Monday, December 21, 2015

Quantifying "Earth-like"

Figure 1. Three perspectives on one small planet.

Writers throughout the twentieth century engaged in speculation on extraterrestrial life: not only life as we know it, but life as we never knew it. Lately, however, interest has shifted away from such intriguing prospects as crystalline colonial organisms, sentient plasma clouds, and the ecologies of gas giant atmospheres, toward the more prosaic question of carbon-based life on Earth-like worlds. Current investigations invoke habitable zones and habitable planets, which are defined by the conditions that enabled our own mundane biosphere to emerge and endure (Figure 1).

The hydrocarbon lakes of Titan and the subsurface oceans of Enceladus and Europa remain topics of keen interest. For all we know, these icy worlds might support exotic biochemistries based on carbon compounds. But we have no secure way of detecting similar environments in exoplanetary systems, whether by present methods or those expected for some time to come (Kasting & Catling 2003, Kasting et al. 2014). This restriction indefinitely tethers our extrasolar speculations to Earth. The best response to such restraint might be simply to look in the mirror. What are the limits of “Earth-like?” Which parameters of our own world can tell us when we’ve found an extrasolar cousin or sibling? 


The most widely discussed feature of exoplanetary systems is the habitable zone – the range of orbits where the host star’s flux would permit surface bodies of water on a rocky planet with the appropriate mass and atmosphere. Obviously, Earth is such a planet, so we know where to look for the Solar System’s habitable zone. Recent research continues to explore this concept (Kopparapu et al. 2013, Zsom et al. 2013, Kasting et al. 2014). Kopparapu & colleagues express the more or less standard view: The habitable zone of a G-type star with the same mass as our Sun extends from about 0.9 to 1.5 astronomical units (AU). For a K-type star of 0.75 Solar masses (Msol) the approximate boundaries are 0.5 – 0.9 AU. For an M dwarf of 0.4 Msol, they shrink to 0.15 – 0.30 AU. 


It is also universally accepted that a planet needs some minimum mass in order to sustain an active rheology, robust atmosphere, and surface water over billions of years. Mars, at 0.11 Earth masses (Mea), is evidently too lightweight to fulfill this condition, whereas Venus, at 0.81 Mea, would be just right if only its orbit were wider. The lower boundary for mass must be somewhere in between. In a classic study, James Kasting and colleagues (1993) defined a habitable planet as “several times more massive than Mars.” They also reasoned that larger planets “have higher internal heat flows and should therefore be able to maintain tectonic activity” for substantial periods. As far as I know, the only researchers who have offered a precise value for the minimum habitable mass are Sean Raymond and colleagues (2006, 2007), who propose 0.3 Mea.

Finding the maximum mass, however, has been contentious. At least two factors are in play: plate tectonics and atmospheric accretion. 


Back in 1993, Kasting & colleagues noted that the carbon-silicate cycle is a necessary enabler of life on Earth. Although they didn’t mention it, this cycle is supported by plate tectonics (Figure 2), a process foregrounded by most subsequent discussions of extrasolar life.

More than a decade later, when theoretical discussions of Super Earths commenced, several studies examined the habitability of planets in the range of 1 to 10 Mea. These objects were typically assigned Earth-like compositions and heavy element atmospheres – rather than, for example, extended hydrogen/helium (H/He) envelopes. At least one study argued that plate tectonics would be “inevitable” on such Super Earths (Valencia et al. 2007). Similar conclusions were implicit in other literature of the time.

Figure 2. Volcanic eruptions are among the most visible indicators of the active geology that sustains habitable conditions on Earth. This photo shows the 1990 eruption of Redoubt Volcano along the coast of Cook Inlet in Alaska. Credit: Wikimedia
Then came the skeptics. Some have argued that solid objects above a few Earth masses are unlikely to support plate tectonics (O’Neill & Lenardic 2007, Korenaga 2010), or at least that the likelihood shrinks as mass increases (Stein et al. 2013, Stamenkovic & Breuer 2014). Others are more optimistic, and the debate is far from over.

In the meantime, it would be helpful to know exactly what “a few Earth masses” means. If there’s an upper mass limit on tectonics, what is it?

A clue just came in a preprint by Cayman Unterborn & colleagues (hereafter U15). Seeking to define “Earth-like,” they make plate tectonics the primary criterion. To generalize across star systems, they propose a three-layer model for terrestrial planets: a mostly iron core surrounded by magnesium silicate minerals, in proportions that will vary from world to world, distributed in a dense lower mantle that transitions to a lighter upper mantle. In Earth, the bulk mass percentage for each layer is respectively 32%, 51%, and 17%. U15 argue that planets with mantle fractions and convective regimes similar to Earth will experience plate tectonics.

To illustrate (their Figure 8), they offer a schema of masses, radii, and chemical compositions that meet this condition, extending as high as an object of 5 Mea and 1.5 Rea. Although they don’t explicitly discuss a mass limit for mantle convection, they do argue that plate tectonics is possible on all the rocky planets encompassed by their schema. As a real-life example they offer Kepler-36b, a classic Hot Super Earth on a 14-day orbit with an approximate mass and radius of 4.45 Mea and 1.49 Rea, respectively.

The findings of U15 invite comparison with the recent work of Courtney Dressing & colleagues (hereafter D15) on the structure of terrestrial planets. D15 tried to find a single composition that could explain the masses and radii of Earth, Venus, and five well-characterized extrasolar terrestrials (CoRoT-7b; Kepler-10b, -36b, -78b, and -93b). By contrast, U15 tried to generalize from Earth’s parameters to construct a flexible model that would encompass all potentially Earth-like planets. This difference in goals might explain why D15 used a simpler model of planet structure, with only two layers: a pure iron core accounting for 17% of the total mass and a magnesium silicate mantle accounting for 87%.

Notably, their solution involves a much less massive core than the one proposed by U15. Hence they find a smaller planetary mass at each radius than do U15. As their real-life example of a terrestrial planet, D15 offer Kepler-93b, for which they prefer a mass of about 4 Mea and a radius of 1.48 Rea. For the same radius, U15 provide a mass 10% higher. However, the mismatch between the two studies falls within the range of uncertainties for the parameters of the five exoplanets modeled by D15. For Kepler-93b, they defined the mass range as 3.34-4.70 Mea, while for the same mass range U15 provided a range in radius of about 1.35-1.50 Rea, which encompasses D15’s preferred value.

Happily, then, the results of U15 appear consistent with the those of D15, whose model has been widely endorsed. Their conclusions have the further appeal of supporting plate tectonics on selected planets massing up to 5 Mea. This is a more generous upper bound than I’ve imagined in recent years.

Nonetheless, it’s clear that an appropriate mantle structure is insufficient by itself to guarantee either plate tectonics or life. Water and atmosphere make critical contributions.

Figure 3. The total water content of Earth and Europa compared. Even though Europa is only about 1% of Earth’s mass, it contains more water by mass than Earth. Credit: K. P. Hand


The factors governing Earth-like physical conditions and carbon-based life have complex interdependencies. Life needs water, and oceans need plate tectonics, but tectonics also needs oceans (Korenaga 2010, Lammer et al. 2009, 2010). Lubrication is required to facilitate plate movement, and apparently ice won’t do: the eternal wandering of continents is borne by water.

For truly Earth-like conditions, however, the amount of water requires titration. Too much is just as bad as not enough, and an excess of water seems quite easy for young planets to accrete, at least according to simulations of Solar System history (Raymond et al 2007). Helmut Lammer & colleagues (2009) pointed out that a rocky planet covered by a water layer 100 km deep, as modeled by studies of “Ocean Planets” (e.g., Leger et al. 2004), would be unsuitable for the development of life. Regardless of temperature, high-pressure ices would form at the bottom of such an ocean and prevent interaction between the water layer and chemicals in the crust. Without this interaction, life could not arise, and the carbon-silicate cycle would not emerge.

Even though we were told as children that three-quarters of our planet is covered by oceans, that water layer is remarkably thin (Figure 3). The total water inventory of Earth, including water in the mantle, is quite small: just 0.05% Mea (Raymond et al. 2014). Yann Alibert (2014) found that a planet of Earth mass and composition can maintain both a global ocean and a carbon cycle only if its water content is 2% or less by mass, with the maximum percentage falling rapidly with rising planet mass. So far, that seems to be the best available constraint on water inventory.

It’s worth noting that although Alibert’s upper limit is 40 times greater than Earth’s current reservoir, it’s still 5 to 25 times smaller than the water fractions proposed for Sauna Planets and Water Worlds. 


An atmosphere is generally assumed as a prerequisite for life. One notable exception involves airless bodies like Europa, where biochemistries could evolve in shallow subsurface oceans. However, our focus is Earth-like planets with masses that exceed Europa’s by more than an order of magnitude. These objects are believed to accrete or outgas significant atmospheres during their formation and early evolution.

Any planet above the minimum Earth-like mass (defined here as 0.3 Mea) will likely retain its gas envelope as long as it can withstand the high levels of extreme ultraviolet (XUV) flux emitted by young stars. Planets orbiting near the star are the most vulnerable to atmospheric erosion, while those with masses of several Mea have the best protection.

Pre-Kepler discussions took it for granted that any object under 10 Mea would be incapable of supporting an H/He envelope. Then in 2011 came the discovery of the Kepler-11 system, where at least four planets with masses between 2 and 8 Mea revealed puffy silhouettes consistent with deep H/He atmospheres. Many comparable Kepler planets have been characterized since then. It is now understood that rocky cores similar in mass to Earth can have radically more extensive envelopes. Since H2 is a greenhouse gas, and H/He atmospheres produce surface pressures far in excess of Earth’s, even relatively low-mass planets with deep gas envelopes will not sustain surface bodies of water. By definition, they are not Earth-like.

Two recent studies, led respectively by Rebekah Dawson and Helmut Lammer, explored the conditions needed for terrestrial planets up to 5 Mea to accrete and sustain puffy envelopes. While taking different approaches, these two groups found a similar lower boundary for envelope survival: 2 Mea. Above that mass, unless they orbit very close to their host stars, most planets will accrete and retain H/He atmospheres. Below that mass, even on cooler orbits, primordial H/He will dissipate.

Dawson & colleagues (hereafter D15) conducted N-body simulations to study the in situ accretion of rocky objects in protoplanetary disks of varying metallicity, along with their atmospheric evolution up to the dispersion of the nebular gas. They did not address the photoevaporation of primitive atmospheres, a factor that becomes significant only after the nebula disperses. Nonetheless, they cited Lopez & Fortney (2014) for a discussion of atmospheric loss at later stages of system evolution.

The central aim of D15 was to investigate the relationship between the surface density of solid materials suspended in the primordial nebula and the mass and atmospheric composition of the resulting planets. Their simulations followed the evolution of rocky embryos orbiting a Sun-like star between 0.04 and 1 AU in the presence of a dusty H/He nebula that dissipated after 1 million years. The embryos grew by mergers and accreted gas according to their mass. D15 found that 1) planetary cores smaller than 2 Mea did not accrete substantial envelopes from the nebula, and 2) cores of 2 Mea or more could form within 1 million years only in protoplanetary disks with a high surface density of solids. Such environments are typical of highly metallic stars even without significant gas-driven migration of planetesimals or embryos.

D15 also concluded that the inner nebulae of stars with ordinary or depleted metallicity can still achieve a sufficient surface density to build gas dwarfs if they experience migration of solids from outer orbits. In other words, D15’s approach does not require “strict” in situ accretion (Chiang & Laughlin 2013). They even permit the migration of full-formed gas dwarfs from outside 1 AU, as in Lee & colleagues (2014, 2015).

This study raises interesting questions on many points, including realistic timeframes for nebula dispersion and the effect of mixed migration pathways on the composition of planets that achieve habitable orbits. Nevertheless, the answers probably wouldn’t change D15’s most salient findings on envelope accretion by young planets. They conclude that planets under 2 Mea are unlikely to capture or sustain H/He atmospheres, while planets above that mass will do so under typical conditions. This result provides a clear constraint on definitions of “Earth-like.”
Figure 4. Since massive rocky objects are likely to accrete deep hydrogen atmospheres, the range of Earth-like planets is narrow, extending from about one-third to twice the mass of Earth. Shown above are four NASA photographs of our planet scaled according to the mass-radius relationships of Unterborn et al. (2015). For perspective, one photogenic but out-of-range object – Mars – is included at far left.

An earlier study by Lammer & colleagues (hereafter L14) had already looked at the evolution of gas-enveloped planets after nebula dispersion. They studied a single phase of the process under idealized conditions, modeling rocky planets with masses between 0.1 and 5 Mea orbiting a Solar twin at 1 AU. All planets were assumed to reach their final core masses in the presence of the primordial nebula, and to accrete H/He envelopes in proportion to their mass.

L14 differed sharply from D15 in their handling of gas accretion, since even sub-Earth objects in their model captured H/He. Nor did they advance any argument regarding surface densities of solids or the formation pathways of their theoretical planets. Their approach appears agnostic to these factors.

In the most favorable variations on their model, some Super Earths of 5 Mea reached the threshold of runaway gas accretion. In real systems these would become gas giants. The rest, along with all other planets of lower mass, captured smaller but still substantial H/He envelopes before the nebula dispersed. With the loss of this protective cloud, however, the effects of XUV flux became significant.

L14 found that rocky planets down to 2 Mea –in some cases even as low as 1 Mea – suffered minimal atmospheric loss during the phase of “saturated” flux in the first 100 million years of stellar evolution. These planets were able to retain their H/He envelopes indefinitely, resembling typical Kepler planets with masses of 2-5 Mea and radii of 2-4 Rea. Less massive planets, however, lost their envelopes.

Considering L14’s findings alongside the other constraints discussed so far, we see some major shrinkage in the likely mass range of Earth-like planets around Sun-like stars. Evidently it’s about 0.3 to 2 Mea (Figure 4), with potential outliers at slightly higher masses. Given the level of XUV flux typical of the habitable zones of late F, G, and early K-type stars, all planets under 1 Mea lose their primordial H/He, whereas most planets over 2 Mea retain it.

L14 proposed that the relative dustiness of the nebula would be a critical factor determining the survival of H/He envelopes around planets over 1 Mea. Some fraction of rocky planets between 2 and ~2.5 Mea might end up with friendly atmospheres of nitrogen and carbon dioxide, but that outcome becomes vanishingly less likely with increasing mass.

The only exception might be planets that reached their final bulk during a phase of giant impacts after the nebula dissipated. This is actually how the Solar System’s small planets formed, but recent work suggests that our system’s evolutionary history is unique. In other potential evolutionary scenarios, we could imagine a collision between two Super Earths of 1.8 Mea each, which (after the dust settles) would create a single gas-free planet of about 3.5 Mea. Since the collision happens after nebula dispersion, the new planet cannot accrete any additional atmosphere. Thus it evolves into a truly super-sized Earth with volcanoes, oceans, and all the rest. Scenarios like that might be rare, though.

On firmer ground, the complementary results of D15 and L14 indicate that rocky planets of 0.3–2.0 Mea and 0.7–1.2 Rea will be free of troublesome H/He envelopes. Depending on the specifics of formation pathways and XUV flux, these planets would make plausible candidates for Earth 2. 

spectral type 

But don’t forget about that X-ray and XUV flux. Indeed, it’s been getting a lot of attention over the past year or so. Most of the studies discussed in this posting used stars of the same mass and effective temperature as our Sun for their standard. The underlying assumption is that factors relevant to thermal environment, such as the location of the system habitable zone, can be scaled to fit stars of different effective temperatures, luminosities, and colors.

But stellar evolution places a limit on such scaling. The earliest discussions of extrasolar life noted that stars above a certain mass – around 1.5 times Solar (1.5 Msol) – have a main sequence lifetime too brief to permit the evolution of life. Even if a planet of the right mass and composition were to orbit comfortably in the habitable zone of an A-type star of 1.8 Msol, it would barely have time to cool down and recover from asteroid bombardment before its parent star began expanding and reddening into the subgiant stage. Rising temperatures would then boil off the planetary ocean and sterilize any emergent biosphere.

Fortunately, high-mass stars of spectral types A, B, and O represent less than 1% of the stellar population in our region of the Milky Way. What about M dwarfs, which account for three-quarters of all main sequence stars? Questions regarding the habitability of their planets are getting complicated.

Earlier studies noted that the close-in habitable zones of M dwarfs would constrain planets with the appropriate insolation to be tidally locked, without benefit of a day/night cycle. M dwarfs are also more likely than higher-mass stars to erupt in intense flares that could destroy volatiles and erode the atmospheres of close-in planets. Nevertheless, neither factor seems an insurmountable barrier to the emergence of life. Presumably organisms could evolve in non-stop daylight and eventually colonize darker longitudes, while robust atmospheres would shield surface ecologies against occasional flares.

Recent literature, however, finds new causes for doubt. A study by Luger & Barnes (2015) provides several reasons for pessimism about the habitability of M dwarf planets. In addition to the propensity of red stars to undergo extreme flaring events, Luger & Barnes note the antagonistic qualities of their evolutionary history. Newborn stars under about 0.65 Msol spend several hundred million years at luminosities one to two orders of magnitude higher than their main sequence brightness. Yet planet formation around any star happens on a much shorter timescale, within a few tens of millions of years after stellar ignition. Luger & Barnes demonstrate what that mismatch in developmental histories means for water and life. Planets that form in a young M dwarf’s habitable zone will freeze out once the star matures, whereas planets that end up in the mature star’s habitable zone will have been roasted for a billion years by intense X-ray and XUV flux, including violent flares. The cool planets will be too cold, while the warm planets will be stripped of volatiles.

These results make M dwarf stars less attractive as potential hosts of Earth-like planets than they looked just a few years ago. It seems that sensibly Sun-like stars in the approximate range of 0.7–1.3 Msol are once again the best choice of parents, at least if you want to grow up to be green. 

perspectives on known space 

In the context of the other constraints outlined here, the findings of Luger & Barnes also call for a harder look at the clutch of small Kepler planets proposed over the past few years as potential Earth analogs. Among the six candidates confirmed to date, four orbit stars less massive than 0.65 Msol – M dwarfs by any other name. Ironically, these are the four smallest planets of the lot (Kepler-438b, -186f, -395c, -442b), with radii ranging from 1.12 to 1.34 Rea. Since all have periods shorter than 130 days, and two have periods shorter than 40 days, they all might have suffered complete desiccation. In fact, a new study just reported that Kepler-438b, with a semimajor axis of only 0.17 AU, experiences powerful flares from its host star that make it vulnerable to complete loss of atmosphere (Armstrong et al. 2015). None of these candidates seem truly Earth-like.

The other two planets (Kepler-62f, -452b) orbit hotter stars on longer orbital periods, so they appear to occupy their systems’ long-term habitable zones. By some definitions their radii place both of them at or near the upper edge of the Earth-like range, but according to the criteria established in this discussion, 452b is definitely, and 62f is probably, just too big.

Assuming the proposed radius of 1.41 Rea, an Earth-like composition would confer a mass of 3.5–4 Mea on Kepler-62f, following the models of Dressing et al. (2015) and Unterborn et al. (2015), respectively. Either value would be consistent with plate tectonics (at least according U15). But we have no constraints on this object’s true mass. It seems just as likely to be a less dense and thus less massive planet with a large volatile content: either a deep global ocean, failing the criterion for water, or a remnant H/He envelope, failing the criterion for atmosphere. According to the results of Lammer et al. (2014) and Dawson et al. (2015), it could be a relatively hospitable rocky planet of 3.5–4 Mea only if it formed by giant impacts after the system’s protoplanetary nebula dissipated. Because 62f is part of compact multiplanet system with four inner companions, however, a history of dynamical upset seems unlikely.

Assuming a radius 1.63 Rea, Kepler-452b has a similar range of potential compositions, although the all-rocky option is even more unlikely. At 5–6 Mea, an ice- and hydrogen-free planet would also need to be iron-free to achieve a radius so large. Such a composition isn’t plausible, and even if it were, it’s hard to see how plate tectonics might develop. A water world or an unlucky Earth-mass object with a residual H/He shroud seems more likely.

Given all these disappointing candidates, imaginary Earth-like planets (Figure 5) will have to do for a while longer.

Figure 5. Concocted Earth-like planets across an order of magnitude in mass, following the mass-radius curves of Unterborn et al. 2015. For rocky worlds, radius increases more slowly than mass, so the most massive object pictured here is not quite double the radius of the least massive. Note that the fanciful world of 3 Mea is very likely an outlier; most hydrogen-free planets are expected to be under 2 Mea and 1.2 Rea, like the other five examples. (In the diagram, RE = Earth radius, ME = Earth mass.) My gratitude to everyone who lives in these worlds.


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