Tidal deformation of Ganymede: Sensitivity of Love numbers on the interior structure

We assess the gravity constraints on the interior structure of Europa in anticipation of the Europa Clipper mission.Moore and Schubert (2000) illustrated that the diurnal tide amplitude, quantified by the diurnal (tidal) Love numbers, k2d and h2d, can be used to determine the presence of a subsurface liquid ocean due to the significant increase in tidal amplitudes associated with the mechanical decoupling of the shell with a subsurface ocean.&#160; However, they considered a limited range of possible interior parameters except the ice shell rigidity, which was assumed to be in the range of 1-10 GPa. We consider a wider range of possible interior structure parameters and a more realistic ice shell rigidity range of 1-4 GPa. Inferring the presence of a subsurface ocean is slightly easier than previously thought (Verma & Margot 2018), with required absolute precisions of 0.08 for k2d , and 0.44 for h2d .Previous work have considered diurnal (tidal) gravity constraints alone or static gravity constraints alone using a forward modeling approach (e.g.&#160; Anderson et al., 1998; Moore and Schubert, 2000; Wahr et al., 2006). We evaluate constraints on interior structure parameters using Bayesian inversion with the mass, static gravity, and diurnal gravity as constraints, allowing a probabilistic view of Europa's interior structure. Given the same relative uncertainties, the static Love numbers provide stronger constraints on the interior structure relative to those from the mean moment of inertia (MOI). Additionally, the static Love numbers can be inferred directly from the static gravity field whereas inferring the MOI requires the Radau-Darwin approximation.Jointly considered with the static shape, the static gravity field can constrain the average and long-wavelength thickness of the shell. For an isostatically compensated shell, it is usual to conceptualize the crust as a series of independently floating columns of equal cross-sectional area which, by application of Archimedes' principle, should have equal mass above the depth of compensation. However, this approach is unphysical in the presence of curvature and self-gravitation. We consider alternative prescriptions of Airy isostasy: the equal-pressure prescription (Hemingway and Matsuyama, 2017), and the minimum-stress prescription (Dahlen 1982; Beuthe et al., 2016; Trinh et al., 2019).&#160; The gravitational coefficients are more sensitive to shell thickness than would be expected from the classical (equal-mass) approach, illustrating that the equal-mass prescription can lead to large errors in the inferred average shell thickness and its lateral variations.Diurnal gravity data alone can only constrain the product of the shell rigidity and thickness (Moore and Schubert, 2000; Wahr et al., 2006). An additional observational constraint that is sensitive to these parameters is the libration amplitude, which can be obtained from direct imaging or from altimeter data. We show that a joint gravity and libration analysis is able to separately constrain the shell thickness and rigidity.

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Constraints on the interior structure of Phobos from tidal deformation modelling

Icarus ◽

10.1016/j.icarus.2021.114714 ◽

2021 ◽

pp. 114714

Author(s):

Andrei A. Dmitrovskii ◽

Amir Khan ◽

Christian Boehm ◽

Amirhossein Bagheri ◽

Martin van Driel

Keyword(s):

Interior Structure ◽

Tidal Deformation

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Interior structure models and tidal Love numbers of Titan

Journal of Geophysical Research Atmospheres ◽

10.1029/2003je002044 ◽

2003 ◽

Vol 108 (E12) ◽

Cited By ~ 87

Author(s):

F. Sohl ◽

H. Hussmann ◽

B. Schwentker ◽

T. Spohn ◽

R. D. Lorenz

Keyword(s):

Interior Structure ◽

Love Numbers

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Matrix-propagator approach to compute fluid Love numbers and applicability to extrasolar planets

Astronomy and Astrophysics ◽

10.1051/0004-6361/201834181 ◽

2018 ◽

Vol 620 ◽

pp. A178 ◽

Cited By ~ 5

Author(s):

S. Padovan ◽

T. Spohn ◽

P. Baumeister ◽

N. Tosi ◽

D. Breuer ◽

...

Keyword(s):

Density Profile ◽

Extrasolar Planets ◽

Nonlinear Effects ◽

Extrasolar Planet ◽

Interior Structure ◽

Love Number ◽

Hot Jupiters ◽

Linear Approach ◽

Love Numbers ◽

The Matrix

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 numberk2provides such a measure, and estimates ofk2for 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 numbersknof 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 ofk2can 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 whichk2will be measured.

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Interior structure models and fluid Love numbers of exoplanets in the super-Earth regime

Astronomy and Astrophysics ◽

10.1051/0004-6361/201731775 ◽

2018 ◽

Vol 615 ◽

pp. A39 ◽

Cited By ~ 8

Author(s):

C. Kellermann ◽

A. Becker ◽

R. Redmer

Keyword(s):

Extrasolar Planets ◽

Water Layer ◽

Parameter Study ◽

Interior Structure ◽

Tidal Interactions ◽

Love Numbers ◽

Mass Fractions ◽

Successful Technique ◽

Outer Water ◽

Measured Mass

Space missions such as CoRoT and Kepler have made the transit method the most successful technique in observing extrasolar planets. However, although the mean density of a planet can be derived from its measured mass and radius, no details about its interior structure, such as the density profile, can be inferred so far. If determined precisely enough, the shape of the transiting light curve might, in principle, reveal the shape of the planet, and in particular, its deviation from spherical symmetry. These deformations are caused, for instance, by the tidal interactions of the planet with the host star and by other planets that might orbit in the planetary system. The deformations depend on the interior structure of the planet and its composition and can be parameterized as Love numbers kn. This means that the diversity of possible interior models for extrasolar planets might be confined by measuring this quantity. We present results of a wide-ranging parameter study in planet mass, surface temperature, and layer mass fractions on such models for super-Earths and their corresponding Love numbers. Based on these data, we find that k2 is most useful in assessing the ratio of rocky material to iron and in ruling out certain compositional configurations for measured mass and radius values, such as a prominent core consisting of rocky material. Furthermore, we apply the procedure to exoplanets K2-3b and c and predict that K2-3c probably has a thick outer water layer.

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Impact of tides on the transit-timing fits to the TRAPPIST-1 system

Astronomy and Astrophysics ◽

10.1051/0004-6361/202037546 ◽

2020 ◽

Vol 635 ◽

pp. A117 ◽

Cited By ~ 3

Author(s):

E. Bolmont ◽

B.-O. Demory ◽

S. Blanco-Cuaresma ◽

E. Agol ◽

S. L. Grimm ◽

...

Keyword(s):

General Relativity ◽

Love Number ◽

The Past ◽

Tidal Deformation ◽

Love Numbers ◽

Rotational Deformation ◽

Standard Values ◽

Using Data ◽

The Masses ◽

Liquid Magma

Transit timing variations (TTVs) can be a very efficient way of constraining masses and eccentricities of multi-planet systems. Recent measurements of the TTVs of TRAPPIST-1 have led to an estimate of the masses of the planets, enabling an estimate of their densities and their water content. A recent TTV analysis using data obtained in the past two years yields a 34 and 13% increase in mass for TRAPPIST-1b and c, respectively. In most studies to date, a Newtonian N-body model is used to fit the masses of the planets, while sometimes general relativity is accounted for. Using the Posidonius N-body code, in this paper we show that in the case of the TRAPPIST-1 system, non-Newtonian effects might also be relevant to correctly model the dynamics of the system and the resulting TTVs. In particular, using standard values of the tidal Love number k2 (accounting for the tidal deformation) and the fluid Love number k2f (accounting for the rotational flattening) leads to differences in the TTVs of TRAPPIST-1b and c that are similar to the differences caused by general relativity. We also show that relaxing the values of tidal Love number k2 and the fluid Love number k2f can lead to TTVs which differ by as much as a few 10 s on a 3−4-yr timescale, which is a potentially observable level. The high values of the Love numbers needed to reach observable levels for the TTVs could be achieved for planets with a liquid ocean, which if detected might then be interpreted as a sign that TRAPPIST-1b and TRAPPIST-1c could have a liquid magma ocean. For TRAPPIST-1 and similar systems the models to fit the TTVs should potentially account for general relativity, for the tidal deformation of the planets, for the rotational deformation of the planets, and to a lesser extent for the rotational deformation of the star, which would add up to 7 × 2 + 1 = 15 additional free parameters in the case of TRAPPIST-1.

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