Issue 
A&A
Volume 528, April 2011



Article Number  A18  
Number of page(s)  7  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201015803  
Published online  18 February 2011 
On the degeneracy of the tidal Love number k_{2} in multilayer planetary models: application to Saturn and GJ 436b
^{1}
Institute of Physics, University of Rostock,
18051
Rostock,
Germany
email: ulrike.kramm2@unirostock.de
^{2}
Department of Astronomy and Astrophysics, University of
California, Santa
Cruz, CA
95064,
USA
^{3}
Division of Geological and Planetary Sciences, California
Institute of Technology, Pasadena, CA
91125,
USA
Received:
22
September
2010
Accepted:
23
December
2010
Context. In order to accurately model giant planets, a whole set of observational constraints is needed. As the conventional constraints for extrasolar planets like mass, radius, and temperature allow for a large number of acceptable models, a new planetary parameter is desirable in order to further constrain planetary models. Such a parameter may be the tidal Love number k_{2}.
Aims. In this paper we aim to study the capability of k_{2} to reveal further information about the interior structure of a planet.
Methods. With theoretical planetary models we investigate how the tidal Love number k_{2} responds to the internal density distribution of a planet. In particular, we demonstrate the effect of the degeneracy of k_{2} due to a density discontinuity in the envelope of a threelayer planetary model.
Results. The effect of a possible outer density discontinuity masks the effect of the core mass on the Love number k_{2}. Hence, there is no unique relationship between the Love number k_{2} and the core mass of a planet. We show that the degeneracy of k_{2} with respect to a layer boundary in the envelope also occurs in existing planets, e.g. Saturn and the Hot Neptune GJ 436b. As a result of the degeneracy, the planetary parameter k_{2} cannot be used to further constrain the core mass of stateoftheart Saturn models and for GJ 436b only a maximum possible core mass can be derived from a given k_{2}. To significantly narrow the uncertainty about the core mass of GJ 436b the combined knowledge of k_{2} and atmospheric metallicity and temperature profile is necessary.
Key words: planets and satellites: interiors / planets and satellites: individual: Saturn / planets and satellites: individual: GJ 436b / methods: numerical
© ESO, 2011
1. Introduction
Ever since the first extrasolar planet around a solartype star was detected (Mayor & Queloz 1995), questions about the composition and origin of extrasolar planetary objects (exoplanets) have been of major interest. Models of exoplanets are often little constrained based on the observable parameters mass, radius, and effective temperature, in particular metalrich planets (Adams et al. 2008). For solar planets additional constraints are provided by the gravitational moments which have been measured by spacecraft or Earthbased observations of the motion of satellites and hence are not accessible for extrasolar planets. However, a similar quantity does exist: the tidal Love number k_{2}. To first order in the dimensionless number that describes the effect of rigid rotation or degree 2 tidal distortion k_{2} is equivalent to J_{2} (see e.g. Hubbard 1984).
The tidal Love number k_{2} is a potentially observable parameter. Ragozzine & Wolf (2009) showed that the dominant source of apsidal precession of Hot Jupiters is the tidal interaction between the planet and its star. This tidally induced apsidal precession creates a unique variation in the transit light curve which is detectable by spacebased missions like Kepler. Another possibility of determining k_{2} is the measurement of the orbital parameters of a twoplanet system in apsidal alignment (Batygin et al. 2009). Due to tidal dissipation a coplanar twoplanet system can evolve into a tidal fixed point which is characterized by the alignment of the apsidal lines (Mardling 2007) and both orbits precess with the same rate. Batygin et al. (2009) showed that in this state the Love number k_{2} is a function of the inner planet’s eccentricity.
Like J_{2} for the solar system planets, k_{2}, if known, can be used to further constrain the models of extrasolar planets as it is sensitive to the internal density distribution of the planet. Understanding the planetary interior is important for determining not only physical processes but also the formation history. Hence, it is crucial to analyze what information can be extracted from a measured k_{2} and its implications on the planetary interior.
First, we will describe the definition and calculation of the Love numbers in Sect. 2. We also confirm the correlation between the central condensation of a planet and its Love number k_{2} within a simple twolayer model. In Sect. 3 we introduce a more sophisticated threelayer planetary model and demonstrate the degeneracy of k_{2} with respect to the density discontinuity in the envelope. We apply these results to Saturn and to the Hot Neptune GJ 436b in Sect. 4. The main results of this paper are summarized in Sect. 5.
2. The Love number k_{2}
2.1. Definition and calculation
Love numbers quantify the deformation of the gravity field of a planet in response to an external perturbing body of mass M, which can be the parent star, another planet or a satellite. M, moving in a circular orbit of radius a around a planet, causes a tideraising potential (Zharkov & Trubitsyn 1978) (1)where s is the radial coordinate of the point under consideration inside the planet, θ′ the angle between the planetary mass element at s and the center of mass of M at a, and P_{n} are Legendre polynomials. Due to the tidally induced mass shift the planet’s potential changes by , where K_{n}(s) is the Love function (Love 1911). Thus, at the planet’s surface the definition of the Love numbers k_{n} reads (2)As we are interested in low eccentricity synchronous orbits, we concentrate on the purely hydrostatic tides. For the calculation of the Love numbers we follow the approach by Zharkov & Trubitsyn (1978), see also Gavrilov & Zharkov (1977) and Gavrilov et al. (1975). A Love number of degree n is obtained from (3)where T_{n}(R_{p}) is the value of the function T_{n}(s) at the planet’s surface, R_{p} is the radius of the planet and g_{0} the surface gravity for the unperturbed planet. The function T_{n}(s) satisfies the following second order differential equation: (4)The radial coordinate is represented by s and ρ(s) and V(s) give the unperturbed density distribution and potential of the planet, respectively. The primes denote first and second differentiation with respect to the radius s. If the planet has an internal density jump then the jump condition for the function T is Here b is the radial position of the density jump and the “+” (“−”) denotes a place just outside (inside) the discontinuity. This procedure assumes linear response, i.e. the tidal distortion is assumed to be small. Note that the only input needed for the calculation of a Love number k_{n} is the radial density distribution ρ(s) of the planet. Hence, the Love numbers contain important information about the interior structure of a planet. In this paper we focus on the Love number k_{2} (twice the apsidal motion constant described by Sterne 1939).
2.2. Central condensation and twolayer model
The Love number k_{2} is of special interest. It is a measure for the level of central condensation of an object: the more homogeneous the planet in mass distribution, the bigger the Love number k_{2}. Maximum homogeneity is represented by a planet of constant density, yielding the maximum value of k_{2} of 1.5. If the planet is more centrally condensed, the Love number decreases. A planet with a density distribution of a n = 1 polytrope has a value of k_{2} = 15/π^{2} − 1 = 0.5198... Planets with a core can have an even smaller Love number due to a stronger central condensation, e.g. Saturn interior models with a ~10 M_{⊕} core give k_{2} = 0.32. That is why it has been suggested to use k_{2} to infer the presence of a massive core (Ragozzine & Wolf 2009).
First, we investigate the dependence of k_{2} on the density distribution within a simple twolayer model, consisting of a core with constant density A and an n = 1 polytropic envelope: The functions ρ_{1}(x) and ρ_{2}(x) give the radial density distribution of the envelope and core, respectively. The radius coordinate has been scaled such that x = 1 is the surface of the planet and x_{c} gives the radius of the core in planet radii (x = r/R_{p}). The core density A is always greater than ρ_{1}(x_{c}). The free parameters of this model are the core radius x_{c} and the quantity q. While x_{c} only influences the core density, q influences both the core density and the density in the envelope. The core density A is determined by the choice of x_{c} and q under the condition that just outside of the core hydrostatic equilibrium is satisfied. Due to the strong influence of q on the core density A, a change in q is equivalent with changing the ratio of core mass to total mass M_{core}/M_{total}^{1}. It is necessary to choose q ≤ π since we must have ρ_{1}(x) > 0. At the surface ρ_{1}(1) = 0.
The use of an n = 1 polytrope is motivated by two considerations. First, it is a good approximation for the behavior of hydrogenhelium mixtures for the range of pressures and temperatures of interest and is even still roughly correct if modest amounts of ice or rock are mixed with hydrogen and helium. Second, it yields an analytic form (given above) for the density profile, thus making the Love number calculations straightforward. The parameter q is directly related to the proportionality constant K in the assumed polytropic equation of state P = Kρ^{(1 + 1/n)}. Since our intent here is to understand the general nature of the dependence of the Love number on density structure and not to derive highly specific and precise models, it serves our purpose well.
Fig. 1
Love number k_{2} in dependence on the ratio of core mass to total mass for theoretical twolayer models consisting of a constant density core and an n = 1 polytropic envelope, see Eqs. (7) and (8). The lines show solutions for core radii of x_{c} = 0.05 (solid), 0.10 (dashed), 0.15 (dotted) and 0.20 (dashdotted). The black diamond shows the solution for the coreless polytropic planet. To obtain different values for M_{core}/M_{total} the parameter q has been varied. q = π gives the minimum value of M_{core}/M_{total}. Thick lines show the region where 0.9π < q < π which corresponds to realistic ρ_{1}(x_{c})/A ratios. 
By choosing x_{c} and q we can construct different twolayer models. Figure 1 shows the Love number k_{2} as function of M_{core}/M_{total} for different core radii of respectively x_{c} = 0.05, 0.10, 0.15 and 0.20, and in the coreless case. To obtain different values for M_{core}/M_{total} the parameter q has been varied. A value of q = π gives the minimum ratio M_{core}/M_{total} as the core density A decreases with q. Reducing the q parameter causes a rapid increase of the core density A but only a modest increase of ρ_{1}(x_{c}). For the q parameter range of 0.9π < q < π the ratio of the envelope density at the coremantle boundary and the core density has values of ρ_{1}(x_{c})/A ≳ 0.01. Models with too low q become unrealistic as they have an enormously small ρ_{1}(x_{c})/A. Note that the choice of the core radius x_{c} has a compensating effect and q could be choosen lower if x_{c} is bigger. In reality, lower k_{2} can be obtained in particular if the mantle density distribution does not follow an n = 1 polytrope and a compressive equation of state for the core material is used. As can be seen from Fig. 1, k_{2} decreases with increasing ratio of core mass to total mass. This behavior applies for all choosen core radii. For a planet of a given ratio of core mass to total mass, a bigger core radius x_{c} implies a lower density for that core and hence less central condensation and a larger Love number, as Fig. 1 shows. The data points for the smallest mass ratios have been generated with the maximum value of q = π. This means that for a twolayer model with a fixed core radius there is a minimum core mass.
These calculations within a simple twolayer model give an intuitive interpretation of the characteristic of k_{2} of being a measure for the level of central condensation: the bigger the core mass, the more centrally condensed and hence the smaller the Love number k_{2}. However, as we will show in the next section this simple deduction is no longer possible when there is another density discontinuity in the envelope of the planet.
3. Degeneracy in threelayer models
In this section we will investigate the behavior of k_{2} in a more complicated model. In addition to the density discontinuity at the coremantle boundary we introduce another discontinuity in the envelope of the planet. A three layer structure is a common assumption in planet modeling and has been used for modeling the solar system giants Jupiter and Saturn (see e.g. Guillot 1999; Saumon & Guillot 2004). Such a separation of layers in the planetary envelope can occur as a result of demixing of hydrogen and helium (Stevenson & Salpeter 1977). It could also arise from double diffusive convection in the presence of a density gradient that is introduced during accretion or because of subsequent core erosion (Stevenson 1982). Layer boundaries are also compatible with standard models of planet formation (Hubbard et al. 1995). We define our theoretical threelayer model as follows: It consists of a core with constant density A and two polytropic envelopes described by ρ_{1}(x) (outer envelope) and ρ_{2}(x) (inner envelope). The same characteristics as for the twolayer model apply. The layer boundary in the envelope is placed at x_{m}. We choose the parameters q_{1} and q_{2} (near but smaller than π and not much different from each other). The location of the density discontinuities x_{c} and x_{m} are also free parameters. The core density A is determined as for the twolayer model. The continuity of pressure and gravity at x_{m} is used to calculate x_{a}. We can then choose B to get a specified nondimensionalized size of the density jump Δρ = (ρ_{2} − ρ_{1})/ρ_{1}_{x = xm}. Summarizing, in this threelayer model we can vary the parameters x_{c}, x_{m}, q_{1}, q_{2} and Δρ while A, B and x_{a} are determined by the parameters chosen.
3.1. Love number and core mass
The parameters characterizing the density discontinuity in the envelope are x_{m} and Δρ, position and size of the discontinuity. Thus, in order to investigate the influence of the outer discontinuity on the Love number k_{2} we varied the parameters x_{m} and Δρ from 0.5 to 0.9 planet radii (increment 0.01) and from 0.01 to 0.5 (increment 0.01), respectively. For the example we give here, the other free parameters are fixed at q_{1} = 0.98π, q_{2} = 0.99π and x_{c} = 0.1. This choice is in order to keep the envelope structure close to an n = 1 polytrope. Together with a moderate value for the core size this example for the density distribution mimics a Jupiterlike planet.
Fig. 2
Lines of equal k_{2} (blue) and equal M_{core}/M_{total} (red) in Δρx_{m} parameter space for a theoretical threelayer model, see Eqs. (9)–(11) with the fixed parameters q_{1} = 0.98π, q_{2} = 0.99π and x_{c} = 0.1. Numbers give the corresponding values of k_{2} and M_{core}/M_{total}, respectively. The equik_{2}lines demonstrate the degeneracy of k_{2} with respect to the position (x_{m}) and size (Δρ) of a density discontinuity in the envelope and the ratio of core mass to total mass (a blue line intersects several red lines). 
The change of the Love number k_{2} in dependence on the parameters of the outer density discontinuity is shown by Fig. 2. Lines of equal k_{2} and their values are given. This demonstrates a degeneracy of the Love number k_{2} with respect to the outer density discontinuity. One can always find many different x_{m}Δρpairs (that means different threelayer planetary models) that give the same k_{2}. These models lie on one of the equik_{2}lines and hence one cannot distinguish between these models by a measurement of k_{2}. In addition to the Love number k_{2} we calculated the ratio of core mass to total mass M_{core}/M_{total} in the same parameter space. Lines of equal mass ratio are also shown in Fig. 2. For a specific value of k_{2} planetary models with different mass ratios are possible, see intersections of equik_{2} and equiM_{core}/M_{total}lines. (Note that the resulting M_{core}/M_{total} have a very limited range because only x_{m} and Δρ change and M_{core}/M_{total} is more sensitive to the q parameters which are fixed in our example here. Anyhow, other choices of the q parameters and/or x_{c} would yield qualitatively similar results.) In the same manner, planetary models with identical mass ratios can have different k_{2} values. This demonstrates that a unique relationship between the core mass and the Love number k_{2} is no longer valid in a threelayer model. It can also be seen from Fig. 2 that planets with a more massive core can have a bigger Love number than planets with a smaller core (compare e.g. left lower part with right upper part of Fig. 2). This is in contrast to the results that were suggested by the simple twolayer model where more massive cores lead to smaller k_{2}. In conclusion, the effect of a possible density discontinuity in the envelope of a planet on the Love number k_{2} can mask the effect of the core mass.
Even though we found that there is a nonunique relation between the Love number k_{2} and the core mass, Fig. 2 is still consistent with the assumption of k_{2} being a measure for the level of “central condensation” of a planet as the term “central condensation” cannot be defined by the core mass alone. A big density jump Δρ means that there is a significantly higher density in the inner than in the outer envelope. This constitutes a higher level of central condensation than an envelope without a discontinuity. That is why k_{2} is generally smaller in regions with bigger Δρ. On the other hand, a very small Δρ means that there is a small difference in density at the layer boundary^{2}. Hence, the planet is more homogeneous and the Love number k_{2} tends to bigger values. In the region of small Δρ the equik_{2} lines also become flatter, because the position x_{m} of the discontinuity becomes less important for Δρ → 0, as the distinguishability between a twolayer and a threelayer model disappears.
3.2. Moment of inertia
Another quantity closely related to the structure of a planet is its moment of inertia C, or its nondimensional form . While it has been possible to obtain moments of inertia from measurements of precession for Earth and Mars (see e.g. Folkner et al. 1997), it is more difficult for the giant planets. For the modeling of the internal structure of the giants, the gravitational moments have been used instead. The firstorder response of a planet to rotational distortion (quantified by J_{2}) can be related to the planet’s nondimensional polar moment of inertia with the RadauDarwin equation (Hubbard 1984): (12)To linear order with , the ratio of centrifugal force to gravity. This equation implies that the response coefficient is a function of the moment of inertia only.
Since the tidal potential (1) has the same functional form as the rotational potential, the tidal response of a planet can be treated analogously to the rotational one. For a liquid or perfectly elastic planet the tidal response coefficient is the same as the rotational one (Hubbard 1984). Substituting for in Eq. (12), we find: (13)With this relation we can calculate the nondimensional moment of inertia from our previously generated values of the Love number k_{2}. Another, more direct, method to obtain the moment of inertia for our models is to integrate over the density distribution: (14)We calculated moments of inertia for the reference threelayer model of Sect. 3.1. In analogy to Fig. 2 the characteristics of the envelope layer boundary are varied. Figure 3 shows lines of equal moment of inertia, obtained from Eq. (13) or (14).
Fig. 3
Lines of equal moment of inertia C_{nd} for the same threelayer model as in Sect. 3.1 and in the same Δρx_{m} parameter space. Blue lines are for the moments of inertia obtained directly from the Love numbers k_{2} via Eq. (13). Green lines show the moments of inertia calculated from the density distribution with Eq. (14). 
Of course the equiC_{nd}lines generated from (13) are equivalent to the equik_{2}lines of Fig. 2, as (13) is a onetoone correspondence. Obviously, they are different from the results obtained from (14). This is due to the approximations made for the derivation of Eq. (13).
The RadauDarwin Eq. (12) itself is an approximation. It only approaches a correct result C_{nd} → 0.4 for the limit of a completeley homogeneous planet. It is not true in the limit C_{nd} → 0 (see e.g. Murray & Dermott 1999). This feature of the RadauDarwin equation can be seen in Fig. 3. The difference between the equiC_{nd}lines becomes bigger with increasing Δρ, corresponding to stronger central condensation.
We conclude that the RadauDarwin relation is a too crude approximation to describe the moment of inertia of gas giant planets.
4. Application to existing planets
In this section we apply the results obtained for the simple models shown above to interprete interior models for real planets. We will show that the degeneracy of k_{2} is not just an artifact of our theoretical threelayer model but actually occurs in existing solar as well as extrasolar planets.
4.1. Saturn
As the second largest planet in our solar system, Saturn has attracted much attention in the past. With the help of space missions like e.g. Cassini observational data have been derived that provides important constraints for modeling Saturn. Voyager measurements have yielded a depletion of helium compared to the solar abundance (Conrath & Gautier 2000). In conclusion, the missing helium must be hidden deep inside the planet. The helium depletion in the outer part of the envelope can be explained by a phase separation of hydrogen and helium. Lorenzen et al. (2009) and Morales et al. (2009) showed recently that demixing occurs on large scale at standard Saturn interior model conditions, motivating a layeredenvelope assumption for Saturnian models. Furthermore, the demixing theory allows to reproduce the correct age of Saturn which was not possible without demixing (Fortney & Hubbard 2003).
As Saturn can be modeled in a three or more layer approach, it is also affected by the k_{2}degeneracy. We demonstrate this effect by examining selected Saturn models from Nettelmann (2009). Those models have a rocky core based on the rock equation of state (EOS) by Hubbard & Marley (1989) and two isentropic envelopes of hydrogen, helium and water (as representative for metals) based on the linear mixing Rostock equation of state (LMREOS), see Holst et al. (2008), Kietzmann et al. (2007) and French et al. (2009), respectively. The pressure at the layer boundary in the envelope is called the transition pressure P_{12}. By changing P_{12} the layer boundary can be shifted.
The variation of P_{12} affects the core mass of Saturn. We assumed a layer boundary between 1 and 3.8 Mbar, where P_{12} = 1 Mbar gives a rocky core mass of 14 M_{⊕} and P_{12} = 3.8 Mbar gives the maximum value of P_{12} as the core vanishes when the layer boundary is put that deep inside the planet (Nettelmann 2009). It is important to note that all these models reproduce the observational constraints, including the gravitational moments J_{2}, J_{4} and J_{6}. This means that the parameter P_{12} is not fixed. As a consequence, there is a large uncertainty in the core mass of at least 14 M_{⊕}. We calculated the Love number k_{2} for these models and found for all these Saturn models k_{2} ≃ 0.32. Even though the core mass varies fom 0 to 14 M_{⊕}, no significant difference in the k_{2} values can be seen. This is an expected result because k_{2} is proportional to J_{2} to first order in the expansion of the planet’s potential (see e.g. Hubbard 1984) and all models match the observed J_{2}.
The constant k_{2}value demonstrates the effect described in the previous section: the influence of the outer density discontinuity masks the effect of the core mass. For one k_{2} value different planetary models can be found. Increasing the transition pressure P_{12} is equivalent to shifting the density discontinuity inwards to smaller x_{m} (compare theoretical threelayer model in Sect. 3). At the same time the size of the density jump Δρ grows in order to obtain models that are consistent with the measured gravitational moments. So all Saturn models presented in this work lie on an equik_{2}line (compare Fig. 2), showing the degeneracy of the Love number k_{2} with respect to the outer layer boundary in Saturn.
Figure 2 also predicts a growing core mass with a deeper layer boundary and increasing density jump. However, this is not applicable here because our Saturn models are based on the additional constraints of matching the gravitational moments. These constraints were not considered for the simple models in Sect. 3. In the planet modeling procedure the gravitational moments are adjusted by varying the envelope metallicities. This means that the metallicities in the envelope are determined by the observed gravitational moments for a specified P_{12}. Shifting the layer boundary inwards (bigger P_{12}, smaller x_{m}) forces the metallicity of the inner envelope to increase in order to ensure consistency with the gravitational moments. These “additional” metals for the inner envelope are taken from the core, which is why the core mass shrinks with increasing transition pressure P_{12}.
4.2. GJ 436b
For modeling extrasolar planets the Love number k_{2} can be a valuable constraint because the gravitational moment J_{2} is not accessible to observations. To first order k_{2} is equivalent to J_{2} (see e.g. Hubbard 1984) and is potentially observable due to apsidal precession of the planet’s orbit (Ragozzine & Wolf 2009) or measurement of the orbital parameters of a twoplanet system in apsidal alignment (Mardling 2007; Batygin et al. 2009). However, interior models of extrasolar planets are also subject to k_{2}degeneracy. Here we demonstrate this effect with the example of the Hot Neptune GJ 436b. Numerous investigations about this transiting planet have been made in the past (see e.g. Adams et al. 2008; Figueira et al. 2009; Torres et al. 2008). Based on mass, radius, and temperature measurements alone, a large variety of models is possible, ranging from a “water world” to a SuperEarth. We have performed extensive calculations for modeling the interior structure of GJ 436b (Nettelmann et al. 2010b). We considered twolayer models with a rocky core and one envelope where water is homogeneously mixed into H/He as well as threelayer models consisting of a rocky core, a water layer and an outer H/He layer with metal (water) abundance Z_{1}. For selected models we calculated the Love number k_{2}, for details about the modeling see Nettelmann et al. (2010b). Here we present a new and extended version of Fig. 2 in Nettelmann et al. (2010b) in order to show the degeneracy of k_{2}.
Fig. 4
Love number k_{2} and core masses of twolayer models (squares) and threelayer models (diamonds) of GJ 436b for temperatures of 700 K (cyan) and 1300 K (orange) of the isothermal (1–100 bar) atmosphere. Color coded circles denote models of Jupiter (red), Saturn (yellow), Neptune (blue) and a 20 M_{⊕} water planet (black). Threelayer models with the same metal abundance in the outer envelope are connected with dashed (Z_{1} = 0), dotted (Z_{1} = 0.5) and dashdotted (Z_{1} = 0.7) lines. The grey dashed line marks a highly degenerate area constructed to contain all Z_{1} = 0 threelayer models (the position of this line has been shifted compared to Fig. 2 in Nettelmann et al. (2010b) in order to account for the new 1300 K model series). 
Figure 4 shows Love numbers k_{2} and core masses of various two and threelayer models of GJ 436b. All models shown here have an isothermal outer layer from 1 to 100 bar. For comparison, Fig. 4 also contains the solar planets Jupiter, Saturn and Neptune and a 20 M_{⊕} water planet. For twolayer models solutions are shown which have a metal abundance of Z_{1} = 0, 0.5, and the largest possible value when M_{core} = 0. Model series with three layers are displayed for constant outer envelope metallicities of Z_{1} = 0, 0.5, and 0.7. Solutions are shown for two different surface temperatures of 700 and 1300 K. Even though possible, we consider colder models to be unrealistic as recent results by Spiegel et al. (2010) and Lewis et al. (2010) predict 1300 K and 1100 K, respectively. In such warm models water is in the plasma phase and miscible with H/He. Hence, the water EOS applied to the inner envelope of such warm threelayer models represents a mean density of a H/He/H_{2}O/rock mixture.
For GJ 436b we find k_{2} = 0−0.72. Models of GJ 436b with k_{2} = 0.02−0.24 can have any core mass between 0 and ~0.9 planet masses. This means that a measurement of k_{2} in this regime would not help constraining the interior further. This highly degenerate area includes all Z_{1} = 0 threelayer models (note that this area has been extended compared to Fig. 2 in Nettelmann et al. 2010b, in order to account for the new 1300 K model series). However, one can predict maximum possible core masses for specified k_{2}. The uncertainty in the core mass decreases with increasing k_{2}. The upper limits of possible core masses, if k_{2} and the atmospheric temperature profile are known, are necessarily given by twolayer models with one homogeneous envelope. Any redistribution of metals from an outer to an inner part of the envelope would, if the central condensation is to be kept at constant k_{2} value, require a compensating transport of core material to the envelope. The core mass would therefore decrease and drop below these upper limits. This is shown by the threelayer models with constant envelope metallicity of respectively Z_{1} = 0, 0.5, 0.7, which all have lower core masses than the twolayer models. Threelayer models contain a density discontinuity in the envelope, dividing it into an inner water envelope (Z_{2} = 1) and an outer H/He/H_{2}O envelope (Z_{1} = 0, 0.5 and 0.7). The threelayer model approach gives the additional free parameter P_{12}, which is the pressure at the envelope layer boundary. To construct the equalmetallicity lines the metallicities are kept at constant values (Z_{2} = 1 and Z_{1} = 0, 0.5, 0.7) and P_{12} is varied from a minimum value where M_{core} = 0 to a maximum value where the water layer vanishes and the model is identical to a twolayer model with the same envelope metallicity (for the modeling procedure see also Nettelmann et al. 2010b).
Models of equal envelope metallicity Z_{1} show systematic behavior in M_{core}k_{2} space. For low outer envelope metallicities the Love Number k_{2} shrinks with growing core mass due to the stronger central condensation created by the bigger core. With a high metal abundance in the outer envelope this behavior changes and k_{2} increases slightly with growing core mass. The higher amount of metals in the outer envelope leads to a smaller density gap at the envelope layer boundary so that the planet becomes more homogeneous than planets with a low metallicity envelope. As can be seen from Fig. 4, this effect, and especially the transition between the two different slopes, is strongly temperature dependent. For lower temperatures (see 700 K curves) k_{2} increases with the core mass already at smaller metallicities than for warmer temperatures (compare 1300 K curves). That is because colder envelope temperatures increase their density and hence the homogeneity of the planet. The upper limit of possible core masses increases with decreasing atmospheric temperatures.
Concluding, an observational k_{2} would imply a maximum possible core mass. But, especially for k_{2} < 0.24, it would not help to further constrain interior models of GJ 436b because in that regime the solutions are too degenerate. However, a k_{2} > 0.24, if measured, would indicate a maximum core mass M_{c} < 0.5 M_{p} and large outer envelope metallicities. Only with additional information about the planet’s atmospheric composition (Z_{1}) together with k_{2} interior models of GJ 436b can be further constrained. If we had knowledge about the atmospheric temperature profile and metallicity and the Love number k_{2}, the core mass could be determined or at least strongly narrowed, depending on the slope of the equimetallicitylines. However, not all aspects of metallicity are potentially measurable in an exoplanet, only those that are the consequence of elements which are in the gas phase in the observable part of the atmosphere. As an example in our own solar system, the metallicities of Jupiter and Saturn are not measurable spectroscopically because water forms clouds deep below the photosphere.
The results for GJ 436b are in perfect agreement with our results obtained from Fig. 2. For a given k_{2} value several planetary models and core masses are possible (highly degenerate area in Fig. 4). Even for constant outer envelope metallicity there are also models that have a larger k_{2}, indicating a rather homogeneous planet, but nevertheless a more massive core. This behavior is caused by the outer density discontinuity at the waterH/He layer boundary as discussed in Sect. 3 and predicted by Fig. 2.
5. Summary
In this paper we investigated the effect of the density distribution of a planet on its tidal Love number k_{2} in order to find out what conclusions can be drawn from a measured k_{2} on the internal structure of a planet. We confirmed that the Love number k_{2} is a measure of the level of central condensation of a planet. However, in a three layer model approach k_{2} is not a unique function of the core mass. There is a degeneracy of k_{2} with respect to a density discontinuity in the envelope of the planet. It is possible to have several acceptable models for a given k_{2} value, which can differ significantly in core mass. The effect of the outer density discontinuity on k_{2} is compensating the effect of the core. Furthermore, we showed that the RadauDarwin relation is a too crude approxomation to describe the moment of inertia of gas giant planets.
We verified our results on k_{2} with models of existing planets. For Saturn the freedom to place the layer boundary in the envelope leads to a high uncertainty in the core mass. Regardless of the core mass all Saturn models have the same k_{2}, demonstrating the degeneracy caused by the outer layer boundary.
For extrasolar planets the Love number k_{2} can be an equivalent constraint to J_{2} for the solar system planets. However, one has to be careful with estimates about the core mass derived from k_{2} as degeneracy may also occur in extrasolar planets. For GJ 436b we find a highly degenerate area of k_{2} < 0.24 where a measurement of k_{2} would barely help to further constrain the interior models. Only a maximum possible core mass and for k_{2} > 0.24 a large metallicity can be inferred. With additional knowledge about the atmospheric metal abundance the uncertainty about the core mass could be significantly narrowed.
Tabulating k_{2} values of various planetary models can prove to be very useful once k_{2} is actually measured for extrasolar transiting planets. For instance, for the SuperEarth GJ 1214bNettelmann et al. (2010a) demonstrated that H/He or water envelopes result in significantly different values of k_{2}. Furthermore, we have shown in this paper that even though the Love number k_{2} is a degenerate quantity it can help constraining the core mass of a planet. Knowledge about the core masses of planets is highly desired because it is thought to help to distinguish between the possible planet formation scenarios of core accretion (see e.g. Pollack et al. 1996) and gravitational instability (see e.g. Boss 1997). However, one has to keep in mind that core accretion models can also result in very small cores of 1.7 M_{⊕} or 0.25 M_{⊕} in the case of grainfree or even metalfree envelopes, respectively (Hori & Ikoma 2010). On the other hand, gravitational instability models allow the formation of a massive core as well if the protoplanet is cold enough for grain settling to take place (Helled & Schubert 2008). A clear distinction between the two formation models can only be made for massive extrasolar giant planets ≥5 M_{J}. Helled & Schubert (2008) showed that such massive protoplanets formed by disk instability cannot build up a core at all due to their high internal temperatures and evaporation of the grains.
Acknowledgments
We acknowledge helpful discussions with Ralph Neuhäuser. U.K. and R.R. acknowledge support from the DFG SPP 1385 “Young Planets”.
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All Figures
Fig. 1
Love number k_{2} in dependence on the ratio of core mass to total mass for theoretical twolayer models consisting of a constant density core and an n = 1 polytropic envelope, see Eqs. (7) and (8). The lines show solutions for core radii of x_{c} = 0.05 (solid), 0.10 (dashed), 0.15 (dotted) and 0.20 (dashdotted). The black diamond shows the solution for the coreless polytropic planet. To obtain different values for M_{core}/M_{total} the parameter q has been varied. q = π gives the minimum value of M_{core}/M_{total}. Thick lines show the region where 0.9π < q < π which corresponds to realistic ρ_{1}(x_{c})/A ratios. 

In the text 
Fig. 2
Lines of equal k_{2} (blue) and equal M_{core}/M_{total} (red) in Δρx_{m} parameter space for a theoretical threelayer model, see Eqs. (9)–(11) with the fixed parameters q_{1} = 0.98π, q_{2} = 0.99π and x_{c} = 0.1. Numbers give the corresponding values of k_{2} and M_{core}/M_{total}, respectively. The equik_{2}lines demonstrate the degeneracy of k_{2} with respect to the position (x_{m}) and size (Δρ) of a density discontinuity in the envelope and the ratio of core mass to total mass (a blue line intersects several red lines). 

In the text 
Fig. 3
Lines of equal moment of inertia C_{nd} for the same threelayer model as in Sect. 3.1 and in the same Δρx_{m} parameter space. Blue lines are for the moments of inertia obtained directly from the Love numbers k_{2} via Eq. (13). Green lines show the moments of inertia calculated from the density distribution with Eq. (14). 

In the text 
Fig. 4
Love number k_{2} and core masses of twolayer models (squares) and threelayer models (diamonds) of GJ 436b for temperatures of 700 K (cyan) and 1300 K (orange) of the isothermal (1–100 bar) atmosphere. Color coded circles denote models of Jupiter (red), Saturn (yellow), Neptune (blue) and a 20 M_{⊕} water planet (black). Threelayer models with the same metal abundance in the outer envelope are connected with dashed (Z_{1} = 0), dotted (Z_{1} = 0.5) and dashdotted (Z_{1} = 0.7) lines. The grey dashed line marks a highly degenerate area constructed to contain all Z_{1} = 0 threelayer models (the position of this line has been shifted compared to Fig. 2 in Nettelmann et al. (2010b) in order to account for the new 1300 K model series). 

In the text 
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