Improving tidal modeling for rocky worlds
Authors: Revol Alexandre , Bolmont Émeline , Tobie Gabriel , Dumoulin Caroline , Musseau Yann , Mathis Stéphane , Strugarek Antoine , Brun Allan Sacha 
Published: September 2022

&lt;p&gt;The high number of discovered close-in planets motivates the improvement of tidal modeling.Among the five thousand exoplanets discovered up to now, half of them have an orbital period of less than 10 days and therefore experience some form of tidal evolution. Besides, the habitable zone of ultra-cool dwarfs is located close-in (e.g. TRAPPIST-1, Gillon et al. 2017), so planets in the habitable zone of these dwarfs are expected to undergo tidal evolution. By impacting the orbital and rotational dynamics of close-in planets, tides have an impact on their habitability. Indeed, tides drive the spin evolution of planets, influence their thermal state by internal friction (Henning &amp; Hurford 2014, Bolmont et al. 2020), and drive their orbital evolution by exchange of angular momentum. This leads to the migration of the planets, their eccentricity and obliquity damping, and the precession of their orbits ( e.g Hut 1981, Bolmont et al. 2011, 2012, Ragozzine &amp; Wolf 2009).&lt;/p&gt;
&lt;p&gt;&amp;#160;&lt;/p&gt;
&lt;p&gt;Solid planetary tides also act to synchronize the spin of planets on circular orbits with their orbital motion. It is generally assumed that close-in planets (i.e. experiencing strong tidal interactions) have reached this state of synchronization, also called the 1:1 spin-orbit resonance (hereafter SOR). However, in the case of an eccentric or inclined orbit, tides can trap the spin in higher rotation rates, e.g. in the 3:2 or the 2:1 SOR. If a planet has an atmosphere, another tidal mechanism should be taken into account: the thermal tide, which is caused by the differential heating between day and night sides (Correia &amp; Laskar 2001, Auclair-Desrotour 2017a). In the case of thick atmospheres such as that of Venus, the thermal tides acting on the atmosphere can be as strong as the gravitational tides acting on the solid core (Auclair-Desrotour et al. 2017b) and can both de-synchronize the planet and increase the spin inclination. The competition between the two tides, solid and thermal can be seen in figure [1].&lt;/p&gt;
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&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;
&lt;p&gt;Figure [1]:&lt;/p&gt;
&lt;p&gt;The two tidal contributions, gravitational and thermal, acting in opposition to each other. The solide tide drives the spin to synchronization, while the thermal tide drives it away from it.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Tidal model&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;To understand the evolution of exoplanets, we need a consistent tidal model which encapsulates the complex response of rocky planets to stress. Most models assume that planets are made of weakly viscous fluid (e.g. Hut 1981, Goldreich &amp; Soter 1966) even for rocky planets. However, it has been shown that they do not reproduce the correct behavior for highly viscous solid bodies (Henning et al. 2009; Efroimsky &amp; Makarov 2013). In contrast, we use here a model which accounts for more realistic anelastic behavior for the rocks (such as the Andrade rheology) for a Venus-like planet (Dumoulin et al. 2017). We choose here an approach developed by Kaula (1964), which consists in using a decomposition of the tidal potential into Fourier harmonic modes. For the thermal tide, we use the analytical formulation of Leconte et al (2015), which aims at reproducing Venus today. Our developments were added to the ESPEM code (Benbakoura et al. 2019) to compute the orbital and rotational evolution of a Venus-like planet around a Sun-like star.&lt;/p&gt;
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&lt;p&gt;&lt;strong&gt;2) A complex rotational evolution:&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;Figure [2] shows the evolution of a Venus-like planet, with and without an atmosphere, orbiting a Sun-like planet with an initial obliquity of 50 degrees and a rotation of about 100 days and on a circular orbit.&amp;#160;&lt;/p&gt;
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&lt;p&gt;&lt;strong&gt;No atmosphere (red curve):&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;When the planet does not have an atmosphere, the rotation spins down to synchronization in 100 million years. Interestingly and despite the absence of an eccentricity, Figure [2] shows that the obliquity of the planet allows the presence of the 2:0 spin-orbit resonance in which the planet spends a few 10 million years. The planet leaves this resonance when the obliquity becomes too low to maintain this configuration stable.&lt;/p&gt;
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&lt;p&gt;&lt;strong&gt;With atmosphere (blue curve):&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;The blue curve of Fig. [2] shows the effect of a strong CO2 atmosphere on the evolution of the spin and the obliquity. The evolution here is more complex. First, the combination of the two tides damps the spin toward the 2:1 SOR while increasing the obliquity. When the obliquity becomes high enough, the thermal torque begins to counter-balance the solid torque. The thermal tides continue to increase the obliquity and the spin of the planet until an equilibrium state between the solid and thermal tides is found.&lt;/p&gt;
&lt;p&gt;&amp;#160;&lt;/p&gt;
&lt;p&gt;Depending on the atmospheric parameters, the evolutions we obtain are complex, with spin-orbit resonances, coming either from the eccentricity or from the obliquity of the planet. I will show the variety of different spin evolutions of a Venus-like planet, and how it depends on the presence of an eccentricity or obliquity and the presence or not of a thick atmosphere.&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;
&lt;p&gt;Figure [2]&lt;/p&gt;
&lt;p&gt;Simulation of a Venus-like planet orbiting a Sun-like star, without an atmosphere (in red) and with an atmosphere (in blue). Top panel: evolution of the ratio of the planetary spin &amp;#937; and the mean motion n. Bottom panel: evolution of the obliquity of the planet (defined as the angle between the equatorial plane and the orbital plane).&lt;/p&gt;