Planets

by Paul Gilster on November 14, 2014

As I decompress from the Tennessee Valley Interstellar Workshop (and review my notes for next week’s report), I have the pleasure of bringing you Andrew LePage’s incisive essay into a key exoplanet question. Are some of the planets now considered potentially habitable actually unlikely to support life? Recent work gives us some hard numbers on just how large and massive a planet can be before it is more likely to be closer to Neptune than the Earth in composition. The transition from rocky to non-rocky planets is particularly important now, when our instruments are just becoming able to detect planets small enough to qualify as habitable. LePage, who writes the excellent Drew ex Machina, remains optimistic about habitable planets in the galaxy, but so far the case for many of those identified as such may be weaker than we had thought. A prolific writer, Drew is also a Senior Project Scientist at Visidyne, Inc., where he specializes in the processing and analysis of remote sensing data.

by Andrew LePage

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For much of the modern era, astronomy has benefitted greatly from the efforts of amateur scientists. But while amateur astronomers equipped with telescopes have certainly filled many important niches left by the far less numerous professionals in the field, others interested in astronomy equipped with nothing more than a computer and an Internet connection are capable of making important contributions as well. One project taking advantage of this resource is Planet Hunters.

The Planet Hunters project was originally started four years ago by the Zooinverse citizen science program to enlist the public’s help in searching through the huge photometric database of NASA’s Kepler mission looking for transits caused by extrasolar planets. While automated systems have been able to uncover thousands of candidate planets, they are limited to finding only what their programmers designed them to find – multiple, well defined transits occurring at regular intervals. The much more adaptable human brain is able to spot patterns in the changes in the brightness of stars that a computer program might miss but could still indicate the presence of an extrasolar planet. Currently in Version 2.0, the Planet Hunters project has uncovered 60 planet candidates to date through the efforts of 300,000 volunteers worldwide.

A paper by a team of astronomers with Joseph Schmitt (Yale University) as the lead author was just published in The Astrophysical Journal which describes the latest find by Planet Hunters. The target of interest for this paper is a billion year old, Sun-like star called Kepler 289 located about 2,300 light years away. Automated searches of the Kepler data had earlier found two planets orbiting this distant star: a large super-Earth with a radius 2.2 times that of the Earth (or RE) in a 34.5-day orbit originally designated Kepler 289b (called PH3 b in the new paper) and a gas giant with a radius of 11.6 RE in 125.8-day orbit, Kepler 289c (now also known as PH3 d). The new planet, PH3 c, has a radius of 2.7 RE and a mean orbital period of 66.1 days. With a mean stellar flux about 11 times that of Earth, this planet is highly unlikely to be habitable but its properties have profound implications for assessing the potential habitability of other extrasolar planets.

The planet had been missed by earlier automated searches because its orbital period varies regularly by 10.5 hours over the course of ten orbits due to its strong interactions with the other two planets, especially PH3 d. Because of this strong dynamical interaction, it was possible for Schmitt et al. to use the Transit Timing Variations or TTVs observed in the Kepler data to compute the masses of these three planets much more precisely than could be done using precision radial velocity measurements. The mass of the outer planet, PH3 d, was found to be 132±17 times that of Earth (or ME) or approximately equivalent to that of Saturn. The mass of the inner planet, PH3 b, was poorly constrained with a value of 7.3±6.8 ME. The newest discovery, PH3 c, was found to have a mass of 4.0±0.9 ME which, when combined with the radius determined using Kepler data, yields a mean density of 1.2±0.3 g/cm3 or only about one-fifth that of the Earth. Models indicate that this density is consistent with PH3 c possessing a deep, hot atmosphere of hydrogen and helium making up about half of its radius or around 2% of its total mass.

PH3 c is yet another example of a growing list of known low-density planets with masses just a few times that of the Earth that are obviously not terrestrial or rocky in composition. Before the Kepler mission, such planets were thought to exist but their exact properties were unknown because none are present in our solar system. As a result, the position in parameter space of the transition from rocky to non-rocky planets and the characteristics of this transition were unknown. So when astronomers were developing size-related nomenclature to categorize the planets they expected to find using Kepler, they somewhat arbitrarily defined “super-Earth” to be any planet with a radius in the 1.25 to 2.0 RE range regardless of its actual composition. Planets in the 2.0 to 4.0 RE range were dubbed “Neptune-size”. This has generated some confusion over the term “super-Earth” and has led to claims about the potential habitability of these planets being made in the total absence of an understanding of the true nature of these planets. Now that Kepler has found planets in this size range, astronomers have started to examine the mass-radius relationship of super-Earths.

The first hints about the characteristics of this transition from rocky to non-rocky planets were discussed in a series of papers published earlier this year. Using planetary radii determined from Kepler data and masses found by precision radial velocity measurements and analysis of TTVs, it was found that the density of super-Earths tended to rise with increasing radius as would be expected of rocky planets. But somewhere around the 1.5 to 2.0 RE range, a transition is passed where larger planets tended to become less dense instead. The interpretation of this result is that planets with radii greater than about 1.5 RE are increasingly likely to have substantial envelopes of various volatiles such as water (including high pressure forms of ice at high temperatures) and thick atmospheres rich in hydrogen and helium that decrease a planet’s bulk density. As a result, these planets can no longer be considered terrestrial or rocky planets like the Earth but would be classified as mini-Neptunes or gas dwarfs depending on the exact ratios of rock, water and gas.

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Image: It now appears that many of the fanciful artist depictions of super-Earths are wrong and that most of these planets are more like Neptune than the Earth (NASA Ames/JPL-Caltech).

A detailed statistical study of this transition was submitted for publication this past July by Leslie Rogers (a Hubble Fellow at the California Institute of Technology) who is also one of the coauthors of the PH3 c discovery paper. In her study, Rogers confined her analysis to transiting planets with radii less than 4 RE whose masses had been constrained by precision radial velocity measurements. She excluded planets with masses determined by TTV analysis since this sample may be affected by selection biases that favor low-density planets (for a planet of a given mass, a large low-density planet is more likely to produce a detectable transit event than a smaller high-density planet). Rogers then determined the probability that each of the 47 planets in her Kepler-derived sample were rocky planets by comparing the properties of those planets and the associated measurement uncertainties to models of planets with various compositions. Next, she performed a statistical analysis to assess three different models for the mass-radius distribution for the sample of planets. One model assumed an abrupt, step-wise transition from rocky to non-rocky planets while the other two models assumed different types of gradual transitions where some fraction of the population of planets of a given radius were rocky while the balance were non-rocky.

Rogers’ analysis clearly showed that a transition took place between rocky and non-rocky planets at 1.5 RE with a sudden step-wise transition being mildly favored over more gradual ones. Taking into account the uncertainties in her analysis, Rogers found that the transition from rocky to non-rocky planets takes place at no greater than about 1.6 RE at a 95% confidence level. Assuming a simple linear transition in the proportions of rocky and non-rocky planets, no more than 5% of planets with radii of about 2.6 RE will have densities compatible with a rocky composition to a 95% confidence level. PH3 c, with a radius of 2.7 RE, exceeds the threshold found by Rogers and, based on its density, is clearly not a terrestrial planet.

An obvious potential counterexample to Rogers’ maximum rocky planet size threshold is the case of Kepler 10c, which made the news early this year. Kepler 10c, with a radius of 2.35 RE determined by Kepler measurements and a Neptune-like mass of 17 ME determined by radial velocity measurements, was found to have a density of 7.1±1.0 g/cm3. While this density, which is greater than Earth’s, might lead some to conclude that Kepler 10c is a solid, predominantly rocky planet, Rogers counters that its density is in fact inconsistent with a rocky composition by more than one-sigma. Comparing the measured properties of this planet with various models, she finds that there is only about a 10% probability that Kepler 10c is in fact predominantly rocky in composition. It is much more likely that it possesses a substantial volatile envelope albeit smaller than Neptune’s given its higher density.

While much more work remains to be done to better characterize the planetary mass-radius function and the transition from rocky to non-rocky planets, one of the immediate impacts of this work is on the assessment of the potential habitability of extrasolar planets. About nine planets found to date in the Kepler data have been claimed by some to be potentially habitable. Unfortunately, all but two of these planets, Kepler 62f and 186f, have radii greater than 1.6 RE and it is therefore improbable that they are terrestrial planets, never mind potentially habitable planets.

This still leaves about a dozen planets that have been frequently cited as being potentially habitable that were discovered by precision radial velocity surveys whose radii are not known. However, we do know their MPsini values where MP is the planet’s actual mass and i is the inclination of the orbit to our line of sight. Since this angle cannot be derived from radial velocity measurements alone, only the minimum mass of the planet can be determined or the probability that the actual mass is in some range. Despite this limitation, the MPsini values can serve as a useful proxy for radius.

Rogers optimistically estimates that her 1.6 RE threshold corresponds to a planet with a mass of about 6 ME assuming an Earth-like composition (which is still ~50% larger than the measured mass of PH3 c, which is now known to be a non-rocky planet). About half of the planets that some have claimed to be potentially habitable have minimum masses that exceed this optimistic 6 ME threshold while the rest have better than even odds of their actual masses exceeding this threshold. If the threshold for the transition from rocky to non-rocky planets is closer to the 4 ME mass of PH3 c, the odds of any of these planets being terrestrial planets are worse still. The unfortunate conclusion is that none of the planets discovered so far by precision radial velocity surveys are likely to be terrestrial planets and are therefore poor candidates for being potentially habitable.

Please do not get me wrong: I have always been a firm believer that the galaxy is filled with habitable terrestrial planets (and moons, too!). But in the rush to find such planets, it now seems that too many overly optimistic claims have been made about too many planets before enough information was available to properly gauge their bulk properties. Preliminary results of the planetary mass-radius relationship now hints that the maximum size of a terrestrial planet is probably about 1½ times the radius of the Earth or around 4 to 6 times Earth’s mass. Any potentially habitable planet, in addition to having to be inside the habitable zone of the star it orbits, must also be smaller than this. Unfortunately, while recent work suggests that planets of this size might be common, our technology is only just able to detect them at this time. With luck, over the coming years as more data come in, we will finally have a more realistic list of potentially habitable planet candidates that will bear up better under close scrutiny.

The discovery paper for PH3 c by Schmitt et al., “Planet Hunters VII: Discovery of a New Low-Mass, Low Density Planet (PH3 c) Orbiting Kepler-289 with Mass Measurements of Two Additional Planets (PH3 b and d)”, The Astrophysical Journal, Vol. 795, No. 2, ID 167 (November 10, 2014) can be found here. The paper by Leslie Rogers submitted to The Astrophysical Journal, “Most 1.6 Earth-Radius Planets are not Rocky”, can be found here.

For a fuller discussion of how Rogers’ work impacts the most promising planets thought by many to be potentially habitable, please refer to Habitable Planet Reality Check: Terrestrial Planet Size Limit on my website Drew Ex Machina.

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andy November 26, 2014 at 3:58

I never said that there was no connection between the partial pressure of atmospheric CO2 and the pH of the ocean.

Apart from the fact you’ve been arguing this from the start…
November 17, 2014 at 18:02

The high CO2 levels one would expect in the outer parts of the HZ as a natural consequence of the carbonate-silicate cycle would NOT result in ocean acidification

Oh well, not sure there’s much point continuing with this anyway. As I said, I accept the point about the calcium carbonate precipitation.

Andrew LePage November 26, 2014 at 12:19

@andy November 26, 2014 at 3:58

(rolling eyes) We are talking about the degree of change here. I have argued from the start that your original speculation was incorrect and that a habitable world’s oceans under equilibrium conditions would never become acidic enough to dissolve carbonate shells (a claim that you now seem to admit is incorrect). That is NOT the same as claiming that the partial pressure of atmospheric CO2 has no effect whatsoever on the pH of a habitable world’s ocean. The effect is just MUCH less than what you wanted to believe.

Ronald November 28, 2014 at 6:52

Another brief note with regard to CO2, greenhouse warming in the HZ and earthlike planets: CO2 only becomes poisonous to us (earthlike animal life) at very high levels. Even if the atmospheric CO2 level were at ten times, or a few tens of times the current level, causing a strong greenhouse warming effect, it would still not be poisonous to animal life on earth.

Sp, I think that considerably more beneficial CO2 caused greenhouse warming can take place on a terrestrial planet with earthlike biology, effectively pushing the limit for such a planet far outward in the HZ.

Undoubtedly studies have been done in this respect.

Ronald November 28, 2014 at 7:07

Ok, I browsed through the mentioned publication “The Occurrence and Architecture of Exoplanetary Systems”. Interesting, but not a lot of new or spectacular discoveries.

One very relevant thing that stood out in the Summary and conclusions was:

“A Sun-like star has a 10% chance to have a giant planet with a period shorter than a few years, and a 50% chance to have a compact system of smaller planets with periods shorter than a year.”

The first fact, that about 10% of sunlike stars would have a warm – hot Jupiter (a few years, is that within about 2 AU?) surprises me a bit, I used to think it was less. Is there observational bias here, which has not been completely compensated for?

The second fact, 50% of sunlike stars having a compact system of medium-sized planets, does not surprise me at all (anymore), this has probably been THE great and surprising exoplanet discovery of recent years, and I actually thought that this percentage is even (much?) higher, rather that at least two-thirds of sunlike stars have such compact systems.

Mr. Lepage, anyone, is there currently more known about this?

Andrew LePage November 28, 2014 at 10:58

@Ronald November 28, 2014 at 6:52

It is certainly true that any habitable world towards the outer part of its sun’s HZ would have large amounts of atmospheric CO2 that would be fatal to terrestrial animal life (i.e. animals from Earth). But that’s because our cellular machinery evolved over the last couple of billion years to live with much lower levels of atmospheric CO2. Organisms that evolved for billions of years with several bars of CO2 in their planet’s atmosphere would thrive under those conditions even though those same conditions would be fatal to us.

The scientific definition of a “habitable world” does NOT mean it would necessarily be someplace we or other terrestrial life forms could survive or thrive. It has historically been defined as a world that has temperatures in the right range to support liquid water on its surface and has the potential of supporting life. The vast majority of “habitable worlds” will have CO2 levels too high for us to withstand. And since we can not assume that forms of autotrophism like oxygen-producing photosynthesis will inevitably evolve on other life bearing planets, it is probable that the majority of “habitable worlds” that have low atmospheric CO2 levels will have little or no O2 in their atmospheres. Despite the picture painted in popular scifi, it is likely that planets that have surface conditions that can support us without some aid will be exceptionally rare.

@Ronald November 28, 2014 at 7:07

The paper I cited earlier whose result you discuss, Joshua N. Winn and Daniel C. Fabrycky, “The Occurrence and Architecture of Exoplanetary Systems”, arXiv: 1410.4199 (submitted to The Annual Review of Astronomy & Astrophysics), posted October 15, 2014 http://arxiv.org/abs/1410.4199, is a thorough review of the current state of the subject of extrasolar planetary system architecture. The only other recent paper on the topic whose results are not included in this review is one by Sarah Ballard and John Asher Johnson, “The Kepler Dichotomy Among the M Dwarfs: Half of Systems Contain Five or More Coplanar Planets” that was submitted at about the same time. This paper looks at just M dwarf planetary systems and finds pretty much the same thing: Kepler results are best explained by a 50-50 mix of coplanar multiplanet systems and systems with big planets in orbits with high mutual inclination. I talk about this newer paper in detail here:

http://www.drewexmachina.com/2014/10/24/architecture-of-m-dwarf-planetary-systems/

As for you final question, “is there currently more known about this?”, the short answer is, no, the cited papers the latest of what has been published or submitted for publication in peer-reviewed journals to date. However, this is a very active field of investigation and there is much more to be discovered from the ongoing analysis of Kepler data and newer data constantly coming in. More papers dealing with analyses of extrasolar planetary systems with new findings are almost surely in the pipeline.

Eniac November 28, 2014 at 20:56

Andrew LePage:

And since we can not assume that forms of autotrophism like oxygen-producing photosynthesis will inevitably evolve on other life bearing planets, it is probable that the majority of “habitable worlds” that have low atmospheric CO2 levels will have little or no O2 in their atmospheres.

Looking at the enormous variety of autotrophic metabolism amongst Earth organisms, it would seem very unlikely that any form of carbon-based life would not find a way to access the carbon reservoir of a CO2 atmosphere. By necessity any such metabolism would be oxygen-producing, and require a source of energy, such as light.

I would predict that any habitable world with sufficiently advanced life on it would have a CO2 depleted atmosphere, i.e. one depleted to the limit where further reduction would make fixation uneconomical. On the other hand, any world without life would likely have a CO2 rich atmosphere, as do Mars and Venus. Or, a reducing one, as seen on Titan.

Eniac November 28, 2014 at 20:57

Andrew LePage:

And since we can not assume that forms of autotrophism like oxygen-producing photosynthesis will inevitably evolve on other life bearing planets, it is probable that the majority of “habitable worlds” that have low atmospheric CO2 levels will have little or no O2 in their atmospheres.

Looking at the enormous variety of autotrophic metabolism amongst Earth organisms, it would seem very unlikely that any form of carbon-based life would not find a way to access the carbon reservoir of a CO2 atmosphere. By necessity any such metabolism would be oxygen-producing, and require a source of energy, such as light.

I would predict that any habitable world with sufficiently advanced life on it would have a CO2 depleted atmosphere, i.e. one depleted to the limit where further reduction would make fixation uneconomical. On the other hand, any world without life would likely have a CO2 rich atmosphere, as do Mars and Venus. Or, a reducing one, as seen on Titan.

Michael November 28, 2014 at 22:44

It is very unlikely there will be planets with thick CO2 atmospheres together with a lot of liquid water, the CO2 will be sequestered more than likely in hundreds of millions of years of the planets formation. Nitrogen and hydrogen/helium will most likely be the main constituents, the latter will be more dependant on the gravity of the worlds.

Andrew LePage November 29, 2014 at 22:25

@Michael November 28, 2014 at 22:44

“It is very unlikely there will be planets with thick CO2 atmospheres together with a lot of liquid water, the CO2 will be sequestered more than likely in hundreds of millions of years of the planets formation. ”

This could not happen on a habitable world where carbonate-silicate cycle is maintaining habitable temperatures. The key parts of this cycle that prevent the permanent sequestration of carbon are the subduction of carbonate deposits into the mantle followed by their breakdown into CO2 which is then released back into the atmosphere via vulcanism. A simple tutorial on the carbonate-silicate cycle developed over two decades ago by Dr. James Kasting (which represents the core of our modern understanding of planetary habitability) can be found at the following web site:

http://www3.geosc.psu.edu/~jfk4/PersonalPage/ResInt2.htm

A more detailed, readable description of the carbonate-silicate cycle and its role in the long term stability of habitable conditions can be found in Dr. Kasting’s recent book, “How to Find a Habitable Planet”.

Andrew LePage November 29, 2014 at 23:04

@Eniac November 28, 2014 at 20:56

“Looking at the enormous variety of autotrophic metabolism amongst Earth organisms, it would seem very unlikely that any form of carbon-based life would not find a way to access the carbon reservoir of a CO2 atmosphere. ”

Why? There are ecosystems that manage fine without tapping into atmospheric CO2 today and such ecosystems were even more prevalent earlier in in Earth’s history.

“By necessity any such metabolism would be oxygen-producing, and require a source of energy, such as light. ”

Not true. There are anoxygenic forms of photosynthesis where water does not act at the electron donor in the production of ATP and therefore do not produce oxygen. Green sulfur bacteria, purple bacteria and acidobacteria are examples of extant organisms that use anoxygenic photosynthesis. And there are many forms of energy available other than light that autotrophs can take advantage of.

“I would predict that any habitable world with sufficiently advanced life on it would have a CO2 depleted atmosphere”

This is highly unlikely. There is a vast reservoir of climatically inert carbonates locked up in Earth’s crust (estimated to be on the order of 60 bars) that would be released into the atmosphere in cases where there is excessive fixation of CO2 into the biosphere as a natural consequence of the carbonate-silicate cycle. And the growth of the biosphere is likely to be limited by the finite supply of other resources (nitrogen, phosphorus, etc.) and it is highly unlikely to ever deplete a planets supply of CO2 and carbonates.

Ronald December 1, 2014 at 10:52

@ Andrew LePage;
ok, thanks for your enlightening replies, most of which I agree with (or gladly accept as new knowledge).
However, one point I do not agree with;

Andrew LePage November 28, 2014 at 10:58:
“The vast majority of “habitable worlds” will have CO2 levels too high for us to withstand”.
In an earlier comment you stated that terrestrial planets would have to be close to the inner edge of the HZ for earthlike animal life, or otherwise, in order to be warm enough for liquid water, the CO2 level has to be too high (and hence poisonous) for earthly life.

This is what I meant to contest in my previous comment: what you say may be true toward the outer limit of the HZ. However, even earthly animal life can easily tolerate, or adapt to, a CO2 level about 10x, or even tens of times, the present level without being poisonous. CO2 only becomes poisonous to (most) animals at very high levels (many % points).
At 10 or a few tens of times present CO2 level (i.e. not nearly poisonous yet) there is a significant greenhouse effect, especially in combination with water vapor.
Hence my idea that terrestrial planets with earthlike biology can probably exist in a significant part (how much?) of the HZ, not just near the inner edge.

Michael December 1, 2014 at 14:55

@Andrew LePage November 29, 2014 at 22:25

‘This could not happen on a habitable world where carbonate-silicate cycle is maintaining habitable temperatures. The key parts of this cycle that prevent the permanent sequestration of carbon are the subduction of carbonate deposits into the mantle followed by their breakdown into CO2 which is then released back into the atmosphere via vulcanism.’

There is bound to be less subduction as a planet ages due to cooling, the natural consequence is that carbonates will form faster than they are been subducted ‘released back out’ leading to an overall reduction of CO2 in the atmosphere even early in the planets history.

Andrew LePage December 2, 2014 at 0:39

@Michael December 1, 2014 at 14:55

“There is bound to be less subduction as a planet ages due to cooling, the natural consequence is that carbonates will form faster than they are been subducted ‘released back out’ leading to an overall reduction of CO2 in the atmosphere even early in the planets history.”

This is why a one of the necessary conditions for habitable worlds is sufficient geologic activity to drive the recycling of carbonates. As is already acknowledged in the professional literature, without this activity, there is no long-term habitability. It is also appreciated that a world’s geologic activity typically decreases over time and will eventually fade to the point where key parts of the carbonate-silicate cycle break down rendering a previously habitable world uninhabitable. But for Earth-size planets, that process takes in excess of 4.6 billion years.

Andrew LePage December 2, 2014 at 10:22

@Ronald December 1, 2014 at 10:52

I am fully aware of terrestrial animal’s tolerance to CO2 and stand by my statement: CO2 levels in the atmospheres of the overwhelming majority of habitable planets will be lethal to terrestrial life. While water vapor is a more potent greenhouse gas than CO2 in *OUR* atmosphere (which contains just ~0.0004 bars of CO2), atmospheric models show that it quickly condenses out of the atmosphere as the temperatures drops. Once the effective stellar flux decreases by several percent, water vapor is insufficient in maintaining habitable surface conditions even with atmospheric CO2 levels a few times that of our atmosphere today. By the time the effective stellar flux drops to ~0.80 of Earth’s (equivalent to a distance of just ~1.1 AU from the Sun), models suggest that the CO2 partial pressure would need to be ~0.1 bars (on the order of double the level toxic to humans), give or take depending on the details, to maintain above-freezing surface temperatures even with an atmosphere saturated with water vapor. This level rises quickly to ~10 bars of CO2 at an effective stellar flux of ~0.36 (equivalent to a distance of ~1.7 AU) near the outer edge of the HZ. Beyond this, adding more CO2 to the atmosphere does not increase temperatures because the atmosphere starts to become too opaque in the visible and NIR due to absorption and excessive scattering.

Assuming a logarithmic spacing for planets and that the habitable zone for a Sun-like star extends from 0.95 to 1.67 AU (as defined in work by Kasting et al. and more recently by Kopparapu et al.), something on the order of >3/4 of habitable worlds will have CO2 levels that are toxic to humans and other forms of terrestrial animal life.

Ronald December 3, 2014 at 7:26

@Andrew LePage December 2, 2014 at 10:22:

Very enlightening (though disappointing), thank you!

“(…) models suggest that the CO2 partial pressure would need to be ~0.1 bars (…)”

Could you mention a good source for this modeling? Thank you in advance.

Andrew LePage December 3, 2014 at 11:27

@Ronald December 3, 2014 at 7:26

I wrote a fully-referenced review article 16 years ago for the now-defunct SETIQuest Magazine, “The Extremes of Habitability”, that included a detailed discussion of the limits of the habitable zone as derived from the (then) latest models by James Kasting et al. that may be of use (Figure 4 even shows modeled CO2 pressure as a function of insolation which I borrowed from data in one of the papers by Kasting et al. that is referenced).

http://www.drewexmachina.com/download-pdf/SQ_V4_N2_article_001.pdf

There are a slew of more recent work by Ravi Kopparapu and his colleagues that builds on this earlier work which can be accessed via his web site:

http://www3.geosc.psu.edu/~ruk15/planets/

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