Bayesian Analysis of Hot Jupiter Radius Anomalies: Evidence for Ohmic Dissipation?
Daniel P. Thorngren, Jonathan J. Fortney
备注：
14 pages, 14 figures, accepted to The Astronomical Journal. This revision revises the description of statistical methods for clarity, but the conclusions remain the same
The cause of hot Jupiter radius inflation, where giant planets with $T_{\rm eq}$ $>1000$ K are significantly larger than expected, is an open question and the subject of many proposed explanations. Rather than examine these models individually, this work seeks to characterize the anomalous heating as a function of incident flux, $ε(F)$, needed to inflate the population of planets to their observed sizes. We then compare that result to theoretical predictions for various models. We examine the population of about 300 giant planets with well-determined masses and radii and apply thermal evolution and Bayesian statistical models to infer the anomalous power as a function of incident flux that best reproduces the observed radii. First, we observe that the inflation of planets below about $M=0.5 \;\rm{M}_\rm{J}$ appears very different than their higher mass counterparts, perhaps as the result of mass loss or an inefficient heating mechanism. As such, we exclude planets below this threshold. Next, we show with strong significance that $ε(F)$ increases with $T_{\rm{eq}}$ towards a maximum of $\sim 2.5\%$ at $T_{\rm{eq}} \approx 1500$ K, and then decreases as temperatures increase further, falling to $\sim0.2\%$ at $T_\rm{eff}= 2500$ K. This high-flux decrease in inflation efficiency was predicted by the Ohmic dissipation model of giant planet inflation but not other models. We also explicitly check the thermal tides model and find that it predicts far more variance in radii than is observed. Thus, our results provide evidence for the Ohmic dissipation model and a functional form for $ε(F)$ that any future theories of hot Jupiter radii can be tested against.
Hot Jupiter atmospheres exhibit fast, weakly-ionized winds. The interaction of these winds with the planetary magnetic field generates drag on the winds and leads to ohmic dissipation of the induced electric currents. We study the magnitude of ohmic dissipation in representative, three-dimensional atmospheric circulation models of the hot Jupiter HD 209458b. We find that ohmic dissipation can reach or exceed 1% of the stellar insolation power in the deepest atmospheric layers, in models with and without dragged winds. Such power, dissipated in the deep atmosphere, appears sufficient to slow do…
We present calculations of thermal evolution of Hot Jupiters with various masses and effective temperatures under Ohmic dissipation. The resulting evolutionary sequences show a clear tendency towards inflated radii for effective temperatures that give rise to significant ionization of alkali metals in the atmosphere, compatible with the trend of the data. The degree of inflation shows that Ohmic dissipation, along with the likely variability in heavy element content can account for all of the currently detected radius anomalies. Furthermore, we find that in absence of a massive core, low-mass …
We present models of ohmic heating in the interiors of hot jupiters in which we decouple the interior and the wind zone by replacing the wind zone with a boundary temperature Tiso and magnetic field Bphi0. Ohmic heating influences the contraction of gas giants in two ways: by direct heating within the convection zone, and by heating outside the convection zone which increases the effective insulation of the interior. We calculate these effects, and show that internal ohmic heating is only able to slow the contraction rate of a cooling gas giant once the planet reaches a critical value of inter…
We study the radius evolution of close-in extra-solar jupiters under Ohmic heating, a mechanism that was recently proposed to explain the large observed sizes of many of these planets. Planets are born with high entropy and they subsequently cool and contract. We focus on two cases: first, that ohmic heating commences when the planet is hot (high entropy); and second, that it commences after the planet has cooled. In the former case, we use analytical scalings and numerical experiments to confirm that Ohmic heating is capable of suspending the cooling as long as a few percent of the stellar ir…
We revisit the calculation of the Ohmic dissipation in a hot Jupiter presented in Laine et al. (2008) by considering more realistic interior structures, stellar obliquity, and the resulting orbital evolution. In this simplified approach, the young hot Jupiter of one Jupiter mass is modelled as a diamagnetic sphere with a finite resistivity, orbiting across tilted stellar magnetic dipole fields in vacuum. Since the induced Ohmic dissipation occurs mostly near the planet's surface, we find that the dissipation is unable to significantly expand the young hot Jupiter. Nevertheless, the planet insi…
We present a new, magnetohydrodynamic mechanism for inflation of close-in giant extrasolar planets. The idea behind the mechanism is that current, which is induced through interaction of atmospheric winds and the planetary magnetic field, results in significant Ohmic dissipation of energy in the interior. We develop an analytical model for computation of interior Ohmic dissipation, with a simplified treatment of the atmosphere. We apply our model to HD209458b, Tres-4b and HD189733b. With conservative assumptions for wind speed and field strength, our model predicts a generated power that appea…
The inflated radii of giant short-period extrasolar planets collectively indicate that the interiors of hot Jupiters are heated by some anomalous energy dissipation mechanism. Although a variety of physical processes have been proposed to explain this heating, recent statistical evidence points to the confirmation of explicit predictions of the Ohmic dissipation theory, elevating this mechanism as the most promising candidate for resolving the radius inflation problem. In this work, we present an analytic model for the dissipation rate and derive a simple scaling law that links the magnitude o…
Many giant exoplanets in close orbits have observed radii which exceed theoretical predictions. One suggested explanation for this discrepancy is heat deposited deep inside the atmospheres of these "hot Jupiters". Here, we study extended power sources which distribute heat from the photosphere to the deep interior of the planet. Our analytical treatment is a generalization of a previous analysis of localized "point sources". We model the deposition profile as a power law in the optical depth and find that planetary cooling and contraction halt when the internal luminosity (i.e. cooling rate) o…
Using semi-analytical, one-dimensional models, we elucidate the influence of scattering and absorption on the degree of Ohmic dissipation in hot Jovian atmospheres. With the assumption of Saha equilibrium, the variation in temperature is the main driver of the variations in the electrical conductivity, induced current and Ohmic power dissipated. Atmospheres possessing temperature inversions tend to dissipate most of the Ohmic power superficially, at high altitudes, whereas those without temperature inversions are capable of greater dissipation deeper down. Scattering in the optical range of wa…
The atmosphere of a hot jupiter may be subject to a thermo-resistive instability, in which the increasing electrical conductivity with temperature leads to runaway Ohmic heating. We introduce a simplified model of the local dynamics in the equatorial region of a hot jupiter that incorporates the back reaction on the atmospheric flow as the increasing electrical conductivity leads to flux freezing, which in turn quenches the flow and therefore the Ohmic heating. We demonstrate a new time-dependent solution that emerges for a temperature-dependent electrical conductivity (whereas a temperature-i…
We present the first three-dimensional circulation models for extrasolar gas giant atmospheres with geometrically and energetically consistent treatments of magnetic drag and ohmic dissipation. Atmospheric resistivities are continuously updated and calculated directly from the flow structure, strongly coupling the magnetic effects with the circulation pattern. We model the hot Jupiters HD 189733b (Teq \approx 1200 K) and HD 209458b (Teq \approx 1500 K) and test planetary magnetic field strengths from 0 to 30 G. We find that even at B = 3 G the atmospheric structure and circulation of HD 209458…
There have been many proposed explanations for the larger-than-expected radii of some transiting hot Jupiters, including either stellar or orbital energy deposition deep in the atmosphere or deep in the interior. In this paper, we explore the important influences on hot-Jupiter radius evolution of (i) additional heat sources in the high atmosphere, the deep atmosphere, and deep in the convective interior; (ii) consistent cooling of the deep interior through the planetary dayside, nightside, and poles; (iii) the degree of heat redistribution to the nightside; and (iv) the presence of an upper a…
In the presence of a magnetic field and weakly ionizing winds, ohmic dissipation is expected to take place in the envelopes of Jovian and lower-mass planets alike. While the process has been investigated on the former, there have been no studies done on mini-Neptunes so far. From structure and thermal evolution models, we determine that the required energy deposition for halting the contraction of mini-Neptunes increases with planetary mass and envelope fraction. Scaled to the insolation power, the ohmic heating needed is small $\sim10^{-5}$ -- orders of magnitude lower than for exo-Jupiters $…
We present magnetohydrodynamic (MHD) simulations of the atmospheres of hot Jupiters ranging in temperature from 1100-1800K. Magnetic effects are negligible in atmospheres with temperatures $\lesssim$ 1400K. At higher temperatures winds are variable and in many cases, mean equatorial flows can become westward, opposite to their hydrodynamic counterparts. Ohmic dissipation peaks at temperatures $\sim$1500-1600K, depending on field strength, with maximum values $\sim 10^{18}$W at 10bar, substantially lower than previous estimates. Based on the limited parameter study done, this value can not be i…
The atmospheres of hot Jupiters and other strongly-forced exoplanets are susceptible to a thermal instability in the presence of ohmic dissipation, weak magnetic drag and strong winds. The instability occurs in radiatively-dominated atmospheric regions when the ohmic dissipation rate increases with temperature faster than the radiative (cooling) rate. The instability domain covers a specific range of atmospheric pressures and temperatures, typically P ~ 3-300 mbar and T ~ 1500-2500K for hot Jupiters, which makes it a candidate mechanism to explain the dayside thermal "inversions" inferred for …
We study ionization and transport processes in partially ionized multicomponent plasmas. The plasma composition is calculated via a system of coupled mass action laws. The electronic transport properties are determined by the electron-ion and electron-neutral transport cross sections. The influence of electron-electron scattering is considered via a correction factor to the electron-ion contribution. Based on this data, the electrical and thermal conductivity as well as the Lorenz number are calculated. For the thermal conductivity, we consider also the contributions of the translational motio…
We present scaling laws for advection, radiation, magnetic drag and ohmic dissipation in the atmospheres of hot giant exoplanets. In the limit of weak thermal ionization, ohmic dissipation increases with the planetary equilibrium temperature (T_eq >~ 1000 K) faster than the insolation power does, eventually reaching values >~ 1% of the insolation power, which may be sufficient to inflate the radii of hot Jupiters. At higher T_eq values still, magnetic drag rapidly brakes the atmospheric winds, which reduces the associated ohmic dissipation power. For example, for a planetary field strength B=1…
Since the discovery of the first transiting hot Jupiters, models have sought to explain the anomalously large radii of highly irradiated gas giants. We now know that the size of hot Jupiter radius anomalies scales strongly with a planet's level of irradiation and numerous models like tidal heating, ohmic dissipation, and thermal tides have since been developed to help explain these inflated radii. In general however, these models can be grouped into two broad categories: 1) models that directly inflate planetary radii by depositing a fraction of the incident irradiation into the interior and 2…
Causes of Hot Jupiter Inflation from Causal Discovery
Center for Astrophysics and Space Science (CASS), New York University Abu Dhabi（纽约大学阿布扎比分校天体物理学与空间科学中心）；Center for Astronomy and Astrophysics and Department of Physics, Fudan University（复旦大学天文与天体物理系及物理系）；Carnegie Mellon University（卡内基梅隆大学）；INAF IASF-Milano（意大利国家天体物理研究所米兰天文物理研究所）；Ciela, Computation and Astrophysical Data Analysis Institute（Ciela计算与天体物理数据分析研究所）；Department of Physics, Astronomy, & Materials Science, Missouri State University（密苏里州立大学物理、天文与材料科学系）
The small semi-major axes of Hot Jupiters lead to high atmospheric temperatures of up to several thousand Kelvin. Under these conditions, thermally ionised metals provide a rich source of charged particles and thus build up a sizeable electrical conductivity. Subsequent electromagnetic effects, such as the induction of electric currents, Ohmic heating, magnetic drag, or the weakening of zonal winds have thus far been considered mainly in the framework of a linear, steady-state model of induction. For Hot Jupiters with an equilibrium temperature $T_{eq} > 1500$ K, the induction of atmospheric m…
Observations of the flow on Jupiter exists essentially only for the cloud-level, which is dominated by strong east-west jet-streams. These have been suggested to result from dynamics in a superficial thin weather-layer, or alternatively be a manifestation of deep interior cylindrical flows. However, it is possible that the observed winds are indeed superficial, yet there exists deep flow that is completely decoupled from it. To date, all models linking the wind, via the induced density anomalies, to the gravity field, to be measured by Juno, consider only flow that is a projection of the obser…
The radii of hot Jupiters are still not fully understood and all of the proposed explanations are based on the idea that these close-in giant planets possess hot interiors. We approach the radius anomaly problem by adopting a statistical approach. We infer the internal luminosity for the sample of hot Jupiters, study its effect on the interior structure, and put constraints on which mechanism is the dominant one. We develop a flexible and robust hierarchical Bayesian model that couples the interior structure of exoplanets to their observed properties. We apply the model to 314 hot Jupiters and…
Some extrasolar giant planets in close orbits---"hot Jupiters"---exhibit larger radii than that of a passively cooling planet. The extreme irradiation $L_{\rm eq}$ these hot Jupiters receive from their close in stars creates a thick isothermal layer in their envelopes, which slows down their convective cooling, allowing them to retain their inflated size for longer. This is yet insufficient to explain the observed sizes of the most inflated planets. Some models invoke an additional power source, deposited deep in the planet's envelope. Here we present an analytical model for the cooling of suc…
Four Cold Super-Jupiters Revealed by Extended and Complex Microlensing Signals
Chungbuk National University（忠北国立大学）；Korea Astronomy and Space Science Institute（韩国天文与空间科学研究所）；University of Canterbury（坎特伯雷大学）；Ohio State University（俄亥俄州立大学）；University of Science and Technology（科学技术联合大学院大学）；Weizmann Institute of Science（魏茨曼科学研究所）
We report the discovery of MASCARA-3b, a hot Jupiter orbiting its bright (V = 8.33) late F-type host every $5.55149\pm 0.00001$ days in an almost circular orbit ($e = 0.050^{+0.020}_{-0.017}$). This is the fourth exoplanet discovered with the Multi-site All-Sky CAmeRA (MASCARA), and the first of these that orbits a late-type star. Follow-up spectroscopic measurements were obtained in and out of transit with the Hertzsprung SONG telescope. Combining the MASCARA photometry and SONG radial velocities reveals a radius and mass of $1.36\pm 0.05$ $R_{\text{Jup}}$ and $4.2\pm 0.2$ $M_{\text{Jup}}$. I…
KELT-9b is an ultra-hot Jupiter observed to be undergoing extreme mass loss. Its A0-type host star has a radiative envelope, which makes its surface layers prone to retaining recently accreted material. To search for potential signs of planetary material polluting the stellar surface, we carry out the most comprehensive chemical characterisation of KELT-9 to-date. New element detections include Na and Y, which had previously been detected in the ultra-hot Jupiter but not studied in the star; these detections complete the set of nine elements measured in both star and planet. In comparing KELT-…
Migration of giant planets in discs with low viscosity has been studied recently. The proportionality between migration speed and the disc's viscosity is broken by the presence of vortices that appear at the edges of the planet-induced gap. Our goal is to investigate vortex-driven migration in low-viscosity discs in the presence of radial advection of gas, as expected from angular momentum removal by magnetised disc winds. We performed three dimensional simulations using the grid-based code FARGOCA. We mimicked the effects of a disc wind by applying a synthetic torque on a surface layer of the…
We report the discovery of WTS-1b, the first extrasolar planet found by the WFCAM Transit Survey, which began observations at the 3.8-m United Kingdom Infrared Telescope (UKIRT) in August 2007. Light curves comprising almost 1200 epochs with a photometric precision of better than 1 per cent to J ~ 16 were constructed for ~60000 stars and searched for periodic transit signals. For one of the most promising transiting candidates, high-resolution spectra taken at the Hobby-Eberly Telescope (HET) allowed us to estimate the spectroscopic parameters of the host star, a late-F main sequence dwarf (V=…
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Detection of four cold Jupiters through combined analyses of radial velocity and astrometry data
State Key Laboratory of Dark Matter Physics, Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University（暗物质物理国家重点实验室、李政道研究所与物理天文学院，上海交通大学）；Earth and Planets Laboratory, Carnegie Institution for Science（地球与行星实验室，卡内基科学机构）；Observatories of the Carnegie Institution for Science（卡内基科学机构天文台）
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Exoplanet searches have discovered a large number of 'hot Jupiters'--high-mass planets orbiting very close to their parent stars in nearly circular orbits. A number of these planets are sufficiently massive and close-in to be significantly affected by tidal dissipation in the parent star, to a degree parametrized by the tidal quality factor $Q_*$. This process speeds up their stars' rotation rate while reducing the planets' semimajor axis. In this paper, we investigate the tidal destruction of hot Jupiters. Because the orbital angular momenta of these planets are a significant fraction of thei…
We present a systematic evaluation of the agreement between the observed radii of 90 well-characterized transiting extrasolar giant planets and their corresponding model radii. Our model radii are drawn from previously published calculations of core-less giant planets that have attained their asymptotic radii, and which have been tabulated for a range of planet masses and equilibrium temperatures. (We report a two-dimensional polynomial fitting function that accurately represents the models). As expected, the model radii provide a statistically significant improvement over a null hypothesis th…
We report the discovery of a sub-Jupiter mass planet orbiting beyond the snow line of an M-dwarf most likely in the Galactic disk as part of the joint Spitzer and ground-based monitoring of microlensing planetary anomalies toward the Galactic bulge. The microlensing parameters are strongly constrained by the light curve modeling and in particular by the Spitzer-based measurement of the microlens parallax, $π_\mathrm{E}$. However, in contrast to many planetary microlensing events, there are no caustic crossings, so the angular Einstein radius, $θ_\mathrm{E}$ has only an upper limit based on the…
We report the discovery of WTS-1b, the first extrasolar planet found by the WFCAM Transit Survey, which began observations at the 3.8-m United Kingdom Infrared Telescope. Light curves comprising almost 1200 epochs with a photometric precision of better than 1 per cent to J=16 were constructed for 60000 stars and searched for periodic transit signals. For one of the most promising transiting candidates, high-resolution spectra taken at the Hobby-Eberly Telescope allowed us to estimate the spectroscopic parameters of the host star, a late-F main sequence dwarf (V=16.13) with possibly slightly su…
The light curve of an eclipsing system shows anomalies whenever the eclipsing body passes in front of active regions on the eclipsed star. In some cases, the pattern of anomalies can be used to determine the obliquity $Ψ$ of the eclipsed star. Here we present a method for detecting and analyzing these patterns, based on a statistical test for correlations between the anomalies observed in a sequence of eclipses. Compared to previous methods, ours makes fewer assumptions and is easier to automate. We apply it to a sample of 64 stars with transiting planets and 24 eclipsing binaries for which pr…
Binary stars provide an ideal laboratory for investigating the potential effects of planet formation on stellar composition. Assuming the stars formed in the same environment/from the same material, any compositional anomalies between binary components might indicate differences in how material was sequestered in planets, or accreted by the star in the process of planet formation. We present here a study of the elemental abundance differences between WASP-94AB, a pair of stars that each host a hot Jupiter exoplanet. The two stars are very similar in spectral type (F8 and F9), and their ~2700 A…
We present a method to quantify the upper-limit of the energy transmitted from the intense stellar wind to the upper atmospheres of three of the Trappist-1 planets (e, f, and g). We use a formalism that treats the system as two electromagnetic regions, where the efficiency of the energy transmission between one region (the stellar wind at the planetary orbits) to the other (the planetary ionospheres) depends on the relation between the conductances and impedances of the two regions. Since the energy flux of the stellar wind is very high at these planetary orbits, we find that for the case of h…
Planets with several Earth masses and a few day orbital periods have been discovered through radial velocity and transit surveys. Regardless of their formation mechanism, a key evolution issue is the efficiency of their retention near their host stars. If these planets attained their present-day orbits during or shortly after the T Tauri phase of their host stars, a large fraction would have encountered intense stellar magnetic field. Since these planets have a higher conductivity than the atmosphere of their stars, the magnetic flux tube connecting the planet and host star would slip though t…
Thousands of extrasolar planets have been discovered, and it is clear that the galactic planetary census draws on a diversity greatly exceeding that exhibited by the solar system's planets. We review significant landmarks in the chronology of extrasolar planet detection, and we give an overview of the varied observational techniques that are brought to bear. We then discuss the properties of the currently known distribution, using the mass-period diagram as a guide to delineating hot Jupiters, eccentric giant planets, and a third, highly populous, category that we term "ungiants", planets havi…
