We explore the application of machine learning based on mixture density neural networks (MDNs) to the interior characterization of low-mass exoplanets up to 25 Earth masses constrained by mass, radius, and fluid Love number k 2 . We create a dataset of 900 000 synthetic planets, consisting of an iron-rich core, a silicate mantle, a high-pressure ice shell, and a gaseous H/He envelope, to train a MDN using planetary mass and radius as inputs to the network. For this layered structure, we show that the MDN is able to infer the distribution of possible thicknesses of each planetary layer from mass and radius of the planet. This approach obviates the time-consuming task of calculating such distributions with a dedicated set of forward models for each individual planet. While gas-rich planets may be characterized by compositional gradients rather than distinct layers, the method presented here can be easily extended to any interior structure model. The fluid Love number k 2 bears constraints on the mass distribution in the planets' interior and will be measured for an increasing number of exoplanets in the future. Adding k 2 as an input to the MDN significantly decreases the degeneracy of the possible interior structures. In an open repository we provide the trained MDN to be used through a Python Notebook.

Context. The mass and radius of a planet directly provide its bulk density, which can be interpreted in terms of its overall composition. Any measure of the radial mass distribution provides a first step in constraining the interior structure. The fluid Love number k 2 provides such a measure, and estimates of k 2 for extrasolar planets are expected to be available in the coming years thanks to improved observational facilities and the ever-extending temporal baseline of extrasolar planet observations. Aims. We derive a method for calculating the Love numbers k n of any object given its density profile, which is routinely calculated from interior structure codes. Methods. We used the matrix-propagator technique, a method frequently used in the geophysical community. Results. We detail the calculation and apply it to the case of GJ 436b, a classical example of the degeneracy of mass-radius relationships, to illustrate how measurements of k 2 can improve our understanding of the interior structure of extrasolar planets. We implemented the method in a code that is fast, freely available, and easy to combine with preexisting interior structure codes. While the linear approach presented here for the calculation of the Love numbers cannot treat the presence of nonlinear effects that may arise under certain dynamical conditions, it is applicable to close-in gaseous extrasolar planets like hot Jupiters, likely the first targets for which k 2 will be measured.

The evolution of Venus is still in many ways a mystery. Despite its similarity to Earth in terms of size and bulk composition (e.g., Lécuyer et al., 2000), the atmospheric 2 CO E mass of Venus is larger by more than five orders of magnitude (Donahue & Pollack, 1983). Whether Venus' atmosphere was 2 CO E -rich early in its evolution, or whether, and how, it diverged from Earth's atmosphere later in its evolution is a matter of debate (e.g.,

We investigate the encapsulation of CdSe/CdS quantum dots (QDs) in a silica shell by in situ Raman spectroscopy and find a distinct shift of the CdS Raman signal during the first hours of the synthesis. This shift does not depend on the final silica shell thickness but on the properties of the initial core-shell QD. We find a correlation between the Raman shift rate and the speed of the silica formation and attribute this to the changing configuration of the outermost layers of the QD shell, where an interface to the newly formed silica is created. This dependence of Raman shift rate on the speed of silica formation process will give rise to many possible studies concerning the growth mechanism in the water-in-oil microemulsion, rendering in situ Raman a valuable instrument in monitoring this type of reaction.

<p>We explore the application of machine-learning, based on mixture density neural networks (MDNs), to the interior characterization of low-mass exoplanets up to 25 Earth masses constrained by mass, radius, and fluid Love number k<sub>2</sub>. MDNs are a special subset of neural networks, able to predict the parameters of a Gaussian mixture distribution instead of single output values, which enables them to learn and approximate probability distributions. With a dataset of 900,000 synthetic planets, consisting of an iron-rich core, a silicate mantle, a high-pressure ice shell, and a gaseous H/He envelope, we train an MDN using planetary mass and radius as inputs to the network. We show that the MDN is able to infer the distribution of possible thicknesses of each planetary layer from mass and radius of the planet. This approach obviates the time-consuming task of calculating such distributions with a dedicated set of forward models for each individual planet.</p><p>The fluid Love number k<sub>2</sub> bears constraints on the mass distribution in the planets' interior and will be measured for an increasing number of exoplanets in the future. Adding k<sub>2</sub> as an input to the MDN significantly decreases the degeneracy of possible interior structures.</p>

Aims. Characterizing the interior structure of exoplanets is essential for understanding their diversity, formation, and evolution. As the interior of exoplanets is inaccessible to observations, an inverse problem must be solved, where numerical structure models need to conform to observable parameters such as mass and radius. This is a highly degenerate problem whose solution often relies on computationally expensive and time-consuming inference methods such as Markov chain Monte Carlo. Methods. We present ExoMDN, a machine-learning model for the interior characterization of exoplanets based on mixture density networks (MDN). The model is trained on a large dataset of more than 5.6 million synthetic planets below 25 Earth masses consisting of an iron core, a silicate mantle, a water and high-pressure ice layer, and a H/He atmosphere. We employ log-ratio transformations to convert the interior structure data into a form that the MDN can easily handle. Results. Given mass, radius, and equilibrium temperature, we show that ExoMDN can deliver a full posterior distribution of mass fractions and thicknesses of each planetary layer in under a second on a standard Intel i5 CPU. Observational uncertainties can be easily accounted for through repeated predictions from within the uncertainties. We used ExoMDN to characterize the interiors of 22 confirmed exoplanets with mass and radius uncertainties below 10 and 5%, respectively, including the well studied GJ 1214 b, GJ 486 b, and the TRAPPIST-1 planets. We discuss the inclusion of the fluid Love number k2 as an additional (potential) observable, showing how it can significantly reduce the degeneracy of interior structures. Utilizing the fast predictions of ExoMDN, we show that measuring k2 with an accuracy of 10% can constrain the thickness of core and mantle of an Earth analog to ≈13% of the true values.

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<p>Liquid water is generally assumed to be the most important factor for the emergence of life, and so a major goal in exoplanet science is the search for planets with water oceans. On terrestrial planets, the silicate mantle is a large source of water, which can be outgassed into the atmosphere via volcanism. Outgassing is subject to a series of feedback processes between atmosphere and interior, which continually shape both atmospheric composition, pressure, and temperature, as well as interior dynamics. For example, water has a high solubility in surface lava, which can strongly limit its outgassing into the atmosphere even at low atmospheric pressures. In contrast, CO<sub>2</sub> can be easily outgassed. This drives up the surface pressure and temperature, potentially preventing further water outgassing [1].</p> <p>We present the results of an extensive parameter study, where we use a newly developed 1D numerical model to simulate the coupled evolution of the atmosphere and interior of terrestrial exoplanets up to 5 Earth masses around Sun-like stars, with internal structures ranging from Moon- to Mercury-like. The model accounts for the main mechanisms controlling the global-scale, long-term evolution of stagnant-lid rocky planets (i.e. bodies without plate tectonics), and it includes a large number of atmosphere-interior feedback processes, such as a CO<sub>2</sub> weathering cycle, volcanic outgassing based on the pressure-dependent solubility of volatiles in surface lava, a water cycle between ocean and atmosphere, greenhouse heating, as well as the influence of a primordial H<sub>2</sub> atmosphere, which can be lost through escape processes. While many atmosphere-interior feedback processes have been studied before in detail (e.g. [2, 3]), we present here a comprehensive model combining the important planetary processes across a wide range of terrestrial planets.</p> <p>We find that a significant majority of high-density exoplanets (i.e. Mercury-like planets with large cores) are able to outgas and sustain water on their surface. In contrast, most planets with intermediate, Earth-like densities either transition into a runaway greenhouse regime due to strong CO<sub>2</sub> outgassing, or retain part of their primordial atmosphere, which prevents water from being outgassed. This suggests that high-density planets could be the most promising targets when searching for suitable candidates for hosting liquid water. Furthermore, the degeneracy of the interior structures of high-density planets is limited compared to that of planets with Earth-like density, which further facilitates the characterization of these bodies, and our results predict largely uniform atmospheric compositions across the range of high-density planets, which could be verified by future spectroscopic measurements.</p> <p> </p> <p>References:</p> <p>[1] Tosi, N. <em>et al.</em> The habitability of a stagnant-lid earth. <em>A&A</em> <strong>605</strong>, A71 (2017).</p> <p>[2] Noack, L., Rivoldini, A. & Van Hoolst, T. Volcanism and outgassing of stagnant-lid planets: Implications for the habitable zone. <em>Physics of the Earth and Planetary Interiors</em> <strong>269</strong>, 40–57 (2017).</p> <p>[3] Foley, B. J. & Smye, A. J. Carbon Cycling and Habitability of Earth-Sized Stagnant Lid Planets. <em>Astrobiology</em> <strong>18</strong>, 873–896 (2018).</p>