Csizmadia, Szilard

Hellard, Hugo

Padovan, Sebastiano

Rauer, Heike

Sohl, Frank

Institute of Transport Research:Publications

Introduction Shape model    Retrieval of the fluid Love number 𝒌𝟐 from transit light curves Hugo Hellard1,2, Szilárd Csizmadia1, Sebastiano Padovan1, Heike Rauer1,2,3, Juan Cabrera1, Frank Sohl1, Tilman Spohn1, and Doris Breuer1  1Deutsches Zentrum für Luft- und Raumfahrt, Berlin, Germany 2Technische Universität Berlin, Berlin, Germany 3Freie Universität Berlin, Berlin, Germany Knowledge of the planetary radius and mass is not sufficient to infer the interior structure, as different composition and density profiles can lead to the same solution [1], justifying the need for an additional observable.  The second-degree fluid Love number [2], 𝑘2, is proportional to the mass concentration towards the body’s center, hence providing valuable information on the interior. 𝑘2 = 0 is a mass-point, 𝑘2 = 1.5 is a fully homogeneous body, while its full derivation depends on the internal radial density profile [3].  Assuming hydrostatic equilibrium, 𝑘2 is a direct function of the planetary shape.   Planets orbiting close to their Roche limit and outer planets with high rotation rates undergo strong tidal and rotational deformations, respectively, which modify their shape from spherical to more complicated ones.   As a result, the corresponding transit light curve will differ with respect to a transiting sphere.  Question: can we measure the shape of an exoplanet from transit curves ?   Assumptions: spherical star, circular orbit, no interactions between tides and rotation, tilted spin axis with obliquity Θ.  The tidal and rotational potentials can be expanded and expressed in spherical harmonics. Kopal [4] showed that omitting terms with degree j < 4 is equivalent to considering the Roche limit (mass-point surrounded by a massless envelope).  Following Love [2] and Kopal [4], the three-dimensional planetary surface shape is given by            Our model allows direct fitting of the fluid Love numbers and rotational parameters. Planetary mean radius Semi-major axis Feasibility: WASP-121b [5] Application: WASP-103b We considered several white noise levels and injected them into a simulated WASP-121b transit light curve, binned into 2 minute measurements. We assumed a planetary Love number 𝑘2 = 0.5 [6]. Certain noise levels can be achieved by observing facilities with 10 observed transits. Noise level (𝒑𝒑𝒎/ 𝟐𝒎𝒊𝒏 ) Facility (* @ 10,4 Vmag, for 10 observed transits) 23 JWST (NIRSpec)* 45 Kepler* 63 PLATO* 71 CHEOPS* 360 TESS* 600, 1e3, 2e3, 5e3 e.g. Spitzer, ground-based surveys CHEOPS-like noise level @ 10,4 Vmag System: 1.53 𝑅𝐽 hot Jupiter orbiting a 1.44 𝑅⨀ , 12.0 𝑉𝑚𝑎𝑔 star, at roughly 2.3 its Roche limit.  Data: one transit observed in by the GMOS instrument at the 8.1m GEMINI-North telescope [7].  Result: we obtained a Love number equal to 𝑘2 = 0.29−0.18+0.20 Conclusions Reference & Acknowledgements • Our model allows direct fitting of the fluid Love number 𝑘2  • Planets orbiting close to their Roche limit exhibit large tidal surface deformations, making them the best candidates for the retrieval of 𝑘2.  • The precision in 𝑘2 reaches a pleateau where a better photometric precision does not lead to a better precision in 𝑘2. Only an improved knowledge of the planetary mean radius would improve the precision in the Love number.  • Using only one transit observation of WASP-103b from the GEMINI-North telescope, we managed to provide a rough 1.6𝜎 detection of its Love number:    • The current TESS and upcoming CHEOPS missions will help further constrain the interior of exoplanets by providing the first reliable 𝑘2 estimations of exoplanets. [1] Rogers, L.A. & Seager, S. 2010, ApJ, 712, 974 [2] Love, A.E.H. 1911, Some problems of geodynamics (Cambridge University Press) [3] Padovan, S., Spohn, T., Baumeister, P., Tosi, N., Breuer, D., Csizmadia, Sz., Hellard, H., & Sohl, F. 2018, A&A [4] Kopal, Z. 1959, Close Binary Systems (New York: Wiley) [5] Hellard, H., Csizmadia, Sz., Padovan, S., Rauer, H., Cabrera, J., Sohl, F., Spohn, T., & Breuer, D. 2019, ApJ, submitted [6] Wahl, S.M. et al. 2016, ApJ, 831, 1 [7] Lendl, M. et al. 2017, A&A, 606, A18  We acknowledge support from the DFG via the Research Unit FOR 2440 Matter Under Planetary Interior Conditions. Contact: hugo.hellard@dlr.de 𝒌𝟐 = 𝟎. 𝟐𝟗−𝟎.𝟏𝟖+𝟎.𝟐𝟎  

Investigating exoplanet interiors from transit light curves

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Investigating exoplanet interiors from transit light curves

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