Model of Angular Momentum Transport at the Protoplane-tary Disk Evolution and Disk Surface Density
DOI:
https://doi.org/10.15407/ujpe66.11.921Keywords:
protoplanetary disk, viscous torques, angular momentum, planet, accretion disk, surface densityAbstract
We consider how the tidal effect of a protoplanetary disk interaction can be incorporated into calculations of its viscous evolution. The evolution of the disk occurs under the action of both internal viscous torques and external torques resulting from the presence of one or more embedded planets. The planets migrate under the effect of their tidal interaction with the disk (in the type-II migration regime). Torques on a planet are caused by its gravitational interaction with the density waves which occupy the Lindblad resonances in the disk. Our model simplifies the functional form of the rate of injection of the angular momentum Λ(r) to construct and solve the evolution equation for a disk and an embedded protoplanet. The functional Λ(r) depends on the tidal dissipation distribution in the disk which is concentrated in a vicinity of the protoplanet’s orbit. We have found an analytic solution for the disk surface density.
References
D.N.C. Lin, J. Papaloizou. On the tidal interaction between protoplanets and the protoplanetary disc. III - Orbital migration of protoplanets. Astrophysical J. 309, 846 (1986).
https://doi.org/10.1086/164653
P. Goldreich, S. Tremaine. Disk-Satellite interactions. Astrophysical J. 241, 425 (1980).
https://doi.org/10.1086/158356
W. Kley, R.P. Nelson. Planet-disk interaction and orbital evolution. ARA&A 50, 211(2012).
https://doi.org/10.1146/annurev-astro-081811-125523
C. Baruteau, Q. Massey. In: Tidal effects in astronomy and astrophysics. Lecture Notes in Physics 201, 861 (2013).
https://doi.org/10.1007/978-3-642-32961-6_6
C. Baruteau, A. Crida, S.J. Paardekooper, et al. Planetdisc interactions and early evolution of planetary systems. arXiv: 1312.4293 (2013).
https://doi.org/10.2458/azu_uapress_9780816531240-ch029
G. Laughlin, A. Steinacker, F.C. Adams. Type I planetary migration with MHD turbulence. Astrophysical J. 608, 489 (2004).
https://doi.org/10.1086/386316
S.J. Paardekooper, C. Baruteau, A. Crida, W. Kley. A torque formula for non-isothermal type I planetary migration - I. Unsaturated horseshoe drag. MNRAS 401, 1950 (2010).
https://doi.org/10.1111/j.1365-2966.2009.15782.x
S. J. Paardekooper, C. Baruteau, W. Kley. A torque formula for non-isothermal type I planetary migration - II. Effects of diffusion. MNRAS 410, 293 (2011).
https://doi.org/10.1111/j.1365-2966.2010.17442.x
W.R. Ward. Protoplanet migration by nebula tides. ICARUS 126, 261 (1997).
https://doi.org/10.1006/icar.1996.5647
W.R. Ward.Comments of the long-term stability of the Earth's obliquity. ICARUS 50, 444 (1982).
https://doi.org/10.1016/0019-1035(82)90134-8
R.G. Edgar. Giant planet migration in viscous power-law discs. Astrophysical J. 663, 1325 (2007).
https://doi.org/10.1086/518591
R.G. Edgar. Type II migration: Varying planet mass and disc viscosity. arXiv:0807.0625 (2008).
S. J. Paardekooper. Dynamical corotation torques on lowmass planets. MNRAS 444, 2031 (2014).
https://doi.org/10.1093/mnras/stu1542
A. Crida, A. Morbidelli. Cavity opening by a giant planet in a protoplanetary disc and effects on planetary migration. MNRAS 377, 1324(2007).
https://doi.org/10.1111/j.1365-2966.2007.11704.x
D. Lynden-Bell, J. E. Pringle. The evolution of viscous discs and the origin of the nebular variables. MNRAS 168, 603 (1974).
https://doi.org/10.1093/mnras/168.3.603
T. Tanaka. 2011. Exact time-dependent solutions for the thin accretion disc equation: Boundary conditions at finite radius. MNRAS 410, 1007 (2011).
https://doi.org/10.1111/j.1365-2966.2010.17496.x
P.J. Armitage, M. Livio, S.H. Lubow, J.E. Pringle. Predictions for the frequency and orbital radii of massive extrasolar planets. MNRAS 334, 248 (2002).
https://doi.org/10.1046/j.1365-8711.2002.05531.x
D.E. Trilling, W. Benz, T. Guillot, J.I. Lunine, W.B. Hubbard, A. Burrows. Orbital evolution and migration of giant planets: Modeling extrasolar planets. Astrophysical J. 500, 428 (1998).
https://doi.org/10.1086/305711
J. Papaloizou, D.N.C. Lin. On the tidal interaction between protoplanets and the primordial solar nebula. I - Linear calculation of the role of angular momentum exchange. Astrophysical J. 285, 818 (1984).
https://doi.org/10.1086/162561
J.E. Pringle. Accretion discs in astrophysics. ARA&A 19, 137 (1981).
https://doi.org/10.1146/annurev.aa.19.090181.001033
I.S. Gradshteyn, I.M. Ryzhik. Table of Integrals, Series and Products (Academic Press, 1980) [ISBN-13: 978-0-12-373637-6].
P.J. Armitage. Astrophysics of Planet Formation (Cambridge Univ. Press, 2010) [ISBN-13: 978-0-511-69136-2].
D. Lin, J. Papaloizou. On the tidal interaction between protoplanets and the protoplanetary disk. III. Orbital migration of protoplanets. Astrophysical J. 309, 846 (1986).
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