arXiv subjects

Christoph Mordasini

Publications and source records attributed to Christoph Mordasini.

At least 109 records · Page 6Linked to original sources

arXiv2019

A water budget dichotomy of rocky protoplanets from $^{26}$Al-heating

In contrast to the water-poor inner solar system planets, stochasticity during planetary formation and order of magnitude deviations in exoplanet volatile contents suggest that rocky worlds engulfed in thick volatile ice layers are the dominant family of terrestrial analogues among the extrasolar planet population. However, the distribution of compositionally Earth-like planets remains insufficiently constrained, and it is not clear whether the solar system is a statistical outlier or can be explained by more general planetary formation processes. Here we employ numerical models of planet formation, evolution, and interior structure, to show that a planet's bulk water fraction and radius are anti-correlated with initial $^{26}$Al levels in the planetesimal-based accretion framework. The heat generated by this short-lived radionuclide rapidly dehydrates planetesimals prior to accretion onto larger protoplanets and yields a system-wide correlation of planet bulk abundances, which, for instance, can explain the lack of a clear orbital trend in the water budgets of the TRAPPIST-1 planets. Qualitatively, our models suggest two main scenarios of planetary systems' formation: high-$^{26}$Al systems, like our solar system, form small, water-depleted planets, whereas those devoid of $^{26}$Al predominantly form ocean worlds, where the mean planet radii between both scenarios deviate by up to about 10%.

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arXiv2019

Hints for a Turnover at the Snow Line in the Giant Planet Occurrence Rate

The orbital distribution of giant planets is crucial for understanding how terrestrial planets form and predicting yields of exoplanet surveys. Here, we derive giant planets occurrence rates as a function of orbital period by taking into account the detection efficiency of the Kepler and radial velocity (RV) surveys. The giant planet occurrence rates for Kepler and RV show the same rising trend with increasing distance from the star. We identify a break in the RV giant planet distribution between ~2-3 au -- close to the location of the snow line in the Solar System -- after which the occurrence rate decreases with distance from the star. Extrapolating a broken power-law distribution to larger semi-major axes, we find good agreement with the ~ 1% planet occurrence rates from direct imaging surveys. Assuming a symmetric power law, we also estimate that the occurrence of giant planets between 0.1-100 au is 26.6 +7.5 -5.4% for planets with masses 0.1-20MJ and decreases to 6.2 +1.5 -1.2% for planets more massive than Jupiter. This implies that only a fraction of the structures detected in disks around young stars can be attributed to giant planets. Various planet population synthesis models show good agreement with the observed distribution, and we show how a quantitative comparison between model and data can be used to constrain planet formation and migration mechanisms.

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arXiv2018

Microlensing Results Challenge the Core Accretion Runaway Growth Scenario for Gas Giants

We compare the planet-to-star mass-ratio distribution measured by gravitational microlensing to core accretion theory predictions from population synthesis models. The core accretion theory's runaway gas accretion process predicts a dearth of intermediate-mass giant planets that is not seen in the microlensing results. In particular, the models predict $\sim10\,\times$ fewer planets at mass ratios of $10^{-4} \leq q \leq 4 \times 10^{-4}$ than inferred from microlensing observations. This tension implies that gas giant formation may involve processes that have hitherto been overlooked by existing core accretion models or that the planet-forming environment varies considerably as a function of host-star mass. Variation from the usual assumptions for the protoplanetary disk viscosity and thickness could reduce this discrepancy, but such changes might conflict with microlensing results at larger or smaller mass ratios, or with other observations. The resolution of this discrepancy may have important implications for planetary habitability because it has been suggested that the runaway gas accretion process may have triggered the delivery of water to our inner solar system. So, an understanding of giant planet formation may help us to determine the occurrence rate of habitable planets.

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arXiv2018

The formation of Jupiter by hybrid pebble-planetesimal accretion

The standard model for giant planet formation is based on the accretion of solids by a growing planetary embryo, followed by rapid gas accretion once the planet exceeds a so-called critical mass. The dominant size of the accreted solids (cm-size particles named pebbles or km to hundred km-size bodies named planetesimals) is, however, unknown. Recently, high-precision measurements of isotopes in meteorites provided evidence for the existence of two reservoirs in the early Solar System. These reservoirs remained separated from ~1 until ~ 3 Myr after the beginning of the Solar System's formation. This separation is interpreted as resulting from Jupiter growing and becoming a barrier for material transport. In this framework, Jupiter reached ~20 Earth masses within ~1 Myr and slowly grew to ~50 Earth masses in the subsequent 2 Myr before reaching its present-day mass. The evidence that Jupiter slowed down its growth after reaching 20 Earth masses for at least 2 Myr is puzzling because a planet of this mass is expected to trigger fast runaway gas accretion. Here, we use theoretical models to describe the conditions allowing for such a slow accretion and show that Jupiter grew in three distinct phases. First, rapid pebble accretion brought the major part of Jupiter's core mass. Second, slow planetesimal accretion provided the energy required to hinder runaway gas accretion during 2 Myr. Third, runaway gas accretion proceeded. Both pebbles and planetesimals therefore have an important role in Jupiter's formation.

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arXiv2018

The Planet Formation Imager

The Planet Formation Imager (PFI, www.planetformationimager.org) is a next-generation infrared interferometer array with the primary goal of imaging the active phases of planet formation in nearby star forming regions. PFI will be sensitive to warm dust emission using mid-infrared capabilities made possible by precise fringe tracking in the near-infrared. An L/M band combiner will be especially sensitive to thermal emission from young exoplanets (and their disks) with a high spectral resolution mode to probe the kinematics of CO and H2O gas. In this paper, we give an overview of the main science goals of PFI, define a baseline PFI architecture that can achieve those goals, point at remaining technical challenges, and suggest activities today that will help make the Planet Formation Imager facility a reality.

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arXiv2018

Star-planet interactions. V. Dynamical and equilibrium tides in convective zones

When planets are formed from the protoplanetary disk and after the disk has dissipated, the evolution of their orbits is governed by tidal interactions, friction, and gravitational drag, and also by changes in the mass of the star and planet. These interactions may change the initial distribution of the distances between the planets and their host star by expanding the original orbit, by contracting it (which may cause an engulfment of the planet by the star), or by destroying the planet. We study the evolution of the orbit of a planet orbiting its host star under the effects of equilibrium tides, dynamical tides, drag (frictional and gravitational), and stellar mass loss. We used the Geneva stellar evolution code to compute the evolution of stars with initial masses of 1 and 1.5 solar mass with different rotation rates at solar metallicity. The star is evolved from the pre-main-sequence (PMS) up to the tip of the red giant branch. We used these models as input for computing the evolution of the planetary orbits. We explored the effects of changing the planet masses (of 1 Earth mass up to 20 Jupiter masses), the distance between the planet and the star (of 0.015 and more than 3 au), the mass, and the spin of the star. We present results when only the equilibrium tide was accounted for and when both equilibrium and dynamical tides were accounted for. The expression for the dynamical tide is a frequency-averaged dissipation of tidally excited inertial waves, obtained from a piecewise homogeneous two-layer stellar model. Gravity wave damping was neglected. Dynamical tides in convective zones have a significant effect on planetary orbits only during the PMS phase and only for fast-rotating stars. They have no significant effects during the PMS phase for initially slow-rotating stars and during the red giant branch phase, regardless of the initial rotation (abridged).

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arXiv2018

Planetary population synthesis

In stellar astrophysics, the technique of population synthesis has been successfully used for several decades. For planets, it is in contrast still a young method which only became important in recent years because of the rapid increase of the number of known extrasolar planets, and the associated growth of statistical observational constraints. With planetary population synthesis, the theory of planet formation and evolution can be put to the test against these constraints. In this review of planetary population synthesis, we first briefly list key observational constraints. Then, the work flow in the method and its two main components are presented, namely global end-to-end models that predict planetary system properties directly from protoplanetary disk properties and probability distributions for these initial conditions. An overview of various population synthesis models in the literature is given. The sub-models for the physical processes considered in global models are described: the evolution of the protoplanetary disk, the planets' accretion of solids and gas, orbital migration, and N-body interactions among concurrently growing protoplanets. Next, typical population synthesis results are illustrated in the form of new syntheses obtained with the latest generation of the Bern model. Planetary formation tracks, the distribution of planets in the mass-distance and radius-distance plane, the planetary mass function, and the distributions of planetary radii, semimajor axes, and luminosities are shown, linked to underlying physical processes, and compared with their observational counterparts. We finish by highlighting the most important predictions made by population synthesis models and discuss the lessons learned from these predictions - both those later observationally confirmed and those rejected.

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arXiv2018

SUPERSHARP - Segmented Unfolding Primary for Exoplanet Research via Spectroscopic High Angular Resolution Photography

We propose to search for biosignatures in the spectra of reflected light from about 100 Earth-sized planets that are already known to be orbiting in their habitable zones (HZ). For a sample of G and K type hosts, most of these planets will be between 25 and 50 milli-arcsec (mas) from their host star and 1 billion to 10 billion times fainter. To separate the planet's image from that of its host star at the wavelength (763nm) of the oxygen biosignature we need a telescope with an aperture of 16 metres. Furthermore, the intensity of the light from the host star at the position in the image of the exoplanet must be suppressed otherwise the exoplanet will be lost in the glare. This presents huge technical challenges. The Earth's atmosphere is turbulent which makes it impossible to achieve the required contrast from the ground at 763nm. The telescope therefore needs to be in space and to fit the telescope in the rocket fairing it must be a factor of 4 or more times smaller when folded than when operational. To obtain spectroscopy of the planet's biosignature at 763nm we need to use an integral field spectrometer (IFS) with a field of view (FOV) of 1000 x 1000 milli-arcsec (mas) and a spectral resolution of 100. This is a device that simultaneously takes many pictures of the exoplanet each at a slightly different wavelength which are then recorded as a data cube with two spatial dimensions and one wavelength dimension. In every data cube wavelength slice, the background light from the host star at the location of the planet image must be minimised. This is achieved via a coronagraph which blocks the light from the host star and active/adaptive optics techniques which continuously maintain very high accuracy optical alignment to make the images as sharp as possible. These are the technical challenges to be addressed in a design study.

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arXiv2017

Compositional imprints in density-distance-time: a rocky composition for close-in low-mass exoplanets from the location of the valley of evaporation

We use an end-to-end model of planet formation, thermodynamic evolution, and atmospheric escape to investigate how the statistical imprints of evaporation depend on the bulk composition of planetary cores (rocky vs. icy). We find that the population-wide imprints like the location of the "evaporation valley" in the distance-radius plane and the corresponding bimodal radius distribution clearly differ depending on the bulk composition of the cores. Comparison with the observed position of the valley (Fulton et al. 2017) suggests that close-in low-mass Kepler planets have a predominately Earth-like rocky composition. Combined with the excess of period ratios outside of MMR, this suggests that low-mass Kepler planets formed inside of the water iceline, but still undergoing orbital migration. The core radius becomes visible for planets losing all primordial H/He. For planets in this "triangle of evaporation" in the distance-radius plane, the degeneracy in compositions is reduced. In the observed diagram, we identify a trend to more volatile-rich compositions with increasing radius (R/R_Earth<1.6 rocky; 1.6-3.0 ices and/or H/He; >3: H/He). The mass-density diagram contains important information about formation and evolution. Its characteristic broken V-shape reveals the transitions from solid planets to low-mass core-dominated planets with H/He and finally to gas-dominated giants. Evaporation causes density and orbital distance to be anti-correlated for low-mass planets, in contrast to giants, where closer-in planets are less dense, likely due to inflation. The temporal evolution of the statistical properties reported here will be of interest for the PLATO 2.0 mission which will observe the temporal dimension.

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arXiv2017

Retrieval Analysis of the Emission Spectrum of WASP-12b: Sensitivity of Outcomes to Prior Assumptions and Implications for Formation History

We analyze the emission spectrum of the hot Jupiter WASP-12b using our HELIOS-R retrieval code and HELIOS-K opacity calculator. When interpreting Hubble and Spitzer data, the retrieval outcomes are found to be prior-dominated. When the prior distributions of the molecular abundances are assumed to be log-uniform, the volume mixing ratio of HCN is found to be implausibly high. A VULCAN chemical kinetics model of WASP-12b suggests that chemical equilibrium is a reasonable assumption even when atmospheric mixing is implausibly rigorous. Guided by (exo)planet formation theory, we set Gaussian priors on the elemental abundances of carbon, oxygen and nitrogen with the Gaussian peaks being centered on the measured C/H, O/H and N/H values of the star. By enforcing chemical equilibrium, we find substellar O/H and stellar to slightly superstellar C/H for the dayside atmosphere of WASP-12b. The superstellar carbon-to-oxygen ratio is just above unity, regardless of whether clouds are included in the retrieval analysis, consistent with Madhusudhan et al. (2011). Furthermore, whether a temperature inversion exists in the atmosphere depends on one's assumption for the Gaussian width of the priors. Our retrieved posterior distributions are consistent with the formation of WASP-12b in a solar-composition protoplanetary disk, beyond the water iceline, via gravitational instability or pebble accretion (without core erosion) and migration inwards to its present orbital location via a disk-free mechanism, and are inconsistent with both in-situ formation and core accretion with disk migration, as predicted by Madhusudhan et al. (2017). We predict that the interpretation of James Webb Space Telescope WASP-12b data will not be prior-dominated.

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arXiv2017

High signal-to-noise spectral characterization of the planetary-mass object HD 106906 b

We spectroscopically characterize the atmosphere of HD 106906b, a young low-mass companion near the deuterium burning limit. The wide separation from its host star of 7.1" makes it an ideal candidate for high S/N and high-resolution spectroscopy. We aim to derive new constraints on the spectral type, effective temperature, and luminosity of HD106906b and also to provide a high S/N template spectrum for future characterization of extrasolar planets. We obtained 1.1-2.5 $μ$m integral field spectroscopy with the VLT/SINFONI instrument with a spectral resolution of R~2000-4000. New estimates of the parameters of HD 106906b are derived by analyzing spectral features, comparing the extracted spectra to spectral catalogs of other low-mass objects, and fitting with theoretical isochrones. We identify several spectral absorption lines that are consistent with a low mass for HD 106906b. We derive a new spectral type of L1.5$\pm$1.0, one subclass earlier than previous estimates. Through comparison with other young low-mass objects, this translates to a luminosity of log($L/L_\odot$)=$-3.65\pm0.08$ and an effective temperature of Teff=$1820\pm240$ K. Our new mass estimates range between $M=11.9^{+1.7}_{-0.8} M_{\rm Jup}$ (hot start) and $M=14.0^{+0.2}_{-0.5} M_{\rm Jup}$ (cold start). These limits take into account a possibly finite formation time, i.e., HD 106906b is allowed to be 0--3 Myr younger than its host star. We exclude accretion onto HD 106906b at rates $\dot{M}>4.8\times10^{-10} M_{\rm Jup}$yr$^{-1}$ based on the fact that we observe no hydrogen (Paschen-$β$, Brackett-$γ$) emission. This is indicative of little or no circumplanetary gas. With our new observations, HD 106906b is the planetary-mass object with one of the highest S/N spectra yet. We make the spectrum available for future comparison with data from existing and next-generation (e.g., ELT and JWST) spectrographs.

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arXiv2017

HELIOS-Retrieval: An Open-source, Nested Sampling Atmospheric Retrieval Code, Application to the HR 8799 Exoplanets and Inferred Constraints for Planet Formation

We present an open-source retrieval code named HELIOS-Retrieval (hereafter HELIOS-R), designed to obtain chemical abundances and temperature-pressure profiles from inverting the measured spectra of exoplanetary atmospheres. In the current implementation, we use an exact solution of the radiative transfer equation, in the pure absorption limit, in our forward model, which allows us to analytically integrate over all of the outgoing rays (instead of performing Gaussian quadrature). Two chemistry models are considered: unconstrained chemistry (where the mixing ratios are treated as free parameters) and equilibrium chemistry (enforced via analytical formulae, where only the elemental abundances are free parameters). The nested sampling algorithm allows us to formally implement Occam's Razor based on a comparison of the Bayesian evidence between models. We perform a retrieval analysis on the measured spectra of the HR 8799b, c, d and e directly imaged exoplanets. Chemical equilibrium is disfavored by the Bayesian evidence for HR 8799b, c and d. We find supersolar C/O, C/H and O/H values for the outer HR 8799b and c exoplanets, while the inner HR 8799d and e exoplanets have substellar C/O, substellar C/H and superstellar O/H values. If these retrieved properties are representative of the bulk compositions of the exoplanets, then they are inconsistent with formation via gravitational instability (without late-time accretion) and consistent with a core accretion scenario in which late-time accretion of ices occurred differently for the inner and outer exoplanets. For HR 8799e, we find that spectroscopy in the K band is crucial for constraining C/O and C/H. HELIOS-R is publicly available as part of the Exoclimes Simulation Platform (ESP; www.exoclime.org).

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arXiv2017

Two Empirical Regimes of the Planetary Mass-Radius Relation

Today, with the large number of detected exoplanets and improved measurements, we can reach the next step of planetary characterization. Classifying different populations of planets is not only important for our understanding of the demographics of various planetary types in the galaxy, but also for our understanding of planet formation. We explore the nature of two regimes in the planetary mass-radius (M-R) relation. We suggest that the transition between the two regimes of "small" and "large" planets, occurs at a mass of 124 \pm 7, M_Earth and a radius of 12.1 \pm 0.5, R_Earth. Furthermore, the M-R relation is R \propto M^{0.55\pm 0.02} and R \propto M^{0.01\pm0.02} for small and large planets, respectively. We suggest that the location of the breakpoint is linked to the onset of electron degeneracy in hydrogen, and therefore, to the planetary bulk composition. Specifically, it is the characteristic minimal mass of a planet which consists of mostly hydrogen and helium, and therefore its M-R relation is determined by the equation of state of these materials. We compare the M-R relation from observational data with the one derived by population synthesis calculations and show that there is a good qualitative agreement between the two samples.

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arXiv2017

The Planetary Accretion Shock: I. Framework for Radiation-hydrodynamical Simulations and First Results

The key aspect determining the post-formation luminosity of gas giants has long been considered to be the energetics of the accretion shock at the planetary surface. We use 1D radiation-hydrodynamical simulations to study the radiative loss efficiency and to obtain post-shock temperatures and pressures and thus entropies. The efficiency is defined as the fraction of the total incoming energy flux which escapes the system (roughly the Hill sphere), taking into account the energy recycling which occurs ahead of the shock in a radiative precursor. We focus here on a constant equation of state to isolate the shock physics but use constant and tabulated opacities. While robust quantitative results will require a self-consistent treatment including hydrogen dissocation and ionization, the results show the correct qualitative behavior and can be understood semi-analytically. The shock is found to be isothermal and supercritical for a range of conditions relevant to core accretion (CA), with Mach numbers greater than ca. 3. Across the shock, the entropy decreases significantly, by a few entropy units (k_B/baryon). While nearly 100 percent of the incoming kinetic energy is converted to radiation locally, the efficiencies are found to be as low as roughly 40 percent, implying that a meaningful fraction of the total accretion energy is brought into the planet. For realistic parameter combinations in the CA scenario, a non-zero fraction of the luminosity always escapes the system. This luminosity could explain, at least in part, recent observations in the LkCa 15 and HD 100546 systems.

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arXiv2016

The imprint of exoplanet formation history on observable present-day spectra of hot Jupiters

The composition of a planet's atmosphere is determined by its formation, evolution, and present-day insolation. A planet's spectrum therefore may hold clues on its origins. We present a "chain" of models, linking the formation of a planet to its observable present-day spectrum. The chain links include (1) the planet's formation and migration, (2) its long-term thermodynamic evolution, (3) a variety of disk chemistry models, (4) a non-gray atmospheric model, and (5) a radiometric model to obtain simulated spectroscopic observations with JWST and ARIEL. In our standard chemistry model the inner disk is depleted in refractory carbon as in the Solar System and in white dwarfs polluted by extrasolar planetesimals. Our main findings are: (1) Envelope enrichment by planetesimal impacts during formation dominates the final planetary atmospheric composition of hot Jupiters. We investigate two, under this finding, prototypical formation pathways: a formation inside or outside the water iceline, called "dry" and "wet" planets, respectively. (2) Both the "dry" and "wet" planets are oxygen-rich (C/O<1) due to the oxygen-rich nature of the solid building blocks. The "dry" planet's C/O ratio is <0.2 for standard carbon depletion, while the "wet" planet has typical C/O values between 0.1 and 0.5 depending mainly on the clathrate formation efficiency. Only non-standard disk chemistries without carbon depletion lead to carbon-rich C/O ratios >1 for the "dry" planet. (3) While we consistently find C/O ratios <1, they still vary significantly. To link a formation history to a specific C/O, a better understanding of the disk chemistry is thus needed.

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arXiv2016

Formation, Orbital and Internal Evolutions of Young Planetary Systems

The growing body of observational data on extrasolar planets and protoplanetary disks has stimulated intense research on planet formation and evolution in the past few years. The extremely diverse, sometimes unexpected physical and orbital characteristics of exoplanets lead to frequent updates on the mainstream scenarios for planet formation and evolution, but also to the exploration of alternative avenues. The aim of this review is to bring together classical pictures and new ideas on the formation, orbital and internal evolutions of planets, highlighting the key role of the protoplanetary disk in the various parts of the theory. We begin by briefly reviewing the conventional mechanism of core accretion by the growth of planetesimals, and discuss a relatively recent model of core growth through the accretion of pebbles. We review the basic physics of planet-disk interactions, recent progress in this area, and discuss their role in observed planetary systems. We address the most important effects of planets internal evolution, like cooling and contraction, the mass-luminosity relation, and the bulk composition expressed in the mass-radius and mass-mean density relations.

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arXiv2016

Evolution and Magnitudes of Candidate Planet Nine

The recently renewed interest in a possible additional major body in the outer solar system prompted us to study the thermodynamic evolution of such an object. We assumed that it is a smaller version of Uranus and Neptune. We modeled the temporal evolution of the radius, temperature, intrinsic luminosity, and the blackbody spectrum of distant ice giant planets. The aim is also to provide estimates of the magnitudes in different bands to assess whether the object might be detectable. Simulations of the cooling and contraction were conducted for ice giants with masses of 5, 10, 20, and 50 Mearth that are located at 280, 700, and 1120 AU from the Sun. The core composition, the fraction of H/He, the efficiency of energy transport, and the initial luminosity were varied. The atmospheric opacity was set to 1, 50, and 100 times solar metallicity. We find for a nominal 10 Mearth planet at 700 AU at the current age of the solar system an effective temperature of 47 K, much higher than the equilibrium temperature of about 10 K, a radius of 3.7 Rearth, and an intrinsic luminosity of 0.006 Ljupiter. It has estimated apparent magnitudes of Johnson V, R, I, L, N, Q of 21.7, 21.4, 21.0, 20.1, 19.9, and 10.7, and WISE W1-W4 magnitudes of 20.1, 20.1, 18.6, and 10.2. The Q and W4 band and other observations longward of about 13 microns pick up the intrinsic flux. If candidate Planet 9 has a significant H/He layer and an efficient energy transport in the interior, then its luminosity is dominated by the intrinsic contribution, making it a self-luminous planet. At a likely position on its orbit near aphelion, we estimate for a mass of 5, 10, 20, and 50 Mearth a V magnitude from the reflected light of 24.3, 23.7, 23.3, and 22.6 and a Q magnitude from the intrinsic radiation of 14.6, 11.7, 9.2, and 5.8. The latter would probably have been detected by past surveys.

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arXiv2015

Model atmospheres of irradiated exoplanets: The influence of stellar parameters, metallicity, and the C/O ratio

Many parameters constraining the spectral appearance of exoplanets are still poorly understood. We therefore study the properties of irradiated exoplanet atmospheres over a wide parameter range including metallicity, C/O ratio and host spectral type. We calculate a grid of 1-d radiative-convective atmospheres and emission spectra. We perform the calculations with our new Pressure-Temperature Iterator and Spectral Emission Calculator for Planetary Atmospheres (PETIT) code, assuming chemical equilibrium. The atmospheric structures and spectra are made available online. We find that atmospheres of planets with C/O ratios $\sim$ 1 and $T_{\rm eff}$ $\gtrsim$ 1500 K can exhibit inversions due to heating by the alkalis because the main coolants CH$_4$, H$_2$O and HCN are depleted. Therefore, temperature inversions possibly occur without the presence of additional absorbers like TiO and VO. At low temperatures we find that the pressure level of the photosphere strongly influences whether the atmospheric opacity is dominated by either water (for low C/O) or methane (for high C/O), or both (regardless of the C/O). For hot, carbon-rich objects this pressure level governs whether the atmosphere is dominated by methane or HCN. Further we find that host stars of late spectral type lead to planetary atmospheres which have shallower, more isothermal temperature profiles. In agreement with prior work we find that for planets with $T_{\rm eff}$ $<$ 1750 K the transition between water or methane dominated spectra occurs at C/O $\sim$ 0.7, instead of $\sim$ 1, because condensation preferentially removes oxygen.

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