arXiv subjects

Hilke E. Schlichting

Publications and source records attributed to Hilke E. Schlichting.

At least 55 records · Page 3Linked to original sources

arXiv2021

Photoevaporation vs. core-powered mass-loss: model comparison with the 3D radius gap

The EUV/X-ray photoevaporation and core-powered mass-loss models are both capable of reproducing the bimodality in the sizes of small, close-in exoplanets observed by the \textit{Kepler} space mission, often referred to as the "radius gap". However, it is unclear which of these two mechanisms dominates the atmospheric mass-loss which is likely sculpting the radius gap. In this work, we propose a new method of differentiating between the two models, which relies on analysing the radius gap in 3D parameter space. Using models for both mechanisms, and by performing synthetic transit surveys we predict the size and characteristics of a survey capable of discriminating between the two models. We find that a survey of $\gtrsim 5000$ planets, with a wide range in stellar mass and measurement uncertainties at a $\lesssim 5\%$ level is sufficient. Our methodology is robust against moderate false positive contamination of $\lesssim 10\%$. We perform our analysis on two surveys (which do not satisfy our requirements): the \textit{California Kepler Survey} and the \textit{Gaia-Kepler Survey} and find, unsurprisingly, that both data-sets are consistent with either model. We propose a hypothesis test to be performed on future surveys which can robustly ascertain which of the two mechanisms formed the radius gap, provided one dominates over the other.

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arXiv2021

Caught in the Act: Core-powered Mass-loss Predictions for Observing Atmospheric Escape

Past studies have demonstrated that atmospheric escape by the core-powered mass-loss mechanism can explain a multitude of observations associated with the radius valley that separates the super-Earth and sub-Neptune planet populations. Complementing such studies, in this work, we present a shortlist of planets that could be losing their atmospheres today if their evolution is indeed primarily dictated by core-powered mass-loss. We use Bayesian inference analysis on our planet evolution and mass-loss model to estimate the posteriors of the parameters that encapsulate the current state of a given planet, given their published masses, radii and host star properties. Our models predict that the following planets could be losing their atmospheres today at a rate $\gtrsim 10^7$ g/s at 50\% confidence level: pi Men c, Kepler-60 d, Kepler-60 b, HD 86226 c, EPIC 249893012 b, Kepler-107 c, HD 219134 b, Kepler-80 e, Kepler-138 d and GJ 9827 d. As a by-product of our Bayesian inference analysis, we were also able to identify planets that most-likely harbor either secondary atmospheres abundant with high mean-molecular weight species, low-density interiors abundant with ices, or both. The planets belonging to this second category are WASP-47 e, Kepler-78 b, Kepler-10 b, CoRoT-7 b, HD 80653 b, 55 Cnc e and Kepler-36 b. While the aforementioned lists are by no means exhaustive, we believe that candidates presented here can serve as useful input for target selection for future surveys and for testing the importance of core-powered mass-loss in individual planetary systems.

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arXiv2021

To cool is to keep: Residual H/He atmospheres of super-Earths and sub-Neptunes

Super-Earths and sub-Neptunes are commonly thought to have accreted hydrogen/helium envelopes, consisting of a few to ten percent of their total mass, from the primordial gas disk. Subsequently, hydrodynamic escape driven by core-powered mass-loss and/or photo-evaporation likely stripped much of these primordial envelopes from the lower-mass and closer-in planets to form the super-Earth population. In this work we show that after undergoing core-powered mass-loss, some super-Earths can retain small residual H/He envelopes. This retention is possible because, for significantly depleted atmospheres, the density at the radiative-convective boundary drops sufficiently such that thhe cooling time-scale becomes less than the mass-loss time-scale. The residual envelope is therefore able to contract, terminating further mass loss. Using analytic calculations and numerical simulations, we show that the mass of primordial H/He envelope retained as a fraction of the planet's total mass, $f_{ret}$, increases with increasing planet mass, $M_{c}$, and decreases with increasing equilibrium temperature, $T_{eq}$, scaling as $f_{ret} \propto M_{c}^{3/2} T_{eq}^{-1/2} \exp{[M_{c}^{3/4} T_{eq}^{-1}]}$. $f_{ret}$ varies from $< 10^{-8}$ to about $10^{-3}$ for typical super-Earth parameters. To first order, the exact amount of left-over H/He depends on the initial envelope mass, the planet mass, its equilibrium temperature, and the envelope's opacity. These residual hydrogen envelopes reduce the atmosphere's mean molecular weight compared to a purely secondary atmosphere, a signature observable by current and future facilities. These remnant atmospheres may, however, in many cases be vulnerable to long-term erosion by photo-evaporation. Any residual hydrogen envelope likely plays an important role in the long-term physical evolution of super-Earths, including their geology and geochemistry.

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arXiv2020

Where are the Extrasolar Mercuries?

We utilize observations of 16 white dwarf stars to calculate and analyze the oxidation states of the parent bodies accreting onto the stars. Oxygen fugacity, a measure of overall oxidation state for rocks, is as important as pressure and temperature in determining the structure of a planet. We find that most of the extrasolar rocky bodies formed under oxidizing conditions, but approximately 1/4 of the polluted white dwarfs have compositions consistent with more reduced parent bodies. The difficulty in constraining the oxidation states of relatively reduced bodies is discussed and a model for the time-dependent evolution of the apparent oxygen fugacity for a hypothetical reduced body engulfed by a WD is investigated. Differences in diffusive fluxes of various elements through the WD envelope yield spurious inferred bulk elemental compositions and oxidation states of the accreting parent bodies under certain conditions. The worst case for biasing against detection of reduced bodies occurs for high effective temperatures. For moderate and low effective temperatures, evidence for relatively reduced parent bodies is preserved under most circumstances for at least several characteristic lifetimes of the debris disk.

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arXiv2020

Losing Oceans: The Effects of Composition on the Thermal Component of Impact-driven Atmospheric Loss

The formation of the solar system's terrestrial planets concluded with a period of giant impacts. Previous works examining the volatile loss caused by the impact shock in the moon-forming impact find atmospheric losses of at most 20-30 per cent and essentially no loss of oceans. However, giant impacts also result in thermal heating, which can lead to significant atmospheric escape via a Parker-type wind. Here we show that H2O and other high-mean molecular weight outgassed species can be efficiently lost through this thermal wind if present in a hydrogen-dominated atmosphere, substantially altering the final volatile inventory of terrestrial planets. Specifically, we demonstrate that a giant impact of a Mars-sized embryo with a proto-Earth can remove several Earth oceans' worth of H2O, and other heavier volatile species, together with a primordial hydrogen-dominated atmosphere. These results may offer an explanation for the observed depletion in Earth's light noble gas budget and for its depleted xenon inventory, which suggest that Earth underwent significant atmospheric loss by the end of its accretion. Because planetary embryos are massive enough to accrete primordial hydrogen envelopes and because giant impacts are stochastic and occur concurrently with other early atmospheric evolutionary processes, our results suggest a wide diversity in terrestrial planet volatile budgets.

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arXiv2020

Signatures of the Core-Powered Mass-Loss Mechanism in the Exoplanet Population: Dependence on Stellar Properties and Observational Predictions

Recent studies have shown that atmospheric mass-loss powered by the cooling luminosity of a planet's core can explain the observed radius valley separating super-Earths and sub-Neptunes, even without photoevaporation. In this work, we investigate the dependence of this core-powered mass-loss mechanism on stellar mass ($M_\ast$), metallicity ($Z_\ast$) and age ($τ_\ast$). Without making any changes to the underlying planet population, we find that the core-powered mass-loss model yields a shift in the radius valley to larger planet sizes around more massive stars with a slope given by $\text{d log}R_p/\text{d log}M_\ast \simeq 0.35$, in agreement with observations. To first order, this slope is driven by the dependence of core-powered mass-loss on the bolometric luminosity of the host star and is given by $\text{d log}R_p/\text{d log}M_\ast \simeq (3α-2)/36 \simeq 0.33$, where $(L_\ast/L_\odot) = (M_\ast/M_\odot)^α$ is the stellar mass-luminosity relation and $α\simeq 4.6$ for the CKS dataset. We therefore find, in contrast to photoevaporation models, no evidence for a linear correlation between planet and stellar mass, but can't rule it out either. In addition, we show that the location of the radius valley is, to first order, independent of stellar age and metallicity. Since core-powered mass-loss proceeds over Gyr timescales, the abundance of super-Earths relative to sub-Neptunes increases with age but decreases with stellar metallicity. Finally, due the dependence of the envelope's cooling timescale on metallicity, we find that the radii of sub-Neptunes increase with metallicity and decrease with age with slopes given by $\text{d log}R_p/\text{d log}Z_\ast \simeq 0.1$ and $\text{d log}R_p/\text{d log}τ_\ast \simeq -0.1$, respectively. We conclude with a series of observational tests that can differentiate between core-powered mass-loss and photoevaporation models.

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arXiv2019

Using Ice and Dust Lines to Constrain the Surface Densities of Protoplanetary Disks

We present a novel method for determining the surface density of protoplanetary disks through consideration of disk 'dust lines' which indicate the observed disk radial scale at different observational wavelengths. This method relies on the assumption that the processes of particle growth and drift control the radial scale of the disk at late stages of disk evolution such that the lifetime of the disk is equal to both the drift timescale and growth timescale of the maximum particle size at a given dust line. We provide an initial proof of concept of our model through an application to the disk TW Hya and are able to estimate the disk dust-to-gas ratio, CO abundance, and accretion rate in addition to the total disk surface density. We find that our derived surface density profile and dust-to-gas ratio are consistent with the lower limits found through measurements of HD gas. The CO ice line also depends on surface density through grain adsorption rates and drift and we find that our theoretical CO ice line estimates have clear observational analogues. We further apply our model to a large parameter space of theoretical disks and find three observational diagnostics that may be used to test its validity. First we predict that the dust lines of disks other than TW Hya will be consistent with the normalized CO surface density profile shape for those disks. Second, surface density profiles that we derive from disk ice lines should match those derived from disk dust lines. Finally, we predict that disk dust and ice lines will scale oppositely, as a function of surface density, across a large sample of disks.

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arXiv2019

New Constraints From Dust Lines On The Surface Densities Of Protoplanetary Disks

We present new determinations of disk surface density, independent of an assumed dust opacity, for a sample of 7 bright, diverse protoplanetary disks using measurements of disk dust lines. We develop a robust method for determining the location of dust lines by modeling disk interferometric visibilities at multiple wavelengths. The disks in our sample have newly derived masses that are 9-27% of their host stellar mass, substantially larger than the minimum mass solar nebula. All are stable to gravitational collapse except for one which approaches the limit of Toomre-Q stability. Our mass estimates are 2-15 times larger than estimates from integrated optically thin dust emission. We derive depleted dust-to-gas ratios with typical values of ~$10^{-3}$ in the outer disk. Using coagulation models we derive dust surface density profiles that are consistent with millimeter dust observations. In these models, the disks formed with an initial dust mass that is a factor of ~10 greater than is presently observed. Of the three disks in our sample with resolved CO line emission, the masses of HD 163296, AS 209, and TW Hya are roughly 3, 115, and 40 times more massive than estimates from CO respectively. This range indicates that CO depletion is not uniform across different disks and that dust is a more robust tracer of total disk mass. Our method of determining surface density using dust lines is robust even if particles form as aggregates and is useful even in the presence of dust substructure caused by pressure traps. The low Toomre-Q values observed in this sample indicate that at least some disks do not accrete efficiently.

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arXiv2019

Sculpting the Valley in the Radius Distribution of Small Exoplanets as a by-product of Planet Formation: The Core-Powered Mass-Loss Mechanism

Recent observations revealed a bimodal radius distribution of small, short-period exoplanets with a paucity in their occurrence, a radius `valley', around $1.5-2.0$ $R_\oplus$. In this work, we investigate the effect of a planet's own cooling luminosity on its thermal evolution and atmospheric mass-loss (core-powered mass-loss) and determine its observational consequences for the radius distribution of small, close-in exoplanets. Using simple analytical descriptions and numerical simulations, we demonstrate that planetary evolution based on the core-powered mass-loss mechanism alone (i.e., without any photoevaporation) can produce the observed valley in the radius distribution. Our results match the valley's location, shape and slope in planet radius-orbital period parameter space, and the relative magnitudes of the planet occurrence rate above and below the valley. We find that the slope of the valley is, to first order, dictated by the atmospheric mass-loss timescale at the Bondi radius and given by $\text{d log} R_p/ \text{d log} P \simeq 1/(3(1-β))$ which evaluates to $ -0.11$ for $ β\simeq 4$, where $M_c/M_\oplus = (R_c/R_{\oplus})^β (ρ_{c*}/ρ_{\oplus})^{β/3}$ is the mass-radius relation of the core. This choice for $β$ yields good agreement with observations and attests to the significance of internal compression for massive planetary cores. We further find that the location of the valley scales as $ρ_{c*}^{-4/9}$ and that the observed planet population must have predominantly rocky cores with typical water-ice fractions of less than $\sim 20\%$. Furthermore, we show that the relative magnitude of the planet occurrence rate above and below the valley is sensitive to the details of the planet-mass distribution but that the location of the valley is not.

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arXiv2019

Atmospheric Mass Loss from High Velocity Giant Impacts

Using moving mesh hydrodynamic simulations, we determine the shock propagation and resulting ground velocities for a planet hit by a high velocity impactor. We use our results to determine the atmospheric mass loss caused by the resulting ground motion due to the impact shock wave. We find that there are two distinct shock propagation regimes: In the limit in which the impactor is significantly smaller than the target ($R_i<< R_t$), the solutions are self-similar and the shock velocity at a fixed point on the target scale as $m_i^{2/3}$, where $m_i$ is the mass of the impactor. In addition, the ground velocities follow a universal profile given by $v_g/v_i=(14.2x^2-25.3x+11.3)/(x^2-2.5x+1.9) +2\ln{R_i/R_t}$, where $x=\sin\left(θ/2\right)$, $θ$ is the latitude on the target measured from the impact site, and $v_g$ and $v_i$ are the ground velocity and impact velocity, respectively. In contrast, in the limit in which the impactor is comparable to the size of the target ($R_i \sim R_t$), we find that shock velocities decline with the mass of the impactor significantly more weakly than $m_i^{2/3}$. We use the resulting surface velocity profiles to calculate the atmospheric mass loss for a large range of impactor masses and impact velocities and apply them to the Kepler-36 system and the Moon forming impact. Finally, we present and generalise our results in terms of the $v_g/v_i$ and the impactor to target size ratio ($R_i/R_t$) such that they can easily be applied to other collision scenarios.

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arXiv2018

Atmospheric mass loss due to giant impacts: the importance of the thermal component for hydrogen-helium envelopes

Systems of close-in super-Earths display striking diversity in planetary bulk density and composition. Giant impacts are expected to play a role in the formation of many of these worlds. Previous works, focused on the mechanical shock caused by a giant impact, have shown that these impacts can eject large fractions of the planetary envelope, offering a partial explanation for the observed spread in exoplanet compositions. Here, we examine the thermal consequences of giant impacts, and show that the atmospheric loss caused by these effects can significantly exceed that caused by mechanical shocks for hydrogen-helium (H/He) envelopes. When a giant impact occurs, part of the impact energy is converted into thermal energy, heating the rocky core and the envelope. We find that the ensuing thermal expansion of the envelope can lead to a period of sustained, rapid mass loss through a Parker wind, resulting in the partial or complete erosion of the H/He envelope. The fraction of the envelope lost depends on the planet's orbital distance from its host star and its initial thermal state, and hence age. Planets closer to their host stars are more susceptible to thermal atmospheric loss triggered by impacts than ones on wider orbits. Similarly, younger planets, with rocky cores which are still hot and molten from formation, suffer greater atmospheric loss. This is especially interesting because giant impacts are expected to occur $10{-}100~\mathrm{Myr}$ after formation. For planets where the thermal energy of the core is much greater than the envelope energy, the impactor mass required for significant atmospheric removal is $M_\mathrm{imp} / M_p \sim μ/ μ_c \sim 0.1$, approximately the ratio of the heat capacities of the envelope and core. When the envelope energy dominates the total energy budget, complete loss can occur when the impactor mass is comparable to the envelope mass.

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arXiv2018

Core-powered mass loss and the radius distribution of small exoplanets

Recent observations identify a valley in the radius distribution of small exoplanets, with planets in the range $1.5-2.0\,{\rm R}_{\oplus}$ significantly less common than somewhat smaller or larger planets. This valley may suggest a bimodal population of rocky planets that are either engulfed by massive gas envelopes that significantly enlarge their radius, or do not have detectable atmospheres at all. One explanation of such a bimodal distribution is atmospheric erosion by high-energy stellar photons. We investigate an alternative mechanism: the luminosity of the cooling rocky core, which can completely erode light envelopes while preserving heavy ones, produces a deficit of intermediate sized planets. We evolve planetary populations that are derived from observations using a simple analytical prescription, accounting self-consistently for envelope accretion, cooling and mass loss, and demonstrate that core-powered mass loss naturally reproduces the observed radius distribution, regardless of the high-energy incident flux. Observations of planets around different stellar types may distinguish between photoevaporation, which is powered by the high-energy tail of the stellar radiation, and core-powered mass loss, which depends on the bolometric flux through the planet's equilibrium temperature that sets both its cooling and mass-loss rates.

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arXiv2017

Avoiding resonance capture in multi-planet extrasolar systems

A commonly noted feature of the population of multi-planet extrasolar systems is the rarity of planet pairs in low-order mean-motion resonances. We revisit the physics of resonance capture via convergent disk-driven migration. We point out that for planet spacings typical of stable configurations for Kepler systems, the planets can routinely maintain a small but nonzero eccentricity due to gravitational perturbations from their neighbors. Together with the upper limit on the migration rate needed for capture, the finite eccentricity can make resonance capture difficult or impossible in Sun-like systems for planets smaller than ~Neptune-sized. This mass limit on efficient capture is broadly consistent with observed exoplanet pairs that have mass determinations: of pairs with the heavier planet exterior to the lighter planet -- which would have been undergoing convergent migration in their disks -- those in or nearly in resonance are much more likely to have total mass greater than two Neptune masses than to have smaller masses. The agreement suggests that the observed paucity of resonant pairs around sun-like stars may simply arise from a small resonance capture probability for lower-mass planets. Planet pairs that thereby avoid resonance capture are much less likely to collide in an eventual close approach than to simply migrate past one another to become a divergently migrating pair with the lighter planet exterior. For systems around M stars we expect resonant pairs to be much more common, since there the minimum mass threshhold for efficient capture is about an Earth mass.

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arXiv2016

The Formation of Super-Earths and Mini-Neptunes with Giant Impacts

The majority of discovered exoplanetary systems harbour a new class of planets, bodies typically several times more massive than Earth but orbiting their host stars well inside the orbit of Mercury. The origin of these close-in super-Earths and mini-Neptunes is a major unanswered question in planet formation. Unlike Earth, whose atmosphere contains $<10^{-6}$ its total mass, a large fraction of close-in planets have significant gaseous envelopes, containing $1 -10\%$ or more of their total mass. It has been proposed that these close-in planets formed in situ either by delivery of $50-100M_{\oplus}$ of rocky material to the inner disc, or in a disc enhanced relative to the MMSN. In both cases, final assembly of the planets occurs by giant impacts (GIs). Here we test the viability of these scenarios. We show that atmospheres accreted by isolation masses are small ($10^{-3}-10^{-2}$ the core mass) and that atmospheric mass-loss during GIs is significant, with typical post-GI atmospheres that are $8 \times 10^{-4}$ the core mass. Such values are consistent with terrestrial planet atmospheres but more than an order of magnitude below atmospheric masses of $1-10\%$ inferred for many close-in exoplanets. In the most optimistic scenario with no core luminosity, post-GI envelope accretion from a depleted gas disc yields atmospheric masses that are several per cent the core mass. If the gravitational potential energy due to the last mass doubling of the planet by GIs is released over the disc dissipation time-scale as core luminosity, then envelope masses are reduced by about an order of magnitude. Finally we show that radial drift time-scales due to gas drag for many isolation masses are shorter than typical disc lifetimes. Given these challenges, we conclude that most observed close-in planets with envelopes larger than several per cent likely formed at larger separations from their host stars.

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arXiv2016

Super-Earth Atmospheres: Self-Consistent Gas Accretion and Retention

Some recently discovered short-period Earth to Neptune sized exoplanets (super Earths) have low observed mean densities which can only be explained by voluminous gaseous atmospheres. Here, we study the conditions allowing the accretion and retention of such atmospheres. We self-consistently couple the nebular gas accretion onto rocky cores and the subsequent evolution of gas envelopes following the dispersal of the protoplanetary disk. Specifically, we address mass-loss due to both photo-evaporation and cooling of the planet. We find that planets shed their outer layers (dozens of percents in mass) following the disk's dispersal (even without photo-evaporation), and their atmospheres shrink in a few Myr to a thickness comparable to the radius of the underlying rocky core. At this stage, atmospheres containing less particles than the core (equivalently, lighter than a few % of the planet's mass) can be blown away by heat coming from the cooling core, while heavier atmospheres cool and contract on a timescale of Gyr at most. By relating the mass-loss timescale to the accretion time, we analytically identify a Goldilocks region in the mass-temperature plane in which low-density super Earths can be found: planets have to be massive and cold enough to accrete and retain their atmospheres, while not too massive or cold, such that they do not enter runaway accretion and become gas giants (Jupiters). We compare our results to the observed super-Earth population and find that low-density planets are indeed concentrated in the theoretically allowed region. Our analytical and intuitive model can be used to investigate possible super-Earth formation scenarios.

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arXiv2016

CHIMERA: a wide-field, multi-color, high-speed photometer at the prime focus of the Hale telescope

The Caltech HIgh-speed Multi-color camERA (CHIMERA) is a new instrument that has been developed for use at the prime focus of the Hale 200-inch telescope. Simultaneous optical imaging in two bands is enabled by a dichroic beam splitter centered at 567 nm, with Sloan u' and g' bands available on the blue arm and Sloan r', i' and z_s' bands available on the red arm. Additional narrow-band filters will also become available as required. An Electron Multiplying CCD (EMCCD) detector is employed for both optical channels, each capable of simultaneously delivering sub-electron effective read noise under multiplication gain and frame rates of up to 26 fps full frame (several 1000 fps windowed), over a fully corrected 5 x 5 arcmin field of view. CHIMERA was primarily developed to enable the characterization of the size distribution of sub-km Kuiper Belt Objects via stellar occultation, a science case that motivates the frame-rate, the simultaneous multi-color imaging and the wide field of view of the instrument. In addition, it also has unique capability in the detection of faint near-Earth asteroids and will be used for the monitoring of short duration transient and periodic sources, particularly those discovered by the intermediate Palomar Transient Factory (iPTF), and the upcoming Zwicky Transient Facility (ZTF).

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arXiv2015

Stealing the Gas: Giant Impacts and the Large Diversity in Exoplanet Densities

Although current sensitivity limits are such that true Solar System analogs remain challenging to detect, numerous planetary systems have been discovered that are very different from our own Solar System. The majority of systems harbor a new class of planets, bodies that are typically several times more massive than the Earth but that orbit their host stars well inside the orbit of Mercury. These planets frequently show evidence for large Hydrogen and Helium envelopes containing several percent of the planet's mass and display a large diversity in mean densities. Here we show that this wide range can be achieved by one or two late giant impacts, which are frequently needed to achieve long-term orbital stability in multiple planet systems once the gas disk has disappeared. We demonstrate using hydrodynamical simulations that a single collision between similarly sized exoplanets can easily reduce the envelope-to-core-mass ratio by a factor of two and show that this leads to a corresponding increase in the observed mean density by factors of 2-3. In addition we investigate how envelope-mass-loss depends on envelope mass, planet radius, semi-major axis, and the mass distribution inside the envelope. We propose that a small number of giant impacts may be responsible for the large observed spread in mean densities, especially for multiple-planet systems containing planets with very different densities and which have not been significantly sculpted by photo evaporation.

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arXiv2015

A Search for Ringed Exoplanets using Kepler Photometry

Models are developed to simulate lightcurves of stars dimmed by transiting exoplanets with and without rings. These models are then applied to \textit{Kepler} photometry to search for planetary rings in a sample of 21 exoplanets, mostly hot Jupiters, chosen to offer the best observational opportunity for discovering potential rings. We also examine what kinds of rings might be expected for these planets, in terms of both size and orientation, based on arguments involving the host planet's equilibrium temperature, its likely obliquities, and the formation and stability of possible ring systems. Finding no evidence for rings, for each of the 21 studied planets it is determined on an observational basis which potential rings can be rejected out of a representative set of fiducial rings, varying in both size and orientation. For 12 of the 21 planets, we determined that Saturn-like rings could be ruled out for at least certain orientations. Additionally, the detectability of rings is studied, and it is found that ringed planets with small obliquities (roughly $5^{\circ}-10^{\circ}$) can yield large signals, which is encouraging for future work, since such small obliquities are expected for hot Jupiters.

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