The discovery of Exoplanetary Systems has challenged some of the theories of planet formation, which assume unperturbed evolution of the host star and its planets. However, in star clusters the interactions with flyby stars and binaries may be relatively common during the lifetime of a planetary system. Here, via high-resolution $N$-body simulations of star-planet systems perturbed by interlopers (stars and binaries), we explore the reconfiguration to the planetary system due to the encounters. In particular, via an exploration focused on the strong scattering regime, we derive the fraction of encounters which result in planet ejections, planet transfers and collisions by the interloper star/binary, as a function of the characteristics of the environment (density, velocity dispersion), and for different masses of the flyby star/binary. We find that binary interlopers can significantly increase the cross section of planet ejections and collisions, while they only slightly change the cross section for planet transfers. Therefore, in environments with high binary fractions, floating planets are expected to be relatively common, while in environments with low binary fractions, where the cross sections of planet ejection and transfer are comparable, the rate of planet exchanges between two stars will be comparable to the rate of production of free-floating planets.
Planet formation is generally described in terms of a system containing the host star and a protoplanetary disc, of which the internal properties (e.g. mass and metallicity) determine the properties of the resulting planetary system. However, (proto)planetary systems are predicted and observed to be affected by the spatially-clustered stellar formation environment, either through dynamical star-star interactions or external photoevaporation by nearby massive stars. It is challenging to quantify how the architecture of planetary systems is affected by these environmental processes, because stel…
It has recently been shown that stellar clustering plays an important role in shaping the properties of planetary systems. We investigate how the multiplicity distributions and orbital periods of planetary systems depend on the 6D phase space density of stars surrounding planet host systems. We find that stars in high stellar phase space density environments (overdensities) have a factor 1.6 - 2.0 excess in the number of single planet systems compared to stars in low stellar phase space density environments (the field). The multiplicity distribution of planets around field stars is much flatte…
A scatter plot of exoplanet mass against red giant host star radius demonstrates an interesting positive trend: larger stars have more massive planets. This implies that the evolution of a star towards a red giant affects the masses of planets in the system.
Due to the high stellar densities in young clusters, planetary systems formed in these environments are likely to have experienced perturbations from encounters with other stars. We carry out direct $N$-body simulations of multi-planet systems in star clusters to study the combined effects of stellar encounters and internal planetary dynamics. These planetary systems eventually become part of the Galactic field population the parental cluster dissolves, which is where most presently-known exoplanets are observed. We show that perturbations induced by stellar encounters lead to distinct signatu…
Stellar flybys in star clusters may perturb the evolution of young planetary systems in terms of disk truncation, planetary migration and planetary mass accretion. We investigate the feedback of a young planetary system during a single close stellar encounter in a typical open young stellar cluster. We consider 5 masses for the stellar perturbers: 0.5, 0.8, 1, 3 and 8 M$_{Sun}$, in coplanar, prograde and retrograde orbits respect to the planetary disk, varying the pertruber-host star orbital periastron from 100 AU to 500 AU. We have made 3D modelizations with the smooth particle hydrodynamics …
Our improving understanding of the life cycle of planetary systems prompts investigations of the role of the Galactic environment before, during and after Asymptotic Giant Branch (AGB) stellar evolution. Here, we investigate the interplay between stellar mass loss, Galactic tidal perturbations, and stellar flybys for evolving stars which host one planet, smaller body or stellar binary companion and reside in the Milky Way's bulge or disc. We find that the potential evolutionary pathways from a main sequence (MS) to a white dwarf (WD) planetary system are a strong function of Galactocentric dis…
Planet formation theory suggests that planet bulk compositions are likely to reflect the chemical abundance ratios of their host star's photosphere. Variations in the abundance of particular chemical species in stellar photospheres between different galactic stellar populations demonstrate that there are differences among the expected solid planet bulk compositions. We aim to present planetary mass-radius relations of solid planets for kinematically differentiated stellar populations, namely, the thin disc, thick disc, and halo. Using two separate internal structure models, we generated synthe…
We describe how the accurate characterization of exoplanetary atmospheres in the ELT and JWST era will inevitably require taking into consideration the stellar inhomogeneities caused by convection and magnetic fields. The existing evidence that demonstrates the mixture of stellar and planetary signatures in observed transiting spectra is presented. Finally, we discuss how to disentangle these two components through a multipronged approach that includes new solar reference spectra, improved MHD modeling, and synergistic collaborations between the communities involved, from solar to stellar and …
One of the predictions of high eccentricity planetary migration is that many planets will end up plunging into their host stars. We investigate the consequence of planetary mergers on their stellar hosts' spin-period. Energy and angular momentum conservation yield that a planet consumption by a star will spin-up the star. We find that our calculations align with the observed bifurcation in the stellar spin-period in young clusters. For example, after a Sun-like star has eaten a Jupiter-mass planet it will spin up by ~60% (i.e., spin-period is reduced by ~60%), causing an apparent gap in the st…
The ability to observe extrasolar planets transiting their stars has profoundly changed our understanding of these planetary systems. However, these measurements depend on how well we understand the properties of the host star, such as radius, luminosity and limb darkening. Traditionally, limb darkening is treated as a parameterization in the analysis, but these simple parameterizations are not accurate representations of actual center-to-limb intensity variations (CLIV) to the precision needed for interpreting these transit observations. This effect leads to systematic errors for the measured…
In a multi-planet system, a gradual change in one planet's semi-major axis will affect the eccentricities of all the planets, as angular momentum is distributed via secular interactions. If tidal dissipation in the planet is the cause of the change in semi-major axis, it also damps that planet's eccentricity, which in turn also contributes to the evolution of all the eccentricities. Formulae quantifying the combined effects on the whole system due to semi-major axis changes, as well as eccentricity damping, are derived here for a two-planet system. The CoRoT 7 system is considered as an exampl…
We review some of the recent progress on modeling planetary and stellar dynamos. Particular attention is given to the dynamo mechanisms and the resulting properties of the field. We present direct numerical simulations using a simple Boussinesq model. These simulations are interpreted using the classical mean-field formalism. We investigate the transition from steady dipolar to multipolar dynamo waves solutions varying different control parameters, and discuss the relevance to stellar magnetic fields. We show that owing to the role of the strong zonal flow, this transition is hysteretic. In th…
The presence of gaseous giant planets whose orbits lie in extreme proximity to their host stars ("hot Jupiters"), can largely be accounted for by planetary migration, associated with viscous evolution of proto-planetary nebulae. Recently, observations of the Rossiter-McLaughlin effect during planetary transits have revealed that a considerable fraction of detected hot Jupiters reside on orbits that are misaligned with respect to the spin-axes of their host stars. This observational fact has cast significant doubts on the importance of disk-driven migration as a mechanism for production of hot …
We present results from a quantitative spectroscopic analysis conducted on archival Keck/HIRES high-resolution spectra from the California-$Kepler$ Survey (CKS) sample of transiting planetary host stars identified from the $Kepler$ mission. The spectroscopic analysis was based on a carefully selected set of Fe I and Fe II lines, resulting in precise values for the stellar parameters of effective temperature (T$_{\rm eff}$) and surface gravity (log $g$). Combining the stellar parameters with $Gaia$ DR2 parallaxes and precise distances, we derived both stellar and planetary radii for our sample,…
The interaction of escaping upper atmosphere of a hydrogen rich non-magnetized analog of HD209458b with a stellar wind of its host G-type star at different orbital distances is simulated with a 2D axisymmetric multi-fluid hydrodynamic model. A realistic sun-like spectrum of XUV radiation which ionizes and heats the planetary atmosphere, hydrogen photo-chemistry, as well as stellar-planetary tidal interaction are taken into account to generate self-consistently an atmospheric hydrodynamic outflow. Two different regimes of the planetary and stellar winds interaction have been modelled. These are…
The Rossiter-McLaughlin (RM) effect has been widely used to estimate the sky-projected spin-orbit angle, $λ$, of transiting planetary systems. Most of the previous analysis assume that the host stars are rigid rotators in which the amplitude of the RM velocity anomaly is proportional to $v_\star \sin i_\star$. When their latitudinal differential rotation is taken into account, one can break the degeneracy, and determine separately the equatorial rotation velocity $v_\star$ and the inclination $i_{\star}$ of the host star. We derive a fully analytic approximate formula for the RM effect adoptin…
Recent work has demonstrated that exoplanetary system properties correlate strongly with ambient stellar clustering in six-dimensional stellar position-velocity phase space, quantified by dividing planetary systems into sub-samples with high or low phase space densities (`overdensity' and `field' systems, respectively). We investigate the physical origins of the phase space overdensities and, thereby, which environmental mechanisms may have impacted the planetary systems. We consider the galactic-scale kinematic structure of the Milky Way observed with Gaia and show that the overdensities corr…
With the discovery of hundreds of exoplanets and a potentially huge number of Earth-like planets waiting to be discovered, the conditions for their habitability have become a focal point in exoplanetary research. The classical picture of habitable zones primarily relies on the stellar flux allowing liquid water to exist on the surface of an Earth-like planet with a suitable atmosphere. However, numerous further stellar and planetary properties constrain habitability. Apart from "geophysical" processes depending on the internal structure and composition of a planet, a complex array of astrophys…
Lyman $α$ observations of the transiting exoplanet HD 209458b enable the study of exoplanets exospheres exposed to stellar EUV fluxes, as well as the interacting stellar wind properties. In this study we present 3D hydrodynamical models for the stellar-planetary wind interaction including radiation pressure and charge exchange, together with photoionization, recombination and collisional ionization processes. Our models explore the contribution of the radiation pressure and charge exchange on the Ly$α$ absorption profile in a hydrodynamical framework, and for a single set of stellar wind param…
Interactions between the winds of stars and the magnetospheres and atmospheres of planets involve many processes, including the acceleration of particles, heating of upper atmospheres, and a diverse range of atmospheric loss processes. Winds remove angular momentum from their host stars causing rotational spin-down and a decay in magnetic activity, which protects atmospheres from erosion. While wind interactions are strongly influenced by the X-ray and ultraviolet activity of the star and the chemical composition of the atmosphere, the role of planetary magnetic fields is unclear. In this chap…
Hot Jupiters (HJs) are massive gaseous planets orbiting close to their host stars. Due to their physical characteristics and proximity to the central star, HJs are the natural laboratories to study the process of star-planet interaction (SPI). Phenomena related to SPI may include the inflation and the evaporation of planetary atmospheres, the formation of cometary tails and bow shocks and magnetospheric interaction between the magnetic field of the planet and that of the star. Several works suggest that some systems show enhanced stellar activity in phase with the planetary rotation period. In…
Stellar magnetic activity is a source of noise in the study of the transits of extrasolar planets. It induces flux variations which affect significantly the transit depth determination and the derivations of planetary and stellar parameters. Furthermore, the colour dependence of stellar activity may significantly influence the characterization of planetary atmospheres. Here we present a systematic approach to quantify the corresponding stellar flux variations as a function of wavelength bands. We consider a star with spots covering a given fraction of its disc and model the variability in the …
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-…
The transit method, employed by MOST, \emph{Kepler}, and various ground-based surveys has enabled the characterization of extrasolar planets to unprecedented precision. These results are precise enough to begin to measure planet atmosphere composition, planetary oblateness, star spots, and other phenomena at the level of a few hundred parts-per-million. However, these results depend on our understanding of stellar limb darkening, that is, the intensity distribution across the stellar disk that is sequentially blocked as the planet transits. Typically, stellar limb darkening is assumed to be a …
We present initial results of a new campaign of simulations focusing on the interaction of planetary winds with stellar environments using Adaptive Mesh Refinement methods. We have confirmed the results of Stone & Proga 2009 that an azimuthal flow structure is created in the planetary wind due to day/night temperatures differences. We show that a backflow towards the planet will occur with a strength that depends on the escape parameter. When a stellar outflow is included, we see unstable bow waves forming through the outflow's interaction with the planetary wind.
Glorified dimensional analysis is used to derive scaling rules for internal and external magnetic field strengths and various time scales. Naive dimensional analysis is inconclusive because of multiple time scales, but physical arguments serve to weed out irrelevant parameters. Time scales can be derived from linearized instability analysis instead of ill-founded assumptions of Magnetic-Archimedean-Coriolis (MAC) balance. Further relationships can be derived from high-level models of coupled main field components and differential rotation. The ratios of the external dipole field to internal ma…
Intermediate mass stars and stellar remnants often host planets, and these dynamical systems evolve because of mass loss and tides. This paper considers the combined action of stellar mass loss and tidal dissipation on planetary orbits in order to determine the conditions required for planetary survival. Stellar mass loss is included using a so-called Jeans model, described by a dimensionless mass loss rate γand an index β. We use an analogous prescription to model tidal effects, described here by a dimensionless dissipation rate Γand two indices (q,p). The initial conditions are determined by…
It is a truism within the exoplanet field that "to know the planet, you must know the star." This pertains to the physical properties of the star (i.e. mass, radius, luminosity, age, multiplicity), the activity and magnetic fields, as well as the stellar elemental abundances which can be used as a proxy for planetary composition. In this white paper, we discuss important stellar characteristics that require attention in upcoming ground- and space-based missions, such that their processes can be understood and either detangled from that of the planet, correlated with the presence of a planet, o…
Many exoplanets have orbital characteristics quite different from those seen in our own solar system, including planets locked in orbital resonances and planets on orbits that are elliptical or highly inclined from their host star's spin axis. It is debated whether the wide variety in system architecture is primarily due to differences in formation conditions (nature) or due to evolution over time (nurture). Identifying trends between planetary and stellar properties, including stellar age, can help distinguish between these competing theories and offer insights as to how planets form and evol…
Stellar activity has a particularly strong influence on planets at small orbital distances, such as close-in exoplanets. For such planets, we present two extreme cases of stellar variability, namely stellar coronal mass ejections and stellar wind, which both result in the planetary environment being variable on a timescale of billions of years. For both cases, direct interaction of the streaming plasma with the planetary atmosphere would entail servere consequences. In certain cases, however, the planetary atmosphere can be effectively shielded by a strong planetary magnetic field. The efficie…
With the discovery of now more than 500 exoplanets, we present a statistical analysis of the planetary orbital periods and their relationship to the rotation periods of their parent stars. We test whether the structure of planetary orbits, i.e. planetary angular momentum and orbital periods are 'quantized' in integer or half-integer multiples with respect to the parent stars' rotation period. The Solar System is first shown to exhibit quantized planetary orbits that correlate with the Sun's rotation period. The analysis is then expanded over 443 exoplanets to statistically validate this quanti…
The signatures of planets hosted by M dwarfs are more readily detected with transit photometry and radial velocity methods than those of planets around larger stars. Recently, transit photometry was used to discover seven planets orbiting the late-M dwarf TRAPPIST-1. Three of TRAPPIST-1's planets fall in the Habitable Zone, a region where liquid water could exist on the planetary surface given appropriate planetary conditions. We aim to investigate the habitability of the TRAPPIST-1 planets by studying the star's activity and its effect on the planets. We analyze previously-published space- an…
Ultra-short-period planets (USPs) provide important clues to planetary formation and migration. Recently, it is found that the mutual inclinations of the planetary systems are larger if the inner orbits are closer ($\lesssim 5R_*$) and if the planetary period ratios are larger ($P_2/P_1 \gtrsim 5$) (Dai et al. 2018). This suggests that the USPs experienced both inclination excitation and orbital shrinkage. Here we investigate the increase in the mutual inclination due to stellar oblateness. We find that the stellar oblateness (within $\sim 1$Gyr) is sufficient to enhance the mutual inclination…
Photospheric stellar activity might be an important source of noise and confusion in the radial-velocity measurements. RV planet search surveys as well as follow-up of photometric transit surveys require a deeper understanding and characterization of the effects of stellar activities to disentangle it from planetary signals. We simulate dark spots on a rotating stellar photosphere. The variations of the photometry, RV and spectral line shapes are characterized and analyzed according to the stellar inclination, the latitude and the number of spots. The Lomb-Scargle periodograms of the RV variat…
The Kepler mission has discovered thousands of exoplanets around various stars with different spectral types (M, K, G, and F) and thus different masses and effective temperatures. Previous studies have shown that the planet occurrence rate, in terms of average number of planets per star, drops with increasing stellar effective temperature (Teff). In this paper, with the final Kepler Data Release (DR25) catalog, we revisit the relation between stellar effective temperature (as well as mass) and planet occurrence, but in terms of the fraction of stars with planets and the number of planets per p…
A new apodized Keplerian (AK) model is proposed for the analysis of precision radial velocity (RV) data to model both planetary and stellar activity (SA) induced RV signals. A symmetrical Gaussian apodization function with unknown width and center can distinguish planetary signals from SA signals on the basis of the span of the apodization window. The general model for $m$ apodized Keplerian signals includes a linear regression term between RV and the stellar activity diagnostic $\log(R'hk)$, as well as an extra Gaussian noise term with unknown standard deviation. The model parameters are expl…
Observations indicate that intermediate mass stars, binary stars, and stellar remnants often host planets; a complete explanation of these systems requires an understanding of how planetary orbits evolve as their central stars lose mass. Motivated by these dynamical systems, this paper generalizes in two directions previous studies of orbital evolution in planetary systems with stellar mass loss: [1] Many previous treatments focus on constant mass loss rates and much of this work is carried out numerically. Here we study a class of single planet systems where the stellar mass loss rate is time…
In this work we present a mathematical model for the propagation of the shock waves that occur in graded density profiles. These waves can occur in a wide range of astrophysical events, such as collisions in planetary and stellar atmospheres, common envelope explosions and peculiar type Ia supernovae. The behaviour of the shock wave and its evolution can be modelled using type II self similar solutions. In such solutions the evolution of the shock wave is determined by boundary conditions at the shock front and a singular point in the shocked region. We show how the evolution can be determined…
A number of planet-host stars have been observed to rotate with a period equal to an integer multiple of the orbital period of their close planet. We expand this list by analyzing Kepler data of HAT-P-11 and finding a period ratio of 6:1. In particular, we present evidence for a long-lived spot on the stellar surface that is eclipsed by the planet in the same position four times, every sixth transit. We also identify minima in the out-of-transit lightcurve and confirm that their phase with respect to the stellar rotation is mostly stationary for the 48-month timeframe of the observations, conf…
We constrain the planetary atmospheric mass fraction at the time of the dispersal of the protoplanetary disk and the evolution of the stellar rotation rate for a dozen multi-planet systems that host sub-Neptunes and/or super-Earths. We employ a custom-developed Python code that we have dubbed Pasta (Planetary Atmospheres and Stellar roTation rAtes), which runs within a Bayesian framework to model the atmospheric evolution of exoplanets. The code combines MESA stellar evolutionary tracks, a model describing planetary structures, a model relating stellar rotation and activity level, and a model …
We report on the high-resolution spectroscopic observations of two planetary transits of the hot Jupiter KELT-7b ($M_{\rm p}$ $=$ 1.28 $\pm$ 0.17 M$_{\rm Jup}$, $T_{\rm eq}$ $=$ 2028 K) observed with the High Optical Resolution Spectrograph (HORuS) mounted on the 10.4-m Gran Telescopio Canarias (GTC). A new set of stellar parameters are obtained for the rapidly rotating parent star from the analysis of the spectra. Using the newly derived stellar mass and radius, and the planetary transit data of the Transiting Exoplanet Survey Satellite (TESS) together with the HORuS velocities and the photom…
Stellar Wind Driven Alfvén Wing Dynamics in Planetary and Exoplanetary Magnetospheres
Center of Excellence in Space Sciences India, Indian Institute of Science Education and Research Kolkata（印度空间科学卓越中心，印度科学教育研究所加尔各答分校）；Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata（印度科学教育研究所加尔各答分校物理科学系）；Center for High Energy Systems and Sciences, Defence Research and Development Organization（国防研究与发展组织高能系统与科学中心）；Department of Physics, Fakir Chand College（法基尔昌德学院物理系）
It has been suggested that planetary radii increase with the stellar mass, for planets below 6 R$_{\oplus}$ and host below 1 M$_\odot$. In this study, we explore whether this inferred relation between planetary size and the host star's mass can be explained by a larger planetary mass among planets orbiting more massive stars, inflation of the planetary radius due to the difference in stellar irradiation, or different planetary compositions and structures. Using exoplanetary data of planets with measured masses and radii, we investigate the relations between stellar mass and various planetary p…
We present stellar wind modelling of the hot Jupiter host HD189733, and predict radio emission from the stellar wind and the planet, the latter arising from the interaction of the stellar wind with the planetary magnetosphere. Our stellar wind models incorporate surface stellar magnetic field maps at the epochs Jun/Jul 2013, Sep 2014, and Jul 2015 as boundary conditions. We find that the mass-loss rate, angular momentum-loss rate, and open magnetic flux of HD189733 vary by 9%, 40%, and 19% over these three epochs. Solving the equations of radiative transfer, we find that from 10 MHz-100 GHz th…
Young stars are mostly found in dense stellar environments, and even our own Solar system may have formed in a star cluster. Here, we numerically explore the evolution of planetary systems similar to our own Solar system in star clusters. We investigate the evolution of planetary systems in star clusters. Most stellar encounters are tidal, hyperbolic, and adiabatic. A small fraction of the planetary systems escape from the star cluster within 50 Myr; those with low escape speeds often remain intact during and after the escape process. While most planetary systems inside the star cluster remain…
Straight-forward derivation of planetary parameters can only be achieved in transiting planetary systems. However, planetary attributes such as radius and mass strongly depend on stellar host parameters. Discovering a transit host star to be multiple leads to a necessary revision of the derived stellar and planetary parameters. Based on our observations of 14 transiting exoplanet hosts, we derive parameters of the individual components of three transit host stars (WASP-2, TrES-2, and TrES-4) which we detected to be binaries. Two of these have not been known to be multiple before. Parameters of…
No statistically significant evidence for a correlation between stellar and planetary composition
University of Birmingham（伯明翰大学）；University of Cambridge（剑桥大学）；European Space Agency (ESA)（欧洲空间局（ESA））；European Space Research and Technology Centre (ESTEC)（欧洲空间研究与技术中心（ESTEC））；University of Groningen（格罗宁根大学）
The architecture and evolution of planetary systems are shaped in part by stellar flybys. Within this context, we look at stellar encounters which are too weak to immediately destabilize a planetary system but are nevertheless strong enough to measurably perturb the system's dynamical state. We estimate the strength of such perturbations on secularly evolving systems using a simple analytic model and confirm those estimates with direct N-body simulations. We then run long-term integrations and show that even small perturbations from stellar flybys can influence the stability of planetary syste…
Searches for stellar companions to hot Jupiters (HJs) have revealed that planetary systems hosting a HJ are approximately three times more likely to have a stellar companion with a semimajor axis between 50 and 2000 AU, compared to field stars. This correlation suggests that HJ formation is affected by the stellar binary companion. A potential model is high-eccentricity migration, in which the binary companion induces high-eccentricity Lidov-Kozai (LK) oscillations in the proto-HJ orbit, triggering orbital migration driven by tides. A pitfall of this `binary-LK' model is that the observed stel…
We investigate the evolution of the relative angle between the stellar rotation axis and the circumstellar disc axis of a star that forms in a stellar cluster from the collapse of a turbulent molecular cloud. This is an inherently chaotic environment with variable accretion, both in terms of rate and the angular momentum of the material, and dynamical interactions between stars. We find that the final stellar rotation axis and disc spin axis can be strongly misaligned, but this occurs primarily when the disc is truncated by a dynamical encounter so that the final disc rotation axis depends sim…
Proper characterization of the host star to a planet is a key element to the understanding of its overall properties. The star has a direct impact through the modification of the structure and evolution of the planet atmosphere by being the overwhelmingly larger source of energy. The star plays a central role in shaping the structure, evolution, and even determining the mere existence of planetary atmospheres. The vast majority of the stellar flux is well understood thanks to the impressive progress made in the modeling of stellar atmospheres. At short wavelengths (X-rays to UV), however, the …
We investigate the interaction between the magnetized stellar wind plasma and the partially ionized hydrodynamic hydrogen outflow from the escaping upper atmosphere of non- or weakly magnetized hot Jupiters. We use the well-studied hot Jupiter HD 209458b as an example for similar exoplanets, assuming a negligible intrinsic magnetic moment. For this planet, the stellar wind plasma interaction forms an obstacle in the planet's upper atmosphere, in which the position of the magnetopause is determined by the condition of pressure balance between the stellar wind and the expanded atmosphere, heated…
A monomodal model for stellar and planetary convection is derived for the magnitude of the rms velocity, degree of superadiabaticity, and characteristic length scale as a function of rotation rate as well as with thermal and viscous diffusivities. The convection model is used as a boundary condition for a linearization of the equations of motion in the transition region between convectively unstable and stably-stratified regions, yielding the depth to which convection penetrates into the stable region and establishing a relationship between that depth and the local convective Rossby number, di…
