Abstract
We introduce a general mathematical framework to model the internal transport of angular momentum in a star hosting a close-in planetary/stellar companion. By assuming that the tidal and rotational distortions are small and that the deposit/extraction of angular momentum induced by stellar winds and tidal torques are redistributed solely by an effective eddy-viscosity that depends on the radial coordinate, we can formulate the model in a completely analytic way. It allows us to compute simultaneously the evolution of the orbit of the companion and of the spin and the radial differential rotation of the star. An illustrative application to the case of an F-type main-sequence star hosting a hot Jupiter is presented. The general relevance of our model to test more sophisticated numerical dynamical models and to study the internal rotation profile of exoplanet hosts, submitted to the combined effects of tides and stellar winds, by means of asteroseismology are discussed.
Similar content being viewed by others
Notes
For more details, the reader can apply the integral calculator available through the Wolfram website at: http://www.wolframalpha.com/calculators/integral_calculator.
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions (1972)
Aerts, C.: The age and interior rotation of stars from asteroseismology. Astronomische Nachrichten 336:477 (2015). doi:10.1002/asna.201512177, arXiv:1503.06690
Albrecht, S., Winn, J.N., Johnson, J.A., Howard, A.W., Marcy, G.W., et al.: Obliquities of hot jupiter host stars: evidence for tidal interactions and primordial misalignments. Astrophys. J. 757, 18 (2012). doi:10.1088/0004-637X/757/1/18. arXiv:1206.6105
Auclair Desrotour, P., Mathis, S., Le Poncin-Lafitte, C.: Scaling laws to understand tidal dissipation in fluid planetary regions and stars I. Rotation, stratification and thermal diffusivity. Astron. Astrophys. 581, A118 (2015). doi:10.1051/0004-6361/201526246, arXiv:1506.07705
Augustson, K.C., Brun, A.S., Toomre, J.: Dynamo action and magnetic cycles in F-type stars. Astrophys. J. 777, 153 (2013). doi:10.1088/0004-637X/777/2/153
Baglin, A., Auvergne, M., Barge, P., Deleuil, M., Catala, C., COROT Team.: Scientific objectives for a minisat: CoRoT. In: Fridlund, M, Baglin, A, Lochard, J, Conroy, L. (eds.) The CoRoT Mission Pre-Launch Status—Stellar Seismology and Planet Finding, vol 1306, p. 33. ESA Special Publication (2006)
Barker, A.J.: Three-dimensional simulations of internal wave breaking and the fate of planets around solar-type stars. Mon. Not. R. Astron. Soc. 414, 1365–1378 (2011). doi:10.1111/j.1365-2966.2011.18468.x. arXiv:1102.0857
Barker, A.J., Ogilvie, G.I.: On the tidal evolution of Hot Jupiters on inclined orbits. Mon. Not. R. Astron. Soc. 395, 2268–2287 (2009). doi:10.1111/j.1365-2966.2009.14694.x. arXiv:0902.4563
Barker, A.J., Ogilvie, G.I.: On internal wave breaking and tidal dissipation near the centre of a solar-type star. Mon. Not. R. Astron. Soc. 404, 1849–1868 (2010). doi:10.1111/j.1365-2966.2010.16400.x. arXiv:1001.4009
Baruteau, C., Rieutord, M.: Inertial waves in a differentially rotating spherical shell. J. Fluid Mech. 719, 47–81 (2013). doi:10.1017/jfm.2012.605. arXiv:1203.4347
Benomar, O., Takata, M., Shibahashi, H., Ceillier, T., García, R.A.: Nearly uniform internal rotation of solar-like main-sequence stars revealed by space-based asteroseismology and spectroscopic measurements. Mon. Not. R. Astron. Soc. 452, 2654–2674 (2015). doi:10.1093/mnras/stv1493, arXiv:1507.01140
Böhm-Vitense, E.: Über die Wasserstoffkonvektionszone in Sternen verschiedener Effektivtemperaturen und Leuchtkräfte. Mit 5 Textabbildungen. Zeitschrift für Astrophys. 46:108 (1958)
Bolmont, E., Raymond, S.N., Leconte, J., Matt, S.P.: Effect of the stellar spin history on the tidal evolution of close-in planets. Astron. Astrophys. 544, A124 (2012). doi:10.1051/0004-6361/201219645. arXiv:1207.2127
Borucki, W.J., Koch, D., Basri, G., Batalha, N., Brown, T., et al.: Kepler planet-detection mission: introduction and first results. Science 327, 977 (2010). doi:10.1126/science.1185402
Bouvier, J.: Lithium depletion and the rotational history of exoplanet host stars. Astron. Astrophys. 489, L53–L56 (2008). doi:10.1051/0004-6361:200810574. arXiv:0808.3917
Browning, M.K., Miesch, M.S., Brun, A.S., Toomre, J.: Dynamo action in the solar convection zone and tachocline: pumping and organization of toroidal fields. Astrophys. J. Letters 648, L157–L160 (2006). doi:10.1086/507869. arXiv:astro-ph/0609153
Brun, A.S.: Rotation and magnetism of solar-like stars: from scaling laws to spot-dynamos. In: Petit, P., Jardine, M., Spruit, H.C. (eds.) Magnetic Fields Throughout Stellar Evolution, IAU Symposium, vol 302, pp. 114–125 (2014). doi:10.1017/S1743921314001859
Brun, A.S., Toomre, J.: Turbulent convection under the influence of rotation: sustaining a strong differential rotation. Astrophys. J. 570, 865–885 (2002). doi:10.1086/339228. arXiv:astro-ph/0206196
Brun, A.S., Turck-Chièze, S., Zahn, J.P.: Standard solar models in the light of new helioseismic constraints. II. Mixing below the convective zone. Astrophys. J. 525, 1032–1041 (1999). doi:10.1086/307932. arXiv:astro-ph/9906382
Brun, A.S., Miesch, M.S., Toomre, J.: Global-scale turbulent convection and magnetic dynamo action in the solar envelope. Astrophys. J. 614, 1073–1098 (2004). doi:10.1086/423835. arXiv:astro-ph/0610073
Ceillier, T., van Saders, J., García, R.A., Metcalfe, T.S., Creevey, O., et al.: Rotation periods and seismic ages of KOIs - comparison with stars without detected planets from Kepler observations. Mon. Not. R. Astron. Soc. 456, 119–125 (2016). doi:10.1093/mnras/stv2622, arXiv:1510.09023
Chaplin, W.J., Kjeldsen, H., Christensen-Dalsgaard, J., Basu, S., Miglio, A., et al.: Ensemble asteroseismology of solar-type stars with the NASA kepler mission. Science 332, 213 (2011). doi:10.1126/science.1201827. arXiv:1109.4723
Charbonneau, D., Brown, T.M., Latham, D.W., Mayor, M.: Detection of planetary transits across a sun-like star. Astrophys. J. Letters 529, L45–L48 (2000). doi:10.1086/312457. arXiv:astro-ph/9911436
Charbonneau, P.: Dynamo models of the solar cycle. Living Rev. Solar Phys. 7 (2010). doi:10.12942/lrsp-2010-3
Charbonnel, C., Talon, S.: Influence of gravity waves on the internal rotation and Li Abundance of solar-type stars. Science 309, 2189–2191 (2005). doi:10.1126/science.1116849. arXiv:astro-ph/0511265
Damiani, C., Lanza, A.F.: Evolution of angular-momentum-losing exoplanetary systems. Revisiting darwin stability. Astron. Astrophys. 574, A39 (2015). doi:10.1051/0004-6361/201424318. arXiv:1411.3802
Davies, G.R., Chaplin, W.J., Farr, W.M., García, R.A., Lund, M.N., et al.: Asteroseismic inference on rotation, gyrochronology and planetary system dynamics of 16 Cygni. Mon. Not. R. Astron. Soc. 446, 2959–2966 (2015). doi:10.1093/mnras/stu2331. arXiv:1411.1359
Denissenkov, P.A., Pinsonneault, M., MacGregor, K.B.: What prevents internal gravity waves from disturbing the solar uniform rotation? Astrophys. J. 684, 757–769 (2008). doi:10.1086/589502. arXiv:0801.3622
Dobbs-Dixon, I., Lin, D.N.C., Mardling, R.A.: Spin-orbit evolution of short-period planets. Astrophys. J. 610, 464–476 (2004). doi:10.1086/421510. arXiv:astro-ph/0408191
Fabrycky, D.C., Lissauer, J.J., Ragozzine, D., Rowe. J.F., Steffen, J.H., et al.: Architecture of Kepler’s multi-transiting systems. II. New investigations with twice as many candidates. Astrophys. J. 790, 146 (2014). doi:10.1088/0004-637X/790/2/146. arXiv:1202.6328
Favier, B., Barker, A.J., Baruteau, C., Ogilvie, G.I.: Non-linear evolution of tidally forced inertial waves in rotating fluid bodies. Mon. Not. R. Astron. Soc. 439, 845–860 (2014). doi:10.1093/mnras/stu003. arXiv:1401.0643
Ferraz-Mello, S., Tadeu dos Santos, M., Folonier, H., Czismadia, S., do Nascimento, J.D., Jr., Pätzold, M.: Interplay of tidal evolution and stellar wind braking in the rotation of stars hosting massive close-in planets. Astrophys. J. 807, 78 (2015). doi:10.1088/0004-637X/807/1/78. arXiv:1503.04369
Gallet, F., Bouvier, J.: Improved angular momentum evolution model for solar-like stars. Astron. Astrophys. 556, A36 (2013). doi:10.1051/0004-6361/201321302. arXiv:1306.2130
Gallet, F., Bouvier, J.: Improved angular momentum evolution model for solar-like stars. II. Exploring the mass dependence. Astron. Astrophys. 577, A98 (2015). doi:10.1051/0004-6361/201525660. arXiv:1502.05801
García, R.A., Turck-Chièze, S., Jiménez-Reyes, S.J., Ballot, J., Pallé, P.L., et al.: Tracking solar gravity modes: the dynamics of the solar core. Science 316, 1591 (2007). doi:10.1126/science.1140598
García, R.A., Ceillier, T., Salabert, D., Mathur, S., van Saders, J.L., et al.: Rotation and magnetism of Kepler pulsating solar-like stars. Towards asteroseismically calibrated age-rotation relations. Astron. Astrophys. 572, A34 (2014). doi:10.1051/0004-6361/201423888. arXiv:1403.7155
Gizon, L., Ballot, J., Michel, E., Stahn, T., Vauclair, G., et al.: Seismic constraints on rotation of sun-like star and mass of exoplanet. Proc. Natl. Acad. Sci. 110, 13267–13271 (2013). doi:10.1073/pnas.1303291110. arXiv:1308.4352
Goldreich, P., Nicholson, P.D.: Tidal friction in early-type stars. Astrophys. J. 342, 1079–1084 (1989). doi:10.1086/167665
Goodman, J., Dickson, E.S.: Dynamical tide in solar-type binaries. Astrophys. J. 507, 938–944 (1998). doi:10.1086/306348. arXiv:astro-ph/9801289
Gough, D.O., McIntyre, M.E.: Inevitability of a magnetic field in the Sun’s radiative interior. Nature 394, 755–757 (1998). doi:10.1038/29472
Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series and products (1994)
Guenel, M., Baruteau, C., Mathis, S., Rieutord, M.: Tidal inertial waves in the differentially rotating convective envelopes of low-mass stars-I. Free oscillation modes. Astron. Astrophys. 589, A22 (2016)
Guillot, T., Lin, D.N.C., Morel, P., Havel, M., Parmentier, V.: Evolution of exoplanets and their parent stars. In: EAS Publications Series, EAS Publications Series, vol. 65, 327–336 (2014). doi:10.1051/eas/1465009. arXiv:1409.7477
Henry, G.W., Marcy, G.W., Butler, R.P., Vogt, S.S.: A transiting, “51 Peg-like” planet. Astrophys. J. Letters 529, L41–L44 (2000). doi:10.1086/312458
Hut, P.: Stability of tidal equilibrium. Astron. Astrophys. 92, 167–170 (1980)
Hut, P.: Tidal evolution in close binary systems. Astron. Astrophys. 99, 126–140 (1981)
Ivanov, P.B., Papaloizou, J.C.B., Chernov, S.V.: A unified normal mode approach to dynamic tides and its application to rotating sun-like stars. Mon. Not. R. Astron. Soc. 432, 2339–2365 (2013). doi:10.1093/mnras/stt595. arXiv:1304.2027
Kawaler, S.D.: Angular momentum loss in low-mass stars. Astrophys. J. 333, 236–247 (1988). doi:10.1086/166740
Kitchatinov, L.L., Pipin, V.V., Ruediger, G.: Turbulent viscosity, magnetic diffusivity, and heat conductivity under the influence of rotation and magnetic field. Astron. Nachr. 315, 157–170 (1994). doi:10.1002/asna.2103150205
Kurtz, D.W., Saio, H., Takata, M., Shibahashi, H., Murphy, S.J., Sekii, T.: Asteroseismic measurement of surface-to-core rotation in a main-sequence A star, KIC 11145123. Mon. Not. R. Astron. Soc. 444, 102–116 (2014). doi:10.1093/mnras/stu1329. arXiv:1405.0155
Lanza, A.F.: Internal stellar rotation and orbital period modulation in close binary systems. Mon. Not. R. Astron. Soc. 369, 1773–1779 (2006). doi:10.1111/j.1365-2966.2006.10415.x
Lanza, A.F.: Angular momentum conservation and torsional oscillations in the Sun and solar-like stars. Astron. Astrophys. 471, 1011–1022 (2007). doi:10.1051/0004-6361:20077418. arXiv:0706.1623
Lanza, A.F., Damiani, C., Gandolfi, D.: Constraining tidal dissipation in F-type main-sequence stars: the case of CoRoT-11. Astron. Astrophys. 529, A50 (2011). doi:10.1051/0004-6361/201016144. arXiv:1012.5791
Levrard, B., Winisdoerffer, C., Chabrier, G.: Falling transiting extrasolar giant planets. Astrophys. J. Letters 692, L9–L13 (2009). doi:10.1088/0004-637X/692/1/L9. arXiv:0901.2048
Maeder, A., Zahn, J.P.: Stellar evolution with rotation. III. Meridional circulation with MU-gradients and non-stationarity. Astron. Astrophys. 334, 1000–1006 (1998)
Mathis, S.: The variation of the tidal quality factor of convective envelopes of rotating low-mass stars along their evolution. In: Martins, F., Boissier, S., Buat, V., Cambrésy, L., Petit, P., (eds.) SF2A-2015: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, pp 401–405 (2015a). arXiv:1511.01084
Mathis, S.: Variation of tidal dissipation in the convective envelope of low-mass stars along their evolution. Astron. Astrophys. 580, L3 (2015b). doi:10.1051/0004-6361/201526472. arXiv:1507.00165
Mathis, S., Zahn, J.P.: Transport and mixing in the radiation zones of rotating stars. I. Hydrodynamical processes. Astron. Astrophys. 425, 229–242 (2004). doi:10.1051/0004-6361:20040278. arXiv:astro-ph/0406418
Mathis, S., Zahn, J.P.: Transport and mixing in the radiation zones of rotating stars. II. Axisymmetric magnetic field. Astron. Astrophys. 440, 653–666 (2005). doi:10.1051/0004-6361:20052640. arXiv:astro-ph/0506105
Mathis, S., Palacios, A., Zahn, J.P.: On shear-induced turbulence in rotating stars. Astron. Astrophys. 425, 243–247 (2004). doi:10.1051/0004-6361:20040279. arXiv:astro-ph/0403580
Mathis, S., Decressin, T., Eggenberger, P., Charbonnel, C.: Diagnoses to unravel secular hydrodynamical processes in rotating main sequence stars. II. The actions of internal gravity waves. Astron. Astrophys. 558, A11 (2013). doi:10.1051/0004-6361/201321934
Mathis, S., Auclair-Desrotour, P., Guenel, M., Le Poncin-Lafitte, C.: Tidal friction in rotating turbulent convectivestellar and planetary regions. In: Ballet J., Martins F., Bournaud F., Monier R., Reylé C. (eds.) SF2A-2014: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, pp. 251–256 (2014)
Mathis, S., Auclair-Desrotour, P., Guenel, M., Gallet, F., Le Poncin-Lafitte, C.: The impact of rotation on turbulent tidal friction in stellar and planetary convective regions. ArXiv e-prints (2016) arXiv:1604.08570
Mathur, S., Eff-Darwich, A., García, R.A., Turck-Chièze, S.: Sensitivity of helioseismic gravity modes to the dynamics of the solar core. Astron. Astrophys. 484, 517–522 (2008). doi:10.1051/0004-6361:20078839. arXiv:0803.3966
Matt, S.P., Brun, A.S., Baraffe, I., Bouvier, J., Chabrier, G.: The mass-dependence of angular momentum evolution in sun-like stars. Astrophys. J. Letters 799, L23 (2015). doi:10.1088/2041-8205/799/2/L23. arXiv:1412.4786
Mayor, M., Queloz, D.: A Jupiter-mass companion to a solar-type star. Nature 378, 355–359 (1995). doi:10.1038/378355a0
McQuillan, A., Mazeh, T., Aigrain, S.: Stellar rotation periods of the kepler objects of interest: a dearth of close-in planets around fast rotators. Astrophys. J. Letters 775, L11 (2013). doi:10.1088/2041-8205/775/1/L11. arXiv:1308.1845
McQuillan, A., Mazeh, T., Aigrain, S.: Rotation Periods of 34,030 Kepler Main-sequence Stars: The Full Autocorrelation Sample. Astrophys. J. Suppl. 211, 24 (2014). doi:10.1088/0067-0049/211/2/24. arXiv:1402.5694
Morize, C., Le Bars, M., Le Gal, P., Tilgner, A.: Experimental determination of zonal winds driven by tides. Phys. Rev. Lett. 104(21), 214501 (2010). doi:10.1103/PhysRevLett.104.214501
Morse, P.M., Feshbach, H.: Methods of theoretical physics, McGraw-Hill Book Company Inc., New York (1953)
Ogilvie, G.I.: Tides in rotating barotropic fluid bodies: the contribution of inertial waves and the role of internal structure. Mon. Not. R. Astron. Soc. 429, 613–632 (2013). doi:10.1093/mnras/sts362. arXiv:1211.0837
Ogilvie, G.I.: Tidal dissipation in stars and giant planets. Ann. Rev. Astron. Astrophys. 52, 171–210 (2014). doi:10.1146/annurev-astro-081913-035941. arXiv:1406.2207
Ogilvie, G.I., Lesur, G.: On the interaction between tides and convection. Mon. Not. R. Astron. Soc. 422, 1975–1987 (2012). doi:10.1111/j.1365-2966.2012.20630.x. arXiv:1201.5020
Ogilvie, G.I., Lin, D.N.C.: Tidal dissipation in rotating giant planets. Astrophys. J. 610, 477–509 (2004). doi:10.1086/421454. arXiv:astro-ph/0310218
Ogilvie, G.I., Lin, D.N.C.: Tidal dissipation in rotating solar-type stars. Astrophys. J. 661, 1180–1191 (2007). doi:10.1086/515435. arXiv:astro-ph/0702492
Penev, K., Sasselov, D.: Tidal evolution of close-in extrasolar planets: high stellar Q from new theoretical models. Astrophys. J. 731, 67 (2011). doi:10.1088/0004-637X/731/1/67. arXiv:1102.3187
Pillitteri, I., Wolk, S.J., Sciortino, S., Antoci, V.: No X-rays from WASP-18. Implications for its age, activity, and the influence of its massive hot Jupiter. Astron. Astrophys. 567, A128 (2014). doi:10.1051/0004-6361/201423579. arXiv:1406.2620
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in FORTRAN. The art of scientific computing (1992)
Rauer, H., Catala, C., Aerts, C., Appourchaux, T., Benz, W., et al.: The PLATO 2.0 mission. Exp. Astron. 38, 249–330 (2014). doi:10.1007/s10686-014-9383-4. arXiv:1310.0696
Remus, F., Mathis, S., Zahn, J.P.: The equilibrium tide in stars and giant planets. I. The coplanar case. Astron. Astrophys. 544, A132 (2012). doi:10.1051/0004-6361/201118160. arXiv:1205.3536
Ricker, G.R., Winn, J.N., Vanderspek, R., Latham, D.W., Bakos, GÁ., et al.: Transiting Exoplanet Survey Satellite (TESS). In: Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave, Proceedings SPIE, vol. 9143, p. 914320 (2014). doi:10.1117/12.2063489. arXiv:1406.0151
Rogers, T.M.: On the differential rotation of massive main-sequence stars. Astrophys. J. Letters 815, L30 (2015). doi:10.1088/2041-8205/815/2/L30. arXiv:1511.03809
Rüdiger, G., Hollerbach, R.: The magnetic universe: geophysical and astrophysical dynamo theory, Wiley, Weinheim (2004)
Ruediger, G.: Differential rotation and stellar convection. Sun and solar-type stars, Gordon and Breach Science Publishers, New York (1989)
Saio, H., Kurtz, D.W., Takata, M., Shibahashi, H., Murphy, S.J., et al.: Asteroseismic measurement of slow, nearly uniform surface-to-core rotation in the main-sequence F star KIC 9244992. Mon. Not. R. Astron. Soc. 447, 3264–3277 (2015). doi:10.1093/mnras/stu2696. arXiv:1412.5362
Schou, J., Antia, H.M., Basu, S., Bogart, R.S., Bush, R.I., et al.: Helioseismic studies of differential rotation in the solar envelope by the solar oscillations investigation using the Michelson doppler imager. Astrophys. J. 505, 390–417 (1998). doi:10.1086/306146
Skumanich, A.: Time scales for Ca II emission decay, rotational braking, and lithium depletion. Astrophys. J. 171, 565 (1972). doi:10.1086/151310
Song, H.F., Maeder, A., Meynet, G., Huang, R.Q., Ekström, S., Granada, A.: Close-binary evolution. I. Tidally induced shear mixing in rotating binaries. Astron. Astrophys. 556, A100 (2013). doi:10.1051/0004-6361/201321870. arXiv:1306.6731
Song, H.F., Meynet, G., Maeder, A., Ekström, S., Eggenberger, P.: Massive star evolution in close binaries. Conditions for homogeneous chemical evolution. Astron. Astrophys. 585, A120 (2016). doi:10.1051/0004-6361/201526074. arXiv:1508.06094
Spruit, H.C.: Differential rotation and magnetic fields in stellar interiors. Astron. Astrophys. 349, 189–202 (1999). arXiv:astro-ph/9907138
Talon, S., Charbonnel, C.: Hydrodynamical stellar models including rotation, internal gravity waves, and atomic diffusion. I. Formalism and tests on Pop I dwarfs. Astron. Astrophys. 440, 981–994 (2005). doi:10.1051/0004-6361:20053020. arXiv:astro-ph/0505229
Talon, S., Kumar, P.: Dissipation of a tide in a differentially rotating star (1998). doi:10.1086/305959. arXiv:astro-ph/9707309
Talon, S., Zahn, J.P.: Anisotropic diffusion and shear instabilities. Astron. Astrophys. 317, 749–751 (1997). arXiv:astro-ph/9609010
Talon, S., Kumar, P., Zahn, J.P.: Angular momentum extraction by gravity waves in the sun. Astrophys. J. Letters 574, L175–L178 (2002). doi:10.1086/342526. arXiv:astro-ph/0206479
Terquem, C., Papaloizou, J.C.B., Nelson, R.P., Lin, D.N.C.: On the tidal interaction of a solar-type star with an orbiting companion: excitation of g-mode oscillation and orbital evolution. Astrophys. J. 502, 788–801 (1998). doi:10.1086/305927. arXiv:astro-ph/9801280
Tilgner, A.: Zonal wind driven by inertial modes. Phys. Rev. Lett. 99(19), 194501 (2007). doi:10.1103/PhysRevLett.99.194501
Valsecchi, F., Rasio, F.A.: Tidal dissipation and obliquity evolution in hot jupiter systems. Astrophys. J. 786, 102 (2014). doi:10.1088/0004-637X/786/2/102. arXiv:1402.3857
Winn, J.N., Fabrycky, D., Albrecht, S., Johnson, J.A.: Hot stars with hot jupiters have high obliquities. Astrophys. J. Letters 718, L145–L149 (2010). doi:10.1088/2041-8205/718/2/L145. arXiv:1006.4161
Wu, Y.: Origin of tidal dissipation in Jupiter II. The value of Q. Astrophys. J. 635, 688–710 (2005). doi:10.1086/497355. arXiv:astro-ph/0407628
Zahn, J.P.: Les marées dans une étoile double serrée. Ann. d’Astrophys. 29:313 (1966a)
Zahn, J.P.: Les marées dans une étoile double serrée (suite et fin). Ann. d’Astrophys. 29:565 (1966b)
Zahn, J.P.: The dynamical tide in close binaries. Astron. Astrophys. 41, 329–344 (1975)
Zahn, J.P.: Tidal friction in close binary stars. Astron. Astrophys. 57, 383–394 (1977)
Zahn, J.P.: Tidal evolution of close binary stars. I—revisiting the theory of the equilibrium tide. Astron. Astrophys. 220, 112–116 (1989)
Zahn, J.P.: Circulation and turbulence in rotating stars. Astron. Astrophys. 265, 115–132 (1992)
Zahn, J.P.: Rotation and lithium depletion in late-type binaries. Astron. Astrophys. 288, 829–841 (1994)
Zahn, J.P.: Tidal dissipation in binary systems. In: Goupil, M.J., Zahn, J.P. (eds.) EAS Publications Series, EAS Publications Series, vol. 29, pp. 67–90 (2008). doi:10.1051/eas:0829002. arXiv:0807.4870
Zahn, J.P., Talon, S., Matias, J.: Angular momentum transport by internal waves in the solar interior. Astron. Astrophys. 322, 320–328 (1997). arXiv:astro-ph/9611189
Acknowledgments
The authors would like to thank an anonymous Referee for a careful reading of the manuscript and valuable comments that helped to improve their work. They are also grateful to the Editors of this special issue on tides for their kind invitation to contribute. S. M. acknowledges funding by the European Research Council through ERC Grant SPIRE 647383. This work was also supported by the ANR Blanc TOUPIES SIMI5-6 020 01, the Programme National de Planétologie (CNRS/INSU) and PLATO CNES Grant at Service d’Astrophysique (CEA-Saclay). The authors gratefully acknowledge use of the EZ-web stellar evolution resources. Exoplanet studies at INAF-Osservatorio Astrofisico di Catania have been funded also through the Progetti Premiali of the Italian Ministero dell’Istruzione, Università e Ricerca.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lanza, A.F., Mathis, S. Tides and angular momentum redistribution inside low-mass stars hosting planets: a first dynamical model. Celest Mech Dyn Astr 126, 249–274 (2016). https://doi.org/10.1007/s10569-016-9714-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-016-9714-z