Hervé Beust (U Grenoble): Orbital Fitting of Imaged Companions with High Eccentricities and Unbound Orbits. Application to
PZ Telecopii~B
Astrometric follow-up of substellar companions often allows the detection of a projected orbital motion. Markov Chain Monte Carlo (MCMC) has become very popular for fitting their orbits. Some of these companions appear to move on very eccentric, possibly unbound orbits. This is in particular the case for the brown dwarf companion PZ Tel b.
For these cases, standard MCMC codes that assume only bound orbits are inappropriate. I present here a recent MCMC implementation using universal Keplerian variables and Stumpff functions, that is able to continuously handle both bound and unbound orbits.
The code is applied to PZ Tel b, with a first successful fit of its orbit. It shows that while bound and unbound solutions are equally possible, the eccentricity distribution presents a sharp peak near e=1, meaning a quasi-parabolic orbit.
It has been suggested that the presence of unseen inner companions may lead orbital fitting algorithms to artificially give very high eccentricities. Concerning PZ Tel b, we derive a solution involving an inner ~12 Mjup companion, that would mimic an e=1 orbit, despite a real eccentricity of 0.7. However, such a configuration appears dynamically unstable. Our orbital fit appears thus robust.
Raúl Ortega Chametla (ESFM-IPN): Gap Formation by Inclined Massive Planets in Locally Isothermal Three-dimensional Discs
We study gap formation in gaseous protoplanetary discs by a Jupiter mass planet. The planets orbit is circular and inclined relative to the mid-plane of the disc. We use the impulse approximation to estimate the gravitational tidal torque between the planet and the disc, and infer the gap profile. For low-mass discs, we provide a criterion for gap opening when the orbital inclination is ≤ 30°. Using the FARGO3D code, we simulate the disc response to an inclined massive planet. The dependence of the depth and width of the gap obtained in the simulations on the inclination of the planet is broadly consistent with the scaling laws derived in the impulse approximation. Although we mainly focus on planets kept on fixed orbits, the formalism permits to infer the temporal evolution of the gap profile in cases where the inclination of the planet changes with time. This study may be useful to understand the planetary migration of massive planets on inclined orbit, because the strength of the interaction with the disc depends on whether a gap is opened or not.
Adam Dempsey (Northwestern): Effects of High Mass Planets on Low Mass Accretion Disks
Current infrared and submillimeter observatories such as ALMA have the capability to image the deep and wide gaps expected of young, high mass planets embedded in low viscosity protoplanetary disks. Inferring the properties of these unseen planetary mass companions requires detailed hydrodynamical simulations of the planet-disk interactions. The boundary conditions used in these hydrodynamical simulations can have a significant impact on the steady-state disk structure by artificially injecting angular momentum near the boundaries. Using a modified version of the GPU accelerated FARGO3D hydrodynamics code, we solve for the 2D quasi-steady-state solution of a low-mass accretion disk with an embedded planet. We implement angular momentum and mass conserving boundary conditions that allow the density and velocity of the gas to adjust so that the disks attains a spatially constant mass flux throughout the domain. This results in a significant pile-up of gas exterior to the planet that develops over the viscous timescale of the disk. We run a suite of FARGO3D simulations to understand how planet mass and disk viscosity affect the pile-up magnitude and gap properties. We find that the surface density at the outer edge of the gap can be enhanced over the surface density in the absence of the planet by a significant factor. Our results have implications for disk induced planetary migration as this exterior pile-up can enhance the outer disk torques. This speeds up the inwards migration rate of the planet compared to the slow migration rate expected of a high mass planet in a comparatively low mass disk. Additionally, this large surface density of gas at the outer gap edge results in a larger surface density of dust, providing brighter dust emission rings and potentially altering dust growth and fragmentation processes.
Evgeni Grishin (Technion): Stability and Secular Evolution of Exo-moons in Triple and Quadruple Systems
We study the stability and secular chaotic behavior of exo-moons and irregular satellites. The stability of exo-moons is studied by means of the hierarchical three body system. If the exo-moons are a part of a binary, secular evolution could lead chaotic excitations of high inclination and eccentricity. We numerically integrate both direct N-body for hierarchical triples on short timescales, and secular equations for long timescales. For hierarchical triples, we find a complex behavior on the mutual inclination, and explain it using semi-secular theory of Lidov-Kozai cycles and their modification due to evection resonance. We provide polynomial fits to the numerical results to be widely used in any hierarchical triple. Exo-moons in binary systems can exhibit chaotic evolution of inclination and eccentricity excitation due to chaotic resonance overlap. We study the chaotic secular dynamics of quadruple systems with the addition of short range forces that control the level of eccentricity excitation. We find by means of Hamiltonian perturbation theory and numerical simulations that the qualitative chaotic zones depend mainly on the ratio of the innermost and outermost secular timescales. The results apply various systems, such as planets and exo-moons in multiple systems. We demonstrate that exo-moons around gas giants and rocky planets in binary systems can develop high inclinations, even if they are close and remain circular. Many aspects could be also applied to migrating Jupiters in triple stellar systems.
Matthew Heising (Harvard): Resonant Capture of Planetesimals via the Grand Tack with 3D Hydrodynamic Circumstellar
Disk Simulations
The Grand Tack model, which states that Jupiter initially migrated inward to roughly 1.5 AU before migrating outward to its current orbit, has gained popularity for its ability to explain several features of the Solar System's history. In particular, it has been proposed that planetesimals caught in resonances with Jupiter would have fragmented in a collisional cascade, causing them to drift inward via gas drag, shepherding any existing terrestrial planets into the Sun. The current terrestrial planets, then, would had to have formed after the Grand Tack, explaining their relatively low total mass when compared to exoplanetary systems. Here, I present 3D hydrodynamic simulations of a circumstellar disk to explore this scenario. Jupiter's migration is put in "by hand," and the response of the gas disk and the planetesimal population is simulated self-consistently.
Jonathan Jackson (Penn State): Assessing 4-body Tidal Migration in the Kepler-419 System
While hot Jupiters (periods less than 10 days) are some of the most well-studied exoplanets, warm Jupiters (periods between 10 and 200 days) have often been overlooked. Formation and migration theories postulated so far are able to reproduce the observed semi-major axis and eccentricity distributions of warm Jupiters. One possible way to remedy this disconnect is for a secularly perturbing companion to excite the eccentricity of the planet high enough to induce tidal migration. Many warm Jupiters are found in multi-planet systems, so if this mechanism is common among these systems, it could be a prolific warm Jupiter creator. While no known systems have yet been shown to undergo this mechanism, Kepler-419, a system with one warm Jupiter and one eccentric outer companion, is an interesting study case. In its current configuration, the outer perturber cannot excite the eccentricity of the warm Jupiter high enough to undergo tidal circularization; however, if there exists a third giant planet in the system, it could periodically boost the secular eccentricity oscillations of the inner planet enough for migration to occur. We explore the parameter space of this potential third giant planet using a suite of 2420 N-body simulations with a range of initial conditions. Preliminary results rule out this mechanism for much of the parameter space of initial conditions. If and when we are able to determine the area of parameter space most likely to boost the eccentricity without destabilizing the system, we will convert the results to a radial velocity signal. This will either rule out the presence of a third body capable of explaining the warm Jupiter's formation, or inform future observations of the system providing constraints for the detection of a constrained parameter space in which to search for an additional planet.
Jeff Jennings (Universitäts-Sternwarte München): X-ray Photoevaporation and Favorable Conditions for the Streaming Instability
Previous works have considered the onset of the streaming instability in a young planetary disc, a mechanism that fosters rapid planetesimal growth in an environment of simultaneously high dust-to-gas ratio (≥10^-2) and large Stokes number (≥10^-1), due to photoevaporation that selectively removes gas and only the smallest, well-coupled dust grains. We extend these studies that treated EUV-driven photoevaporation to the X-ray driven case, characteristically having a factor of ~100 higher disc mass-loss rate than its EUV counterpart and effectively removing substantial gas from the disc surface at earlier times. For standard disc and dust models, we find that from the onset of a gap opening in the disc due to X-ray photoevaporation, the gas-rich outer edge of the gap experiences a local pressure maximum that moves outward as the disc is dispersed. This region shows sustained high dust-to-gas surface densities and Stokes numbers, favorable for triggering the streaming instability according to current criteria. We discuss the capacity of this process to produce a population of planetesimals between 1 – 10 AU, which may be the precursors of compact terrestrial planet configurations observed in the Kepler catalog.
Mariah MacDonald (Penn State U): Disentangling In-situ Formation Conditions in Establishing the "Dichotomy" in Super-Earth
Orbital Architectures
Studies of the Kepler population have found evidence of a dichotomy in the orbital architectures of super-Earth systems; while many systems have super-Earths with tight spacings, small eccentricities, and small inclinations, others systems display wider spacings, larger eccentricities, and larger mutual inclinations. Many studies have explored the potential cause of this apparent dichotomy. Some have suggested that the less compact systems are a result of dynamical instability or perturbations from outer giant planets, while others look to explain the schism via planetary formation conditions. In particular, Dawson et al. 2016 found that the dichotomy could be explained by the presence or absence of residual gas during the end of the giant impact stage, while Moriarty & Ballard 2016 found that a combination of two different solid surface density profiles can explain it. We aim to disentangle the effects of the gas disk damping and the surface profiles. We find that residual gas and a steep solid density profile are both necessary to build the most compact Kepler systems and that the observed distributions of orbital parameters can arise from a continuum of solid surface density normalizations rather than two distinct sets of formation conditions.
Sean Mills (U Chicago): Mass, Density, and Formation Constraints in the Compact, Sub-Earth Kepler-444 System including Two
Mars-Mass Planets
Kepler-444 is a five planet system around a host-star approximately 11 billion years old. The five transiting planets all have sub-Earth radii and are in a compact configuration with orbital periods between 3 and 10 days. Here we present a transit-timing analysis of the system using the full Kepler data set in order to determine the masses of the planets. Two planets, Kepler-444 d (M_d=0.036+0.065-0.020M_Earth) and Kepler-444 e (M_e=0.034+0.059-0.019M_Earth), have confidently detected masses due to their proximity to resonance which creates transit timing variations. The mass ratio of these planets combined with the magnitude of possible star-planet tidal effects suggests that smooth disk migration over a significant distance is unlikely to have brought the system to its currently observed orbital architecture without significant post-formation perturbations.
Norio Narita (U Tokyo): MuSCAT and MuSCAT2 for Detection and Characterization of Transiting Exoplanets
In this talk, we will introduce 2 multi-color simultaneous cameras, named MuSCAT and MuSCAT2. MuSCAT (Multi-color Simultaneous Camera for studying Atmospheres of Transiting exoplanets) is a 3-color simultaneous camera installed on the 188cm telescope at Okayama Astrophysical Observatory in Japan. MuSCAT2 is a 4-color simultaneous camera under development for the TCS 1.5m telescope in Teide Observatory, Canaries, Spain. First light observations of MuSCAT2 is planned in 2017 summer. I will talk about specifications of those instruments, latest results including the detection of K2-105b (Narita et al. 2017) and the improvement of the transit ephemeris of K2-3d (Fukui et al. 2016), and future plans.
Andrew F. Nelson (Los Alamos National Laboratory): How Do Planetary Envelopes Form?
In the core accretion model for Jovian planet formation, a gaseous envelope grows over time around a heavy element core. To date however, the theoretical work to explain this growth assumes a one-dimensional hydrostatic envelope, cooling from a high-altitude photosphere and adding additional material as that cooling permits. In previous work, we showed that this assumption does not hold: the gas behavior is highly dynamic, with highly inhomogeneous flow into and out of the core's environment (i.e. the Hill and/or Bondi volume) on very short timescales, such that no envelope or photosphere can grow or even be defined. Here, we show that this behavior extends to much lower mass cores than in our original study, and to an exact hydrogen/helium equation of state for the solar nebula gas. We explore some possible solutions, by which the activity may be quenched, but the question of how do planetary envelopes form, remains unanswered.
Larissa Nofi (Lowell Observatory): Searching for Young Hot Jupiters Around T Tauri Stars
Observing and characterizing newly-formed planets around young stars is important for developing planet formation and evolution theory. However, given challenges in detecting young planetary systems, current models are primarily based on systems that are billions of years old. It is therefore unclear which exoplanetary properties are indicators of formation conditions, or of later evolution. We are conducting a survey to detect and confirm young exoplanets around T Tauri stars using the Immersion Grating Infrared Spectrograph (IGRINS) on the 4.3-m Discovery Channel Telescope (DCT). IGRINS simultaneously observes H- and K-bands at a resolution of ~45,000. The IGRINS + DCT system provides a radial velocity (RV) precision of <30 m/s, is sensitive to giant planets within ~3 AU of a host star, and is relatively immune to RV variability triggered by starspots on active young stars. Our sample consists of ~140 T Tauri stars of age <10 Myr in the relatively nearby Taurus star forming region. We aim to 1) detect and confirm young exoplanets; 2) compare hot Jupiter occurrence rates for pre-main sequence stars to those of main sequence stars; 3) investigate interactions between a circumstellar disk and planets; 4) extend these results to planet formation theory.
Henriette Schwarz (UCSC): Spinning Worlds: The Rotation of Young Gas Giants and Brown Dwarf Companions
The spin of an exoplanet is a fundamental observable that affects the atmospheric dynamics and the climate, and which may also hold clues to its formation. I will present the results from a small CRIRES/VLT survey with the aim to study the spin of young directly imaged sub-stellar companions. The observed targets span a range of masses, ages and orbital distances, providing the first opportunity to compare the spin parameters of young exoplanets and brown dwarf companions. Although the observed sample is small, we do see a correlation of spin velocity with age, which we interpret as due to the youngest objects still accreting angular momentum and spinning up through subsequent cooling and contraction.
Darryl Seligman (Yale): Nonlaminar Flow in Protostellar Disks
We examine how type I planetary migration is affected by the presence of nonlaminar flow in the protostellar disk. In the type I regime, a planet does not clear a gap in the disk and its secular motion is driven by torques generated by the wakes it creates in the surrounding disk fluid. Although current numerical and analytical torque calculations generally agree, the inward migration timescales that both approaches predict are much shorter than the average lifetime of protoplanetary disks. We propose that this agreement can be understood by the effective disk viscosity implicitly assumed in both methods. Analytic linear calculations assume steady state laminar flow solutions, while fully nonlinear numerical schemes suffer from significant vortex shedding. We aim to demonstrate that the flow past the planet is unsteady through vorticity preserving numerical schemes, which could significantly alter type I planetary migration timescales.
Robert Stencel (U Denver): Polarimetry of Exoplanets - The Next Frontier
The goals of our efforts include: 1) to develop better understanding of both the detection of exoplanets with polarimetry, as well as analyzing their atmospheres, and 2) create an exoplanet survey polarimeter using Stress-Engineered Optical (SEO) elements. The first goal will be accomplished by refining models of short-period exo-planetary atmospheres (e.g. VLIDORT models by Kopparla, et al. 2016) based on data obtained from observations with the Polarimeter for Inclination Studies of Hot Jupiters (POLISH2, Wiktorowicz et al. 2015). The potential in this approach is that exoplanets with non-transiting inclinations could be detected, as opposed to depending on transit events or direct imaging. From these results and models, a new polarimeter will be designed for the second goal, specifically for the survey and detection of exoplanets with polarimetry. This polarimeter, the Imaging Polarimeter for the Survey of Exoplanets (IMPOLSE), will provide true simultaneous measurements of all four Stokes parameters with a single image by utilizing stress-engineered optics (SEO) that create polarization-dependent point spread functions. We propose to begin surveying selected portions of Kepler space telescope fields of view, focusing on short-period Hot Jupiters, as these fields provide known quantities of exoplanets. This will be done in order to calibrate and refine IMPOLSE via detections of exoplanets, both transiting and non-transiting. At the end this calibration using known exoplanets, studies to adapt IMPOLSE to a larger-scale survey can commence. In addition to its exoplanet data, IMPOLSE will provide an updated polarimetry catalog of each of the stars it measures, in the optical broadband.
Jeff Sudol (West Chester U): On the Possibility of Habitable, Trojan Planets in the Kepler Circumbinary Planetary Systems
We investigate the possibility of habitable, Trojan planets in the Kepler circumbinary planetary systems Kepler-16, Kepler-47, Kepler-453, and Kepler-1647 where a planet resides in the Habitable Zone. For each system, we preformed up to 10,000 separate N-body integrations, each with a one Earth-mass Trojan planet in a random orbit. We find stable orbital configurations are restricted to a narrow range of semi-major axes in all systems. Such stable orbits are restricted to low eccentricities, e < 0.10, for Kepler-16 and Kepler-47, but no such restrictions appear for Kepler-453 and Kepler-1647. No restrictions appear in any other orbital parameters, except in the case of Kepler-16, which exhibits a region of exclusion in inclination. Transit timing variations due to the Trojan planets differ widely. Although Trojan planets may be readily detectable in Kepler-1647 by their magnitude alone, the detection of Trojan planets in Kepler-16 might require decades of transit data?.
Chris Tinney (UNSW Sydney): The Coming Revolution in Exoplanet Observations
The combination of TESS, Gaia and a plethora of new precision Doppler exoplanet follow-up facilities is set to complete change the face of exoplanetary observations. While Kepler has revolutionized our understanding of exoplanets, only a handful of ts system are amenable to the combination of transit and Doppler observation. TESS will deliver thousands of new systems bright enough for 4m and 8m-class telescopes to complement TESS sizes with masses - delivering densities. TESS systems will also be amenable to further searched to find additional planets that don't transit complementing our pictures of these systems architectures. TESS photometry will also deliver the ability (via "flicker" measurements) to find *the most intrinsically stable* systems for blind Doppler planet searches. The result will be an explosion of data on system architectures and planet densities. As TESS is observing the south first, we are prioritizing the implementation of a new laser-comb stabilized precision Doppler system for the Anglo-Australian Telescope called Veloce as well as a spectroscopic survey call FunnelWeb of 3 million bright stars - almost all of the stars that TESS can find planets orbiting, and all 100,000 M dwarfs in the southern sky brighter than G=14.
Su Wang (Purple Mountain Observatory, CAS): The Formation of Planetary Systems in Near Mean Motion Resonances
Plenty of planet pairs in multiple planetary systems detected by the Kepler Mission are trapped near the first order mean motion resonance, 3:2 and 2:1 MMRs. We propose a formation scenario for the planetary configurations near 3:2 and 2:1 MMRs in terrestrial planetary systems. Firstly, the low-mass planets form at a distant region; then they undergo type I migration until they reach the inner region of the gaseous disk and stop migrating. During the migration process, planets are trapped into MMRs. Planet pairs can departure from the exact location of MMRs due to the tidal interactions between the planets and the central star or the depletion of the gas disk. This formation scenario will provide a likely explanation for Kepler candidates involved in 2:1 and 3:2 MMRs. From the statistical results, we find that the proportions near 3:2 and 2:1 MMRs can reach 14.5% and 26.0%, respectively. We also investigate the configuration formation of near 4:2:1 MMRs in the systems with giant planets. Several Earth masses terrestrial planets companion are likely to be produced in these systems from planetesimals. They are finally evolved into near 4:2:1 MMRs. The configurations formed in our work are applicable to the formation of the Kepler-152, 238, and 302 systems. Additionally, considering the effect of mass accretion and different type of gas disk during the planetary formation process, the proportions of planet pairs near 3:2 increases significantly. Based on our simulations, we find that the orbital migration and gas disk depletion play important roles in close-in terrestrial planet formation.
Noriharu Watanabe (SOUKENDAI): The Spin-orbit Misalignment Measured by Doppler Tomography from HDS
The projected-angle between the stellar spin axis and the planetary orbital axis, which is so-called the spin-orbit misalignment, is important to understand how the planetary orbit evolves. Spin-orbit misalignments for transiting exoplanets have been measured either by the Rossiter-Mclaughlin effect or by Doppler Tomography (DT). However, DT has not been tried for data of High Dispersion Spectrograph (HDS) on Subaru telescope. In this research, We have made our own DT code and used it for archive data of HD189733b taken by HDS to measure its misalignment. This work is the first analysis of DT in Japan. We will show the comparison of our result with the previous studies. We also present the status of analysis for other transiting planets taken with the Subaru HDS and future prospects to measure the spin-orbit misalignments of smaller size planets than Neptune around M dwarf stars using Subaru InfraRed Doppler instrument (IRD).
Ka Ho Wong (U Hong Kong): Long-term Stability of Planetary Orbits in the HD 59686 Eccentric Binary System
Despite the large number of extrasolar planets discovered to date, only a few have been found in binary star systems. HD 59686 is special due to the detection of a planet orbiting the primary star of a close and highly eccentric binary system. The binary has orbital semimajor axis of 13.6 AU and eccentricity of 0.73, while the planet with a minimum mass of 7.0 Jupiter mass orbits the primary at 1.1 AU (Ortiz et al. 2016). According to the empirical stability criterion of Holman & Wiegert (1999), the planet may not be stable, if it is in a coplanar prograde orbit. We have explored the stability regions of planetary orbits using numerical orbit integrations of test particles in this binary system. For coplanar prograde orbits, there are extra stable regions beyond the circular stability boundary, if the planetary orbit is eccentric. They are stabilized by alignment with the binary orbit. Some of these stable regions are close to the orbit of the observed planet. For coplanar retrograde orbits, the circular stability boundary extends well beyond the orbit of the observed planet, but there are also extra stable regions due to alignment with the binary orbit.
Ji-Wei Xie (Nanjing U): Where Are The Exomoons?: Impacts Of The Host Stars On Exomoon Systems
Moons are common and outnumber planets in our solar system. In contrast, the detection of an exomoon still remains elusive, though thousands of exoplanets have been detected. Is this just because the signals exomoons are smaller and thus more difficult to detect? Or might there also be a physical reason for fewer exomoons could orbit around the exoplanets we are observing? In this study, we explore a mechanism that could limit the number of exomoons: photo-evaporation. We find that the photoevaporation can trigger the global instability of the moon system. Given the destructive role of photo-evaporation, we speculate that exomoons are less common for close-in planets. The lessons we learn in this study may be helpful for the target selection of on-going/ future exomoon searching programs.
Ji-Wei Xie (Nanjing U): Orbits of Planetary Systems: an Eccentricity Dichotomy, a Common Pattern and the Prevalence of Circular Orbits
While the Solar System planets are on nearly circular (e~0.06) and coplanar ( i~3 deg) orbits, the first several hundred extrasolar planets discovered using the Radial Velocity technique are commonly on eccentric orbits (e~0.3). This raises a fundamental question: Are the Solar System and its formation special? The Kepler mission has found thousands of transiting planets, but most of their orbital eccentricities remain unknown. By using the precise spectroscopic host star parameters from the LAMOST observations, we measure the eccentricity distributions for a large and homogeneous Kepler planet sample. We observe an eccentricity dichotomy, a common orbital pattern and the prevalence of near-circular orbits for planets inside and outside the Solar System, which provide insights to solve the above fundamental question.
Yi Yang (SOKENDAI): Subaru/HiCIAO High-contrast Polarimetry Observations Towards Disks in Binary System GG Tau A
Currently about 200 planets have ever been discovered in binary or multiple systems. Undoubtedly, to understand their formation process, not only theoretical work but also direct observations towards the protoplanetary disks in young binary/multiple star systems is quite necessary. This time I will show the near infrared high-contrast polarimetry observation results towards the circumbinary disk around GG Tau A system by using HiCIAO, a high-contrast observation instrument mounted on the 8.2-m Subaru Telescope. The complicated structures around this young binary system are successfully resolved in our observation, including the famous circumbinary ring, possible circumstellar disk structure around the primary star GG Tau Aa, and a streamer connecting the circumbinary ring and the secondary star, GG Tau Ab. Our observation gives direct evidence that streamers from the outer circumbinary ring could sustain the circumstellar disks around the two binaries. It could be quite beneficial for us to improve current theories of planet formation process in binary systems.
Wei Zhu (OSU): Is There a Planet-Metallicity Correlation for Small Planets?
The planet-metallicity correlation has been well established for giant planets: more metal-rich stars are more likely to host giant planets. However, contradictory results have been found for small planets. In this talk, I will explain why this is the case, by introducing the general form of the planet-metallicity correlation. The ways to uncover this small planet-metallicity correlation will also be discussed.
María Alejandra Jiménez Zúñiga (UNAM): Improved Torque Formula for Type I Planetary Migration
The migration of planets on nearly circular, non-inclined orbits in protoplanetary discs is entirely described by the discs torque. This torque is a complex function of the disc parameters, and essentially amounts to the sum of two components: the Lindblad torque and the corotation torque. Known torque formulae do not reproduce accurately the torque actually experienced in numerical simulations by low-mass planets in radiative discs, the main reason being that these formulae have been worked out by means of two-dimensional analysis. We revisit the torque formula and update many of its dimensionless coefficients by means of tailored, three-dimensional numerical simulations. In particular, we derive the dependence of the Lindblad torque on the temperature gradient, the dependence of the corotation torque on the radial entropy gradient (and work out a suitable expression of this gradient in a three-dimensional disc). We also work out the dependence of the corotation torque on the radial temperature gradient, overlooked so far. Corotation torques are known to scale very steeply with the width of the horseshoe region. We extend the expression of this width to the domain of intermediate mass planets, so that our updated torque formula remains valid for planets up to typically several tens of Earth masses, provided the disc parameters are such that these relatively massive planets do not significantly deplete their coorbital region. Our torque expression can be applied to low- and intermediate-mass planets in protoplanetary discs, as well as protomoons embedded in circumplanetary discs.
