Minimal viscosity discs lock pairs of giant planets in 2:1 resonance with stalled migration.
Authors: Griveaud Philippine , Crida Aurélien , Lega Elena 
Published: September 2022

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&lt;p&gt;Planets form in protoplanetary discs and their interactions with the gas give rise to migration. For a long time, discs were believed to have a non-negligible viscosity to justify the high accretion rates of gas onto the central star. However, it has recently been shown observationally and theoretically that protoplanetary discs are probably much less viscous than previously thought. In our study, we use a new paradigm for the theoretical modelling of discs where the accretion onto the central star is done through the superficial layers while the mid-plane has a close to zero viscosity (Lega et al. 2022). In such discs, the migration of a single giant planet differs from the classical Type-II migration regime and depends on the thickness of the accretion layer. It is therefore interesting to consider the migration of a pair of giant planets in this new model. We have started this project, in the simplified framework of 2D hydrodynamical simulations (using the code FARGOCA) with an&amp;#160;&lt;span class=&quot;math math-inline is-loaded&quot;&gt;&amp;#945;&lt;/span&gt;&amp;#160;viscosity parameter of&amp;#160;&lt;span class=&quot;math math-inline is-loaded&quot;&gt;10&lt;sup&gt;&amp;#8722;5&lt;/sup&gt;&lt;/span&gt;&lt;sup&gt;&amp;#160;&lt;/sup&gt;(in the standard Shakura &amp; Sunyaev 1973 viscosity parametrization). We first consider a pair of Jupiter and Saturn mass planets, then extend our study to a wider range of planetary masses.&lt;/p&gt;
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&lt;p&gt;In classical viscous discs (corresponding to an&amp;#160;&lt;span class=&quot;math math-inline is-loaded&quot;&gt;&amp;#945;=10&lt;sup&gt;&amp;#8722;3&lt;/sup&gt;&lt;/span&gt;), Jupiter and Saturn systems are most often locked in the 3:2 mean motion resonance and may migrate outwards. Instead in our case (&lt;span class=&quot;math math-inline is-loaded&quot;&gt;&amp;#945;=10&lt;sup&gt;&amp;#8722;5&lt;/sup&gt;&lt;/span&gt;), we find that the pair of planets gets locked in the 2:1 resonance and has a stalled or slightly inward migration. We confirmed this result for a range of disc masses and thicknesses as well as different starting positions of Saturn. This result is also independent of the mass of the outer planet. In order to explain our result, we have a used a criterion for resonance crossing based on Batygin, 2015 . Unlike in classical discs, a planet growing and migrating in a low viscosity disc does not reach the migration speed required to cross the 2:1 MMR. The only case in which outward migration is observed, is the &quot;ad-hoc&quot; scenario where Saturn would form inside the 2:1 resonance and get locked in the 3:2. However, owing to the large width of Jupiter's gap, this scenario seems unlikely.&lt;/p&gt;
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&lt;p&gt;If the Solar System formed from such a low-viscosity disc, this result has strong implications for the Grand Tack and Nice models, which both assume Jupiter and Saturn to be inside the 2:1 resonance. The stalled migration could, however, explain the so called warm-Jupiters population among the detected exoplanets, providing that these are in multi-planet systems. Additionally, we remark that the only system with two giant planets observed in a disc of gas, namely PDS70, occurs to be close to the 2:1 resonance.&lt;/p&gt;
&lt;p&gt;&amp;#160;&lt;/p&gt;
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&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;415&quot; height=&quot;373&quot; /&gt;&lt;/p&gt;
&lt;p&gt;&lt;em&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;Figure 1:&lt;/span&gt; &lt;/em&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;Surface density of a 2D simulation after nearly 300 000 years of evolution, &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;displaying Jupiter&lt;/span&gt; &lt;em&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;(filled circle)&lt;/span&gt;&lt;/em&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;and Saturn&lt;/span&gt; &lt;em&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;(empty circle)&lt;/span&gt; &lt;/em&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;in their common gap.&lt;/span&gt; &lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;The &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;planets are stably locked in the 2:1 mean motion resonance and are following a very slow &lt;/span&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;inward migration at a speed of one a.u. per million years.&lt;/span&gt;&lt;/p&gt;
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&lt;p&gt;&lt;span dir=&quot;ltr&quot; role=&quot;presentation&quot;&gt;References:&lt;/span&gt;&lt;/p&gt;
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&lt;p&gt;E. Lega et al., &amp;#8220;Migration of Jupiter Mass Planets in Discs with Laminar Accretion Flows,&amp;#8221;&amp;#160;&lt;em&gt;Astronomy &amp; Astrophysics&lt;/em&gt;&amp;#160;658 (2022)&lt;/p&gt;
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&lt;p&gt;Konstantin Batygin, &amp;#8220;Capture of Planets into Mean-Motion Resonances and the Origins of Extrasolar Orbital Architectures,&amp;#8221;&amp;#160;&lt;em&gt;Monthly Notices of the Royal Astronomical Society&lt;/em&gt;&amp;#160;451, no. 3 (August 11, 2015)&lt;/p&gt;
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&lt;p&gt;&amp;#160;&lt;/p&gt;
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