A torque formula for non-isothermal type I planetary migration – I. Unsaturated horseshoe drag

By Sijme-Jan Paardekooper, C. Baruteau, A. Crida, W. Kley. 2009

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We study the torque on low-mass planets embedded in protoplanetary discs in the two-dimensional approximation, incorporating non-isothermal eects. We couple linear estimates of the Lindblad (or wave) torque to a simple, but non-linear, model of adiabatic corotation torques (or horseshoe drag), resulting in a simple formula that governs Type I migration in non-isothermal discs. This formula should apply in optically thick regions of the disc, where viscous and thermal diusion act to keep the horseshoe drag unsaturated. We check this formula against numerical hydrodynamical simulations, using three independent numerical methods, and nd good agreement.

Subfields: Astronomy and Astrophysics, Instrumentation

View Paper: https://www.semanticscholar.org/paper/19f43c8112695b105804c459cfafb99047903e9d

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Citations in the corpus

Breaking the chains: hot super-Earth systems from migration and disruption of compact resonant chains (2017, background): “(5) Following Paardekooper et al. (2010, 2011) and assuming a gravitational smoothing length for the planet potential of b = 0.4h, the total torque from the gas experienced by a type-I migrating planet can be expressed by tot = L L + C C, (6) where L is the Lindblad torque and C is the corotation…”
A torque formula for non-isothermal Type I planetary migration – II. Effects of diffusion (2010, background): “In Paardekooper et al. (2010), hereafter Paper I, we studied the non-linear corotation torque, or horseshoe drag (Ward 1991), in the two-dimensional adiabatic limit.”
TRANSITIONAL DISKS AS SIGNPOSTS OF YOUNG, MULTIPLANET SYSTEMS (2011, background): “…of planet migration in viscous disks with nonzero thermal diffusivity demonstrates that the corotation torque may halt inward migration or even drive outward migration (Paardekooper & Mellema 2006, Kley & Crida 2008, Kley et al. 2009, Paardekooper et al. 2010, Paardekooper et al. 2011).”
Rapid inward migration of planets formed by gravitational instability (2011, background): “…and driven by a positive entropy gradient in our disc model (Baruteau & Masset 2008a; Paardekooper & Papaloizou c© 2011 RAS, MNRAS 000, 1–13 2008; Masset & Casoli 2009; Paardekooper et al. 2010, in our case, the radial profile of the disc entropy scales in proportion to the orbital separation).”
Characterization of exoplanets from their formation - II. The planetary mass-radius relationship (2012, result): “Based on the results of Paardekooper & Mellema (2006), Baruteau & Masset (2008), Kley & Crida (2008), Kley et al. (2009) and Paardekooper et al. (2010), we now distinguish several sub-types of type I migration which lead to different migration directions and rates.”
THE IMPRINT OF EXOPLANET FORMATION HISTORY ON OBSERVABLE PRESENT-DAY SPECTRA OF HOT JUPITERS (2016, background): “The total torque Γtot can be expressed as (Paardekooper et al. 2010) Γtot = 1 γ (C0 + C1 pΣ + C2 pT) (q h )2 Σa4Ω2.”
A FRAMEWORK FOR CHARACTERIZING THE ATMOSPHERES OF LOW-MASS LOW-DENSITY TRANSITING PLANETS (2013, background): “…combines several simple, standard prescriptions for the growth of the solid core and the gaseous envelope (Pollack et al. 1996), the evolution of the protoplanetary disk (Lynden-Bell & Pringle 1974; Papaloizou & Terquem 1999), and disk migration (Lin & Papaloizou 1986; Paardekooper et al. 2010).”
TOWARD A DETERMINISTIC MODEL OF PLANETARY FORMATION. VI. DYNAMICAL INTERACTION AND COAGULATION OF MULTIPLE ROCKY EMBRYOS AND SUPER-EARTH SYSTEMS AROUND SOLAR-TYPE STARS (2010, background): “However, at special locations where the gas surface density or entropy attains local maxima, these embryos’ type I migration may be stalled due to their tidal interaction with the disk gas near their co-orbital region (Masset et al. 2006; Paardekooper et al. 2010).”
PLANET OCCURRENCE WITHIN 0.25 AU OF SOLAR-TYPE STARS FROM KEPLER (2011, background): “The latest versions of the population synthesis models (Ida & Lin 2010; Alibert et al. 2011) offer improvements, including non-isothermal treatment of the disk (Paardekooper et al. 2010) and multiple, interacting planet embryos per simulation.”
DIRECT IMAGING OF A COLD JOVIAN EXOPLANET IN ORBIT AROUND THE SUN-LIKE STAR GJ 504 (2013, background): “If multiple planets are formed by CA in the inner regions, one or more of them could be scattered outward due to dynamical interactions among the planets (Nagasawa et al. 2008) or between the planets and the protoplanetary disk (Paardekooper et al. 2010; Crida et al. 2009).”
Forming the cores of giant planets from the radial pebble flux in protoplanetary discs (2014, methodology): “Kretke & Lin (2012) give an overview of the migration rates in power-law discs (Tanaka et al. 2002; Paardekooper et al. 2010, 2011).”
TESTING IN SITU ASSEMBLY WITH THE KEPLER PLANET CANDIDATE SAMPLE (2013, background): “…of migration depend critically on the interaction between the planetary forcing and the structure of the disk (e.g., Terquem 2003; Laughlin et al. 2004; Johnson et al. 2006; Paardekooper & Mellema 2006; Paardekooper et al. 2010; Bromley & Kenyon 2011; Hasegawa & Pudritz 2011; Kretke & Lin 2012).”
WIND-DRIVEN ACCRETION IN PROTOPLANETARY DISKS. I. SUPPRESSION OF THE MAGNETOROTATIONAL INSTABILITY AND LAUNCHING OF THE MAGNETOCENTRIFUGAL WIND (2013, background): “When planets have formed, planet migration via planet–disk interaction also depends on the disk radial profile and diffusion processes (e.g., Paardekooper et al. 2010; Baruteau et al. 2011).”
Dynamics of pebbles in the vicinity of a growing planetary embryo: hydro-dynamical simulations (2012, result): “Recent results on migration in radiatively cooling disks (Paardekooper & Mallema 2006; Baruteau & Masset 2008; Kley & Crida 2008; Paardekooper et al. 2010; Masset & Casoli 2010; Bitsch & Kley 2011) show that the embryos migrate from all directions toward an orbital radius where migration is…”
Compositional Imprints in Density–Distance–Time: A Rocky Composition for Close-in Low-mass Exoplanets from the Location of the Valley of Evaporation (2017, background): “The frequency of such planets would thus serve as an important observational constraint for Type I planet migration models (e.g., Paardekooper et al. 2010; Dittkrist et al. 2014).”
Formation and Evolution of Compact-object Binaries in AGN Disks (2019, neutral): “…, 17type I mig cen sat cen disk 2 disk 2 1 ( )⎜ ⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝ ⎞ ⎠ where fmig is a dimensionless factor depending on the local temperature and density profiles (see Paardekooper et al. 2010; Baruteau et al. 2011), hdisk is the half-thickness of the disk, Mcen and Msat are the…”