Editing Planetary migration (section)

=== Disk migration ===
<em>Disk migration</em> arises from the gravitational force exerted by a sufficiently massive body embedded in a disk on the surrounding disk's gas, which perturbs its density distribution.  By the [[Reaction (physics)|reaction]] principle of [[classical mechanics]], the gas exerts an equal and opposite gravitational force on the body, which can also be expressed as a [[torque]].  This torque alters the [[angular momentum]] of the planet's orbit, resulting in a variation of the [[semi-major axis]] and other orbital elements.  An increase over time of the semi-major axis leads to ''outward migration'', i.e., away from the star, whereas the opposite behavior leads to ''inward migration''.

Three sub-types of disk migration are distinguished as Types&nbsp;I, II, and III. The numbering is '''not''' intended to suggest a sequence or stages.

==== Type I migration ====
Small planets undergo <em>Type&nbsp;I disk migration</em> driven by torques arising from Lindblad and co-rotation resonances. [[Lindblad resonance]]s excite [[spiral density wave]]s in the surrounding gas, both interior and exterior of the planet's orbit.  In most cases, the outer spiral wave exerts a greater torque than does the inner wave, causing the planet to lose angular momentum, and hence migrate toward the star. The migration rate due to these torques is proportional to the mass of the planet and to the local gas density, and results in a migration timescale that tends to be short relative to the million-year lifetime of the gaseous disk.<ref name=li2011>{{cite book |author1=Lubow, S.H. |author2=Ida, S. |chapter=Planet Migration |bibcode=2010exop.book..347L |title=Exoplanets |publisher=University of Arizona Press, Tucson, AZ |editor=Seager, S. |pages=347–371 |date=2011 |chapter-url=http://www.uapress.arizona.edu/Books/bid2263.htm |arxiv=1004.4137}}</ref>  Additional co-rotation torques are also exerted by gas orbiting with a period similar to that of the planet. In a reference frame attached to the planet, this gas follows [[horseshoe orbit]]s, reversing direction when it approaches the planet from ahead or from behind. The gas reversing course ahead of the planet originates from a larger semi-major axis and may be cooler and denser than the gas reversing course behind the planet. This may result in a region of excess density ahead of the planet and of lesser density behind the planet, causing the planet to gain angular momentum.<ref name="Paadekooper_Mellema_2006">{{cite journal |last1=Paardekooper |first1=S.-J. |last2=Mellema |first2=G. |title=Halting type&nbsp;I planet migration in non-isothermal disks |journal=Astronomy and Astrophysics |date=2006 |volume=459 |issue=1 |pages=L17–L20 |doi=10.1051/0004-6361:20066304|arxiv=astro-ph/0608658 |bibcode=2006A&A...459L..17P|s2cid=15363298 }}</ref><ref name="Brasser_etal_2017">{{cite journal |last1=Brasser |first1=R. |last2=Bitsch |first2=B. |last3=Matsumura |first3=S. |title=Saving super-Earths: Interplay between pebble accretion and type&nbsp;I migration |date=2017 |arxiv=1704.01962 |doi=10.3847/1538-3881/aa6ba3 |volume=153 |issue=5 |journal=The Astronomical Journal |page=222 |bibcode=2017AJ....153..222B|s2cid=119065760 |doi-access=free }}</ref> 

The planet mass for which migration can be approximated to Type&nbsp;I depends on the local gas pressure [[scale height]] and, to a lesser extent, the kinematic [[viscosity]] of the gas.<ref name="li2011" /><ref name=dangelo_lubow_2010 /> In warm and viscous disks, Type I migration may apply to larger mass planets. In locally isothermal disks and far from steep density and temperature gradients, co-rotation torques are generally overpowered by the [[Lindblad resonance|Lindblad]] torques.<ref name=tanaka_etal_2002>{{cite journal |author1=Tanaka, H. |author2=Takeuchi, T. |author3=Ward, W.R. |title=Three-Dimensional Interaction between a Planet and an Isothermal Gaseous Disk: I. Corotation and Lindblad Torques and Planet Migration
|journal=The Astrophysical Journal |date=2002 |volume=565 |issue=2 |pages=1257–1274 |doi=10.1086/324713 |bibcode=2002ApJ...565.1257T |doi-access=free }}</ref><ref name=dangelo_lubow_2010>{{cite journal |author1=D'Angelo, G. |author2=Lubow, S.H. |title=Three-dimensional disk-planet torques in a locally isothermal disk |journal=The Astrophysical Journal |date=2010 |volume=724 |issue=1 |pages=730–747 |doi=10.1088/0004-637X/724/1/730 |arxiv=1009.4148 |bibcode=2010ApJ...724..730D|s2cid=119204765 }}</ref> Regions of outward migration may exist for some planetary mass ranges and disk conditions in both local isothermal and non-isothermal disks.<ref name=dangelo_lubow_2010 /><ref name="Lega_etal_2014">{{cite journal |last1=Lega |first1=E. |last2=Morbidelli |first2=A. |last3=Bitsch |first3=B. |last4=Crida |first4=A. |last5=Szulágyi |first5=J. |title=Outwards migration for planets in stellar irradiated 3D discs |journal=Monthly Notices of the Royal Astronomical Society |date=2015 |volume=452 |issue=2 |pages=1717–1726 |doi=10.1093/mnras/stv1385 |doi-access=free |arxiv=1506.07348 |bibcode=2015MNRAS.452.1717L|s2cid=119245398 }}</ref> The locations of these regions may vary during the evolution of the disk, and in the local-isothermal case are restricted to regions with large density and/or temperature radial gradients over several pressure scale-heights. Type I migration in a local isothermal disk was shown to be compatible with the formation and long-term evolution of some of the observed [[Kepler (spacecraft)|Kepler]] planets.<ref name=dangelo_bodenheimer_2016>{{Cite journal |author1=D'Angelo, G. |author2=Bodenheimer, P. |title=In-situ and ex-situ formation models of Kepler&nbsp;11 planets |journal=The Astrophysical Journal |year=2016 |volume=828 |issue=1 |at=id. 33 (32 pp.) |doi=10.3847/0004-637X/828/1/33 |arxiv=1606.08088 |bibcode=2016ApJ...828...33D|s2cid=119203398 |doi-access=free }}</ref> The rapid accretion of solid material by the planet may also produce a "heating torque" that causes the planet to gain angular momentum.<ref name="Benitez_Llambay_etal_2015">{{cite journal |last1=Benítez-Llambay |first1=Pablo |last2=Masset |first2=Frédéric |last3=Koenigsberger |first3=Gloria |author3-link=Gloria Suzanne Koenigsberger Horowitz |last4=Szulágyi |first4=Judit |title=Planet heating prevents inward migration of planetary cores |journal=Nature |date=2015 |volume=520 |issue=7545 |pages=63–65 |doi=10.1038/nature14277 |pmid=25832403 |arxiv=1510.01778 |bibcode=2015Natur.520...63B|s2cid=4466971 }}</ref>

==== Type II migration ====
A planet massive enough to open a gap in a gaseous disk undergoes a regime referred to as <em>Type&nbsp;II disk migration</em>. When the mass of a perturbing planet is large enough, the tidal torque it exerts on the gas transfers angular momentum to the gas exterior of the planet's orbit, and does the opposite interior to the planet, thereby repelling gas from around the orbit. In a Type&nbsp;I regime, viscous torques can efficiently counter this effect by resupplying gas and smoothing out sharp density gradients. But when the torques become strong enough to overcome the viscous torques in the vicinity of the planet's orbit, a lower density annular gap is created. The depth of this gap depends on the temperature and viscosity of the gas and on the planet mass. In the simple scenario in which no gas crosses the gap, the migration of the planet follows the viscous evolution of the disk's gas. In the inner disk, the planet spirals inward on the viscous timescale, following the accretion of gas onto the star. In this case, the migration rate is typically slower than would be the migration of the planet in the Type&nbsp;I regime. In the outer disk, however, migration can be outward if the disk is viscously expanding. A Jupiter-mass planet in a typical protoplanetary disk is expected to undergo migration at approximately the Type II rate, with the transition from Type&nbsp;I to Type&nbsp;II occurring at roughly the mass of Saturn, as a partial gap is opened.<ref name=dangelo_etal_2003 /><ref name=dangelo_lubow_2008 /> 

Type II migration is one explanation for the formation of [[hot Jupiter]]s.<ref name="Armitage_2007">{{Cite journal |last1=Armitage |first1=Phillip J. |title=Lecture notes on the formation and early evolution of planetary systems |arxiv=astro-ph/0701485 |bibcode=2007astro.ph..1485A|year=2007 }}</ref> In more realistic situations, unless extreme thermal and viscosity conditions occur in a disk, there is an ongoing flux of gas through the gap.<ref name=lubow_dangelo_2006>{{cite journal |author1=Lubow, S. |author2=D'Angelo, G. |title=Gas flow across gaps in protoplanetary disks |journal=The Astrophysical Journal |date=2006 |volume=641 |issue=1|pages=526–533 |doi=10.1086/500356 |arxiv=astro-ph/0512292 |bibcode=2006ApJ...641..526L|s2cid=119541915 }}</ref> As a consequence of this mass flux, torques acting on a planet can be susceptible to local disk properties, akin to torques at work during Type&nbsp;I migration. Therefore, in viscous disks, Type&nbsp;II migration can be typically described as a modified form of Type I migration, in a unified formalism.<ref name=dangelo_lubow_2008>{{cite journal |author1=D'Angelo, G. |author2=Lubow, S. H. |title=Evolution of migrating planets undergoing gas accretion |journal=The Astrophysical Journal |date=2008 |volume=685 |issue=1 |pages=560–583 |doi=10.1086/590904 |arxiv=0806.1771 |bibcode=2008ApJ...685..560D|s2cid=84978 }}</ref><ref name=dangelo_lubow_2010 /> The transition between Type&nbsp;I and Type&nbsp;II migration is generally smooth, but deviations from a smooth transition have also been found.<ref name=dangelo_etal_2003>{{cite journal |author1=D'Angelo, G. |author2=Kley, W. |author3=Henning T. |title=Orbital migration and mass accretion of protoplanets in three-dimensional global computations with nested grids| journal=The Astrophysical Journal |date=2003 |volume=586 |issue=1 |pages=540–561 |doi=10.1086/367555 |arxiv=astro-ph/0308055 |bibcode=2003ApJ...586..540D|s2cid=14484931 }}</ref><ref name=masset_etal_2006>{{cite journal |author1=Masset, F.S. |author2=D'Angelo, G. |author3=Kley, W. |title=On the migration of protogiant solid cores |journal=The Astrophysical Journal |date=2006 |volume=652 |issue=1 |pages=730–745 |doi=10.1086/507515 |arxiv=astro-ph/0607155 |bibcode=2006ApJ...652..730M|s2cid=17882737 }}</ref> In some situations, when planets induce eccentric perturbation in the surrounding disk's gas, Type&nbsp;II migration may slow down, stall, or reverse.<ref name=dangelo_etal_2006>{{cite journal |arxiv=astro-ph/0608355 |title=Evolution of Giant Planets in Eccentric Disks |journal=The Astrophysical Journal |volume=652 |issue=2 |pages=1698–1714 |last1=D'Angelo |first1=Gennaro |last2=Lubow |first2=Stephen H. |last3=Bate |first3=Matthew R. |year=2006 |doi=10.1086/508451 |bibcode=2006ApJ...652.1698D|s2cid=53135965 }}</ref>

From a physical viewpoint, Type I and Type II migration are driven by the same type of torques (at Lindblad and co-rotation resonances). In fact, they can be interpreted and modeled as a single regime of migration, that of Type I appropriately modified by the perturbed gas surface density of the disk.<ref name=dangelo_lubow_2008 /><ref name=dangelo_lubow_2010 />

==== Type III disk migration ====
<em>Type III disk migration</em> applies to fairly extreme disk / planet cases and is characterized by extremely short migration timescales.<ref name=masset_2003>{{cite journal |author1=Masset, F.S. |author2=Papaloizou, J.C.B. |title=Runaway migration and the formation of hot Jupiters |journal=The Astrophysical Journal |date=2003 |volume=588 |issue=1 |pages=494–508 |doi=10.1086/373892 |arxiv=astro-ph/0301171 |bibcode=2003ApJ...588..494M|s2cid=7483596 }}</ref><ref name=dangelo_2005>{{cite journal |author1=D'Angelo, G. |author2=Bate, M.R.B. |author3=Lubow, S.H. |title=The dependence of protoplanet migration rates on co-orbital torques |journal=Monthly Notices of the Royal Astronomical Society |date=2005 |volume=358 |issue=2 |pages=316–332 |doi=10.1111/j.1365-2966.2005.08866.x |doi-access=free |arxiv=astro-ph/0411705 |bibcode=2005MNRAS.358..316D|s2cid=14640974 }}</ref><ref name="dangelo_lubow_2008"/> Although sometimes referred to as "runaway migration", the migration rate does not necessarily increase over time.<ref name="masset_2003"/><ref name=dangelo_2005 /> '''Type&nbsp;III''' migration is driven by the co-orbital torques from gas trapped in the planet's [[Lagrangian point|libration regions]] and from an initial, relatively fast, planetary radial motion. The planet's radial motion displaces gas in its co-orbital region, creating a density asymmetry between the gas on the leading and the trailing side of the planet.<ref name="dangelo_lubow_2008"/><ref name="li2011"/> Type III migration applies to disks that are relatively massive and to planets that can only open partial gaps in the gas disk.<ref name="li2011"/><ref name="dangelo_lubow_2008"/><ref name="masset_2003"/> Previous interpretations linked Type&nbsp;III migration to gas streaming across the orbit of the planet in the opposite direction as the planet's radial motion, creating a positive feedback loop.<ref name=masset_2003 /> Fast outward migration may also occur temporarily, delivering giant planets to distant orbits, if later Type&nbsp;II migration is ineffective at driving the planets back.<ref name="Pierens_Raymond_2016">{{cite journal |last1=Pierens |first1=A. |last2=Raymond |first2=S.N. |title=Migration of accreting planets in radiative discs from dynamical torques|journal=Monthly Notices of the Royal Astronomical Society |date=2016 |volume=462 |issue=4 |pages=4130–4140 |doi=10.1093/mnras/stw1904 |doi-access=free |arxiv=1608.08756 |bibcode=2016MNRAS.462.4130P|s2cid=119225370 }}</ref>