Issue 
A&A
Volume 538, February 2012



Article Number  A146  
Number of page(s)  8  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201118141  
Published online  16 February 2012 
Constraining the interior of extrasolar giant planets with the tidal Love number k_{2} using the example of HATP13b
^{1} Institute of Physics, University of Rostock, 18051 Rostock, Germany
email: ulrike.kramm2@unirostock.de
^{2} Dept. of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA
^{3} Astrophysical Institute and UniversityObservatory, Schillergässchen 2–3, 07745 Jena, Germany
Received: 23 September 2011
Accepted: 15 November 2011
Context. Transit and radial velocity observations continuously discover an increasing number of exoplanets. However, when it comes to the composition of the observed planets the data are compatible with several interior structure models. Thus, a planetary parameter sensitive to the planet’s density distribution could help constrain this large number of possible models even further.
Aims. We aim to investigate to what extent an exoplanet’s interior can be constrained in terms of core mass and envelope metallicity by taking the tidal Love number k_{2} into account as an additional, possibly observable parameter.
Methods. Because it is the only planet with an observationally determined k_{2}, we constructed interior models for the Hot Jupiter exoplanet HATP13b by solving the equations of hydrostatic equilibrium and mass conservation for different boundary conditions. In particular, we varied the surface temperature and the outer temperature profile, as well as the envelope metallicity within the widest possible parameter range. We also considered atmospheric conditions that are consistent with nongray atmosphere models. For all these models we calculated the Love number k_{2} and compared it to the allowed range of k_{2} values that could be obtained from eccentricity measurements of HATP13b.
Results. We use the example of HATP13b to show the general relationships between the quantities temperature, envelope metallicity, core mass, and Love number of a planet. For any given k_{2} value a maximum possible core mass can be determined. For HATP13b we find M_{core} < 27 M_{⊕}, based on the latest eccentricity measurement. We favor models that are consistent with our model atmosphere, which gives us the temperature of the isothermal region as ~2100 K. With this external boundary condition and our new k_{2}interval we are able to constrain both the envelope and bulk metallicity of HATP13b to 1−11 times stellar metallicity and the extension of the isothermal layer in the planet’s atmosphere to 3−44 bar. Assuming equilibrium tidal theory, we find lower limits on the tidal Q consistent with 10^{3}−10^{5}.
Conclusions. Our analysis shows that the tidal Love number k_{2} is a very useful parameter for studying the interior of exoplanets. It allows one to place limits on the core mass and estimate the metallicity of a planet’s envelope.
Key words: methods: numerical / planets and satellites: interiors / planets and satellites: individual: HATP13b
© ESO, 2012
1. Introduction
Today more than 180 transiting exoplanets have been discovered. Within the exoplanet family planets that are found to transit their host star are especially important. The knowledge of mass and radius enables us to infer the density and bulk composition of a planet. Further, transit and secondary eclipse observations reveal information about an exoplanet’s atmosphere.
Despite all the information given, we are still far away from knowing the composition of the deep interior of exoplanets. Characteristics like the existence and mass of a potential core or the amount and distribution of heavy elements are quite ambiguous. Yet, this information is highly desired in order to understand planet formation.
For the solar system planets, the ambiguity of interior models can be reduced by taking into account information from the gravity field, which is quantified by the gravitational moments J_{2}, J_{4}, and J_{6}. In our solar system these parameters are measureable and improved methods continuously increase their accuracy. However, for an extrasolar planet gravitational moments cannot be determined. Hence, we need a similar parameter that is accessible and will also provide us with information about the interior density distribution of the planet. This can be accomplished with the tidal Love number k_{2} (Love 1911; Gavrilov et al. 1975; Gavrilov & Zharkov 1977; Zharkov & Trubitsyn 1978). As it is equivalent to J_{2} for the solar system planets (Hubbard 1984), it is promising that k_{2} will help us to further constrain the interior models of exoplanets. Recent studies have shown that k_{2} is itself in general a degenerate quantity if considered in a threelayer model (Kramm et al. 2011).
In this study, we focus on the transiting Hot Jupiter HATP13b (Bakos et al. 2009). This planet is of great interest because it is the only planet so far which has an observationally determined value for the Love number k_{2} (Batygin et al. 2009). Hence, it gives us the chance to investigate how much information about the interior structure can be inferred from an actually measured k_{2} and serves as a case study in this work. We especially evaluate the impact of the new eccentricity measurement for HATP13b by Winn et al. (2010) as it gives new information about allowed k_{2} values.
In Sect. 2 we describe our modeling procedure, the equation of state (EOS) used, and our calculation of the Love number. The known observational constraints of the examined planet HATP13b are given in Sect. 3 where we also discuss the effect of different eccentricity measurements, and describe the setup of our interior models. The results are explained in Sect. 4, and in Sect. 5 we compare with previous results. The applicability of the underlying theory is discussed in the Appendix. A summary is given in Sect. 6.
2. Methods
2.1. Modeling procedure
We construct planetary interior models by integrating the equation of hydrostatic equilibrium (1)and the equation of mass conservation (2)inward, where P is the pressure, r the radial coordinate, ρ(r) the mass density at r, m(r) the mass inside the radius r, and M_{p} and R_{p} the total mass and radius of the planet, respectively. We define the surface of the planet R_{p} at the 1bar pressure level P_{1} = 1 bar. For Hot Jupiters the measured transit radius may be at lower pressures of about ~10 mbar (Fortney et al. 2003). For HATP13b, however, the very outermost layer from 1 bar to 10 mbar does not contribute significantly to the radius (within the error bars for the transit radius measurement). Hence, we start our calculation of the interior at the 1bar pressure level as previously done for solar system planets (Guillot 1999). The equations above require the following outer boundary conditions: the radius inside of P_{1} equals the planet radius, so r(P_{1}) = R_{p}, and the mass inside of R_{p} is equal to the planet’s mass, so m(R_{p}) = M_{p}. Furthermore, a planet has to obey the inner boundary condition m(r = 0) = 0. To fulfill this condition, we assume that a core exists and choose the core mass (M_{core} ≥ 0) such that total mass conservation is ensured. Equation (1) needs a P − ρrelation (equation of state). For details about the EOS used, see Sect. 2.2. To obtain ρ_{1} = ρ(P_{1},T_{1}) from the EOS we need the surface temperature T_{1} which we consider as a variable parameter.
2.2. EOS
For the interior models in this work we assume a composition of hydrogen, helium, metals and rocks. The rocks are confined to a core and we apply the pressuredensity relation by Hubbard & Marley (1989) which approximates a mixture of 38% SiO_{2}, 25% MgO, 25% FeS, and 12% FeO. For the envelope we suppose a mixture of H, He, and metals. For H and He we use the interpolated SCvHi EOS (Saumon et al. 1995). To represent elements heavier than He (metals) we scale the density of the HeEOS by a factor of four.
2.3. Love number calculation
For the interior models generated in this work we calculate the Love number k_{2} for a hydrostatic planet. This planetary property quantifies the deformation of the quadrupolic gravity field of the planet in response to an external massive body. In the cases of exoplanetary systems of interest, the parent star of mass M_{∗} causes a tideraising potential (Zharkov & Trubitsyn 1978) (3)where a is the radius of the planet’s orbit, s the radial coordinate of a point inside the planet, θ′ the angle between a planetary mass element at s and M_{∗} at a, and P_{n} are Legendre polynomials. The response of the planet’s potential at the surface is given by (4)where R_{p} is the radius of the planet and k_{n} are its tidal Love numbers. Like the gravitational moments J_{2n} they are determined by the planet’s internal density distribution. For a more detailed description of the method used to calculate Love numbers and an analysis about some general dependences of k_{2} on the density distribution see Kramm et al. (2011) and references therein.
For the solar system giant planets the lowest order gravitational moments J_{2} and J_{4} have proven to be extremely valuable constraints for planet modeling (see e.g. Guillot 1999; Saumon & Guillot 2004; Nettelmann 2011). For extrasolar planets on the other hand, constraints derived from transit and radial velocity observations are often limited to the mass, radius and effective temperature of the planet. In a few cases information about the atmosphere can be obtained from spectroscopy (see e.g. Charbonneau et al. 2002; Pont et al. 2008), which at the current level helps to validate atmosphere models (Fortney et al. 2010).
It can be shown that to first order in the expansion of the planet’s potential, the Love number k_{2} is proportional to J_{2} (see e.g. Hubbard 1984), indicating that a measurement of k_{2} provides us with equivalent information like J_{2} for the solar system giants. In this way, knowledge about the Love number k_{2} can give us some insight into the planet’s interior by constituting an additional constraint if observed. In fact, an estimation of the tidal Love number k_{2} based on observations was made for the Hot Jupiter HATP13b, as it is part of a system in a tidal fixed point (see Sect. 3.1). The consequences of this k_{2}determination for the interior of HATP13b are discussed in Sect. 4. Another possibility to determine k_{2} observationally is to detect the tidally induced apsidal precession that is expected to be seen in the transit light curves of Hot Jupiters (Ragozzine & Wolf 2009).
3. Input data and model assumptions
3.1. Observational constraints
Discovered in 2009 by Bakos et al. (2009), the HATP13 system is the first exoplanetary system that contains a transiting planet that is accompanied by a wellcharacterized longerperiod companion. The planets orbit a metal rich ( [Fe/H] = +0.41 ± 0.08 dex) G4 star with L_{∗} = 2.22 ± 0.31 L_{⊙}. The inner planet HATP13b is a transiting Hot Jupiter which passes in front of its star every P_{b} = 2.916260 ± 0.000010 days, correspondig to an orbital distance of AU. The orbit is nearly circular with e_{b} = 0.021 ± 0.009^{1}. The radius and mass of HATP13b were determined with transit and radial velocity followup observations to be M_{J}, and R_{p} = 1.281 ± 0.079 R_{J}, respectiveley. The companion HATP13c is on a more distant orbit at AU with a period of P_{c} = 428.5 ± 3.0 days. As it has not been observed in transit (Szabó et al. 2010) only the minimum mass Msini = 15.2 ± 1.0 M_{J} is known. An important feature of the system is that the c planet^{2} is on a highly eccentric orbit with e_{c} = 0.691 ± 0.018. For more details about the observed parameters of the system see Bakos et al. (2009).
The high eccentricity of planet c (compared to the inner shortperiod planet) is one prerequisite for the theory of Mardling (2007) on the tidal evolution of a twoplanet system. A further requirement is that the system must be coplanar. Then, according to Mardling’s theory, the system evolves in the following phases: (1) the angle η between the apsidal lines of the two planets circulates and the eccentricities slowly oscillate at constant amplitude accompanied by a decrease of the mean value of the inner planet’s eccentricity. This occurs on the circularization time scale τ_{circ}. For HATP13b τ_{circ} ~ 0.05 Gyr assuming Q_{p} ~ 10^{5} and k_{2} ~ 0.3. (2) η librates around a fixed value η = 0,π. The eccentricities slowly oscillate with declining amplitude but maintain the mean value of the inner planet’s eccentricity. This occurs on twice the circularization time scale. At the end of this phase the system resides in the tidal fixed point characterized by aligned apsidal lines which then precess with the same rate. (3) The subsequent long term evolution proceeds on a time scale several orders of magnitude longer than the circularization time scale. So the tidal fixed point is actually a quasi fixed point. Long term tidal effects will cause a slow nonoscillatory decline of both eccentricities to zero unless the companion is not of very low mass and long period.
Bakos et al. (2009) observed that the pericenter of HATP13b is aligned with that of the outer planet HATP13c to within 4° ± 40°. If the apsidal lines are indeed aligned and if the system is coplanar, it is in the tidal fixed point. Batygin et al. (2009, hereafter BBL) showed that under this assumption one can estimate the tidal Love number k_{2} for the inner planet which in turn will give information about the interior of planet b. They first generated models that satisfy a given choice of the planetary mass, the radius, and an inferred planetary effective temperature T_{eff} = 1649 K. For these models BBL then calculated k_{2} from the density distribution ρ(r). Furthermore, they determined the fixed point eccentricity e_{b} based on the constraint that the orbits of both planets have an identical precession rate when the system is in the tidal fixed point configuration: , where is the total precession of the inner planet incorporating the four most significant contributions arising from secular evolution induced by planetplanet interaction, the tidal and rotational bulges of the planet, and general relativity, where is the precession of the orbit of planet c due to the secular evolution (see also Mardling 2007). BBL found that the different k_{2}values of the inner planet result in significantly different eccentricities e_{b} with larger e_{b} implying a smaller k_{2} and vice versa. They approximated this behavior with a fourthorder polynomial e_{b}(k_{2}). In summary, if a planetary system fulfills the requirements for the tidal evolution theory by Mardling (2007) and has evolved into the tidal fixed point, one can estimate the Love number k_{2} of the inner planet by measuring the orbital parameters of the planets. It is important to note that the estimates for k_{2} are only valid under these specific conditions, which may not apply for HATP13 (see Appendix A for a discussion of the observational evidence of the requirements for the tidal fixed point theory). Nevertheless, HATP13b serves as a good example for us here to investigate what conclusions about the interior of an extrasolar giant planet can be drawn from an inferred k_{2} value.
In the case of HATP13b, BBL derived from the measured e_{b} = 0.021 ± 0.009 an allowed k_{2}interval of 0.116 < k_{2} < 0.425 (hereafter denoted as BBLinterval). They further concluded that the core mass of HATP13b is 0 M_{⊕} < M_{c} < 120 M_{⊕}.
3.2. The importance of the eccentricity measurement
The inner planet’s eccentricity is a crucial observable in our analysis because this quantity determines the allowed k_{2} interval. In fact, the value of the eccentricity is somewhat uncertain. While Bakos et al. (2009) report e_{b} = 0.021 ± 0.009, a later study of new radial velocity measurements by Winn et al. (2010) found an eccentricity of only half that value (e_{b} = 0.0133 ± 0.0041). We favor the eccentricity measurement from Winn et al. (2010), because they include the data from Bakos et al. (2009) and, in addition, use 75 new RV data points which extend the timespan of the data set by about 1 yr, and therefore allow a refinement of the orbital parameters. The consequences for the inferred k_{2} interval are illustrated in Fig. 1.
Fig. 1 Relation between the fixed point eccentricity e_{b} and the Love number k_{2}. The blue line shows the fourth order polynomial fit from Batygin et al. (2009). Different eccentricity measurements from Bakos et al. (2009) and Winn et al. (2010) are plotted in black and red, respectively, with solid lines showing the mean value of the measured eccentricity and dashed lines showing the error bars. The black dotted lines indicate the k_{2} interval inferred from Batygin et al. (2009) based on the eccentricity measurement from Bakos et al. (2009), in the figure also referred to as old k_{2} interval. The gray shaded area marks a region of k_{2} values not possible for models of HATP13b. The vertical black solid line (M_{core} = 0) shows the maximum possible k_{2} value we found based on our interior modeling (see Sect. 4 for details). Combining the information of the e_{b} − k_{2}relation from Batygin et al. (2009), the new eccentricity measurement from Winn et al. (2010), and our interior modeling, we find a new k_{2} interval ranging from 0.265 − 0.379 (rosy shaded). 
Based on the eccentricity measurement from Bakos et al. (2009), BBL found an allowed interval for k_{2} of 0.116−0.425. This result also includes information from their interior modeling, which is why the limits of the interval are not equal to the intersections of their fourth order polynomial fit and the error bars of the eccentricity measurement. Winn et al. (2010) found a significantly smaller eccentricity. As the eccentricity gets smaller, the resulting k_{2} must become larger. As a consequence, the lower limit of the k_{2} interval is raised to 0.265. To find the upper limit, we make use of our interior models. We find the most homogeneous model of HATP13b at a maximum k_{2} of 0.379. At this point the core mass vanishes and there are no solutions beyond that k_{2} value (for details see Sect. 4). This upper limit is also insensitive to variations in mass and radius within the observational error bars (see Sect. 4.2).
By combining the information of the fourth order polynomial fit from BBL, the new eccentricity measurement from Winn et al. (2010), and our interior models, we derived a new k_{2} interval of 0.265 < k_{2} < 0.379. It can be expected that this smaller region of allowed k_{2}values will significantly narrow the allowed range of solutions for the interior of HATP13b.
3.3. Model assumptions
For our models we use the mean values for the mass and radius of HATP13b (Bakos et al. 2009): \arraycolsep1.75ptThe uncertainties introduced by the error bars in mass and radius are discussed in Sect. 4.2. We assume a twolayer structure, consisting of a rocky core and one homogeneous envelope of hydrogen, helium and metals. It is not necessary to consider threelayer models in this work because they are within the range of solutions confined by the twolayer models (see Sect. 4 for a discussion). We set the helium abundance in the envelope to the solar value Y = 0.27 (Bahcall et al. 1995). The amount of heavy elements Z was varied from Z = 0, which gives a model with the biggest possible core when the other parameters are fixed, to a maximum value where the core vanishes.
Due to the proximity to its star, the temperature profile of HATP13b is likely to be strongly influenced by the radiation of the star. The various processes in an exoplanet’s atmosphere influencing its temperature are very complex and beyond the scope of this work. As we aim to study the effects of input parameters important for planet modeling on the structure and Love number of a planet, we vary the envelope temperature within the widest possible range, even though extremely cold or hot models may be unrealistic. For HATP13b we change the 1 kbar temperatures from 3800 to 9000 K. T_{1 kbar} = 3800 K gives the most homogeneous planet with zero core mass and models with T_{1 kbar} > 9000 K would have k_{2} values smaller than the BBLinterval (see also Sect. 4). In the outermost layer of the planet from 1 bar to 1 kbar we assume the temperature profile to be either adiabatic or isothermal. This is supposed to account for the uncertainty of how the star’s radiation influences the outer envelope of the planet. As the planet is very close to its star, it is likely that there will be an isothermal layer to some extent (Fortney et al. 2007). In every case, for pressures P > 1 kbar the planet is adiabatic.
In addition, we compute a model series that is consistent with the nongray model atmospheres from Fortney et al. (2007) in order to further pin down the outer boundary condition. Based on their pressuretemperature profiles of Jupiterlike planets around Sunlike stars (Fig. 3 in Fortney et al. 2007), we interpolated a model atmosphere for HATP13b, assuming that the incoming energy flux remains constant: L_{∗}/(4πa^{2}) = const., where L_{∗} is the stellar luminosity and a is the planet’s semimajor axis. Our interpolation yields T_{1 bar} = 2080 K as new outer boundary condition. It further showed that an isothermal layer may reach down to P_{ad} ~ 100 bar. Our interpolated model atmosphere is in agreement with an atmosphere model specifically calculated for HATP13b, using the methods of Fortney et al. (2007). The thickness of the isothermal layer is a variable parameter (influenced by the age of the planet). Hence, in our model series based on the model atmosphere, we vary P_{ad} as a free parameter while keeping T_{1 bar} fixed at 2080 K.
Our modeling procedure constitutes an improvement compared to the modeling done by Batygin et al. (2009) because we do not approximate the core material by a constant density and we include the effects of different temperatures and temperature profiles in the atmosphere. This allows us to draw some general conclusions about the capability of k_{2} to constrain interior models of extrasolar giant planets.
4. Results
4.1. Love number, metallicity, and core mass
We present in Figs. 2 and 3 results for the calculated core mass, k_{2}, and envelope metallicity. The area between an adiabatic and isothermal line of the same 1 kbar temperature accounts for the uncertainty about the outer temperature profile as the thickness of a potential isothermal layer is not known. Also shown are the models based on our model atmosphere with an isothermal layer of 2080 K, reaching down to 1 (fully adiabatic), 5, 10, 50, or 72 bar, where the core disappears. In Fig. 2, the line connecting zero metallicity fully adiabatic models separates the region of all possible planetary models from the prohibited area which is not accessible for models of HATP13b.
Fig. 2 Love numbers k_{2} and core masses of twolayer models of HATP13b for different 1 kbar temperatures (color coded). For pressures <1 kbar the temperature profile is either adiabatic (colored solid) or isothermal (colored dashed). The metallicity increases along a line, reaching its maximum value for M_{core}/M_{p} = 0. The zero metallicity line for fully adiabatic models (brown solid) separates the region of all possible planetary models (white) from the prohibited area (light gray). Vertical black dashed lines indicate the allowed interval for k_{2}values derived by BBL. Based on that interval we find a maximum possible core mass of 0.32 M_{p} ≈ 87 M_{⊕}. Our new k_{2} interval is shown with vertical black dotdashed lines. It lowers the uncertainty of the planet’s core mass down to 0.1 M_{p} ≈ 27 M_{⊕}. The other black lines show our models that are based on our model atmosphere with a fully adiabatic envelope (solid black) and P_{ad} of 5 bar (dotted black), 10 bar (dashed black), 50 bar (dotdashed black), and 72 bar (black triangle). 
Fig. 3 Love numbers k_{2} as a function of the envelope metallicity. Shown are the same models as in Fig. 2. The numbers close to some models give the core mass in M_{⊕}. The shaded areas demonstrate how models can be ruled out by taking into account the k_{2} intervals and assuming that the envelope metallicity is at least the stellar metallicity. Considering the information from the model atmosphere the favored models reduce to the green shaded area. 
A general trend that can be seen in Figs. 2 and 3 is that the Love number k_{2} increases as more metals are put in the envelope. This is the result of k_{2} being a measure of the level of central condensation of an object. Since the total mass of the planet must always be M_{p}, the enrichment of the envelope with metals leads to a decrease in core mass. A smaller core and higher metal content in the envelope means that the planet is more homogeneous, which is why the Love number grows. The maximum metallicity is reached when the core of the planet vanishes. This marks the maximum possible Love number of a planet for a given temperature.
The temperature itself also has a significant influence on the metallicity, core mass, and consequently Love number of the planet. Higher temperatures reduce the Love number k_{2}. The higher the envelope temperature, the lower its density. Low densities in the envelope require a more massive core in order to ensure mass conservation. High envelope temperatures thus lead to a strong central condensation, reflecting in a small Love number. For HATP13b we find a minimum 1 kbar temperature of 3800 K for fully adiabatic models. At this low temperature the zero metallicity envelope is so dense that the core is almost vanished (M_{core} = 0.2 M_{⊕}). Hence, there is no enrichment of heavy elements possible in the envelope. At this temperature maximum homogeneity is reached. That translates into a maximum possible Love number for HATP13b of k_{2} = 0.379, well below the upper limit given by the BBLinterval. At high temperatures, on the other hand, the density in the envelope is low enough to enable the existence of massive cores. That creates the possibility to enrich the envelope with metals. For instance, at T_{1 kbar} = 9000 K the core disappears at an envelope metallicity as high as 70%. The big cores of the hot models lead to very small Love numbers and tend to lie outside of the BBLinterval.
We note that models with an isothermal atmosphere generally yield smaller Love numbers than their adiabatic counterparts, because they are overall hotter.
As is obvious from Figs. 2 and 3 not all calculated models are placed in the BBLinterval or even in our narrower new k_{2} interval, which means some of the models can be ruled out. The hot and metal poor models cannot fulfill the BBLinterval. Our new k_{2} interval also rules out the metal rich hot models and only leaves us with the colder models.
Further constraints can be made by taking into account model atmospheres in order to get more information about the planet’s outer temperature profile. Our model atmosphere yields a 1 bar temperature of 2080 K. An isothermal layer of P_{ad} > 3 bar is required to produce solutions in our new k_{2} interval.
Another point we can take into account is that planets form out of the same cloud of gas as their host star. Hence, it is reasonable to assume that the planet’s bulk metallicity is at least the stellar metallicity. Of course, a planet may also be enriched in metals due to capture of planetesimals during formation. Enhanced bulk metallicities for extrasolar giant planets over their parent star values were indeed found (Miller & Fortney 2011). We use this argument to also rule out interior models that have less than stellar metallicity (see Fig. 3). Here we can approximate the bulk metallicity with the envelope metallicity as the disputable models have such small cores that most of the planet’s metals are in the envelope. This leaves us only with very few models for the interior of HATP13b, where we favor the ones compatible with the model atmosphere. Combining all the pieces of information, we can conclude from Fig. 3 that HATP13b has an isothermal outer layer with a temperature of 2080 K that may reach down to 3 − 44 bar. Models with isothermal layers that extend to higher pressures have such small cores that their envelope metallicity cannot exceed stellar metallicity. Consequently, our models with P_{ad} ≥ 50 bar are ruled out. The envelope metallicity of the allowed models is 1 − 11 times the stellar metallicity. Also, the bulk metallicity cannot exceed this limit as M_{core} < 27 M_{⊕}.
In order to give a better overview of the different boundary conditions used we plot in Fig. 4P − T profiles of acceptable and some unacceptable models. Those with T_{1 kbar} < 4000 K and T_{1 kbar} > 6500 K are too cold or too hot, respectively, and hence not shown in the plot. We find an area that roughly constrains the P − Tconditions near the atmosphere that produce acceptable models. In addition, a model also needs to have a high enough envelope metallicity in order to be acceptable. For example, our structure models compatible with our model atmosphere with T_{1 bar} = 2080 K and an isothermal layer down to 10 bar lie in the favored area marked by Fig. 4. However, models of this kind with Z = 0 are not acceptable whereas models with a high enough envelope metallicity are, compare also to Fig. 3.
Fig. 4 P − T profiles near the atmosphere of acceptable and unacceptable models. Colors and line styles are the same as in Figs. 2 and 3. For lines with same color and style, models with Z = 0 and Z = Z_{max} are shown. Thick lines indicate acceptable models; compare these also to Fig. 3. The green shaded area marks the P − T regime for acceptable models. However, models in this regime are only acceptable when they also have a large enough amount of metals in their envelopes. 
A very important parameter describing the interior of a planet is its core mass. This quantity is of interest because it may give a hint on the formation history of the planet. Figure 2 shows that the relationship between a k_{2} value and a core mass is nonunique. If, for instance, k_{2} = 0.2 it could either be a very hot T_{1 kbar} = 9000 K, metal rich planet with a very small core or a lower temperature T_{1 kbar} = 6500 K one with a core constituting 10% of the planet’s total mass. The only information that can be concluded is a maximum possible core mass, given by the line consisting of the adiabatic zero metallicity models. The maximum possible core mass of HATP13b is given by the intersection of this line with the lower k_{2} boundary, yielding M_{core} < 0.32 M_{p} ≈ 87 M_{⊕} for the BBLinterval. Taking into account our new k_{2} interval lowers the uncertainty of the planet’s core mass down to M_{core} < 0.1 M_{p} ≈ 27 M_{⊕}.
In principle, the area under each curve can be filled up with threelayer models (not shown in Fig. 2). Every redistribution of metals in the envelope would either result in a stronger central condensation (smaller k_{2}) or, if central condensation is kept constant, require a transport of material from the core to the envelope reducing the core mass. As a result of this degeneracy introduced by a discontinuity in the envelope (see also Kramm et al. 2011) it is only possible to determine a maximum possible core mass. Further narrowing of the maximum core mass could also be achieved with a more detailed measurement of the eccentricity of HATP13b as that would lead to a smaller k_{2} interval.
Finally, from all the interior models we presented here, we favor the ones with T_{1 bar} = 2080 K, an isothermal outer layer of 3−44 bar, and an envelope metallicity of 1 < Z/Z_{∗} < 11 because they (i) are placed in our new k_{2} interval; (ii) are consistent with the model atmospheres from Fortney et al. (2007); and (iii) have an envelope and bulk metallicity that is above the stellar metallicity. Assuming these parameters are “correct” for HATP13b, the planet would have a core mass of 27 M_{⊕} or less. For comparison, Jupiter models from Fortney & Nettelmann (2010) predict M_{Z,tot} = 15−38 M_{⊕}, M_{core} = 0−10 M_{⊕} and Z_{p} = 4−12 Z_{⊙}.
4.2. Mass and radius uncertainties
We now investigate the effect of the measurement uncertainties for the mass and radius of HATP13b. For that purpose, we varied the mass and radius within the 1σ errorbars to obtain different mean densities. As an example, we only look at models consistent with our model atmosphere and P_{ad} = 1 and 50 bar because all other valid models are bounded by these and react the same way on massradius variations.
Fig. 5 Core mass (solid lines) and total mass of metals (dashed lines) as a function of envelope metallicity for different mean densities according to the measurement uncertainties in mass and radius. The numbers are the Love numbers k_{2} for zero and maximum possible metallicity in the envelope. Shown are only models compatible with our model atmosphere (T_{1 bar} = 2080 K) for P_{ad} = 1 bar and P_{ad} = 50 bar. 
Figure 5 shows the effect on core mass and metal enrichment. The uncertainty in mean density results in an uncertainty of about ±20 M_{⊕} in core mass and mass of metals. For a given core mass, the envelope metallicity can vary up to ± 3 Z_{∗}.
There is little effect on the Love number k_{2}. In Fig. 2, the line separating the prohibited area from the possible models consists of zerometallicity models. Since for Z_{env} = 0 the Love number k_{2} only varies by about ± 0.003, only a minor shift of this line can be expected so that all the conclusions drawn for the fiducial M_{p} and R_{p} values remain valid also for other mean densities within the error bars.
4.3. Estimating the tidal Q value
With R_{p} = 1.281 R_{J} HATP13b has a large radius given its mass and age. This implies it possesses an interior energy source, like many other Hot Jupiter planets. Given its eccentric orbit, this energy source could have a strong contribution due to tidal heating.
Based on our analysis of k_{2} and our interior models we found that the adiabatic transition is between 3 and 44 bar. We ran atmosphere models to estimate what values of the intrinsic temperature (T_{int}) these would correspond to. At the adiabatic transitions of 3 and 44 bar we found T_{int} = 750 and 275 K, respectively. This also gives us the intrinsic luminosity L_{int}. Assuming L_{int} is provided only by tidal heating, we estimated Q with the following relation (Miller et al. 2009) (5)This gives us a lower limit on the Q value of HATP13b since tidal heating might not be the only interior energy source. There could be other effects contributing to L_{int}, for instance Ohmic dissipation (Batygin & Stevenson 2010).
Our estimates of the lower limit on Q for the different eccentricity measurements are displayed in Table 1. For comparison, Jupiter’s Q value has been calculated to be between 10^{5} and 10^{6} (Goldreich & Soter 1966).
Lower limits for the tidal Q value of HATP13b for different eccentricity measurements and intrinsic temperatures.
5. Discussion
5.1. Comparison with previous results
In addition to an allowed k_{2}interval, Batygin et al. (2009) also provided a set of possible interior models of HATP13b that give k_{2} values within the given interval. These models range in core mass from 0 to 120 M_{⊕}. However, for the given BBLinterval we find a maximum possible core mass that is significantly smaller (≈87 M_{⊕}). This difference is a result of different model assumptions. The models of Batygin et al. (2009) all have cores with constant densities of 7−12 g/cm^{3} (Batygin, priv. comm.), resembling water cores. In contrast, the models presented in this work were calculated using a compressible rock EOS (Hubbard & Marley 1989), increasing the densities in the cores up to 12 − 40 g/cm^{3}. As a consequence of the lower and constant core densities in the BBL models, a core of a specific mass requires a larger radius than a core of the same mass in our models. This results in a greater Love number than our models with the same core mass. That is why our models need smaller core masses in order to fall in the allowed k_{2}interval. Models of HATP13b in this work with a core mass of 120 M_{⊕} would have a smaller k_{2} than the lower k_{2}boundary of the BBLinterval. Hence, the different maximum core masses in this work and in Batygin et al. (2009) arise from the different core EOS used. This shows the major influence of the EOS. In our example here, the different core EOS cause an uncertainty of about 28% in the maximum possible core mass.
5.2. The eccentricity issue
As mentioned before (see Sect. 3.2) the eccentricity is crucial for determining an allowed k_{2}interval. We used the results from BBL to obtain a new k_{2} interval from another eccentricity measurement (Winn et al. 2010). Our new interval, ranging from 0.265 to 0.379 is significantly smaller than the one previously used by BBL (0.116 < k_{2} < 0.425). This allowed for a more precise estimate of a maximum possible core mass (M_{core} < 27 M_{⊕}).
It of course has to be kept in mind that the ultimate significance of this new eccentricity value (and deduced k_{2}) is unclear. Other effects like the stellar jitter produce a large uncertainty on the eccentricity measurement of the inner planet, as recently pointed out by Payne & Ford (2011).
One way to help to constrain the eccentricity would be a detection of the occultation (secondary eclipse) of the planet (for instance, with Spitzer). The timing of the occultation can be used to place a strong constraint on ecosω, where e is the orbital eccentricity and ω is the longitude of periastron (Charbonneau et al. 2005). Observations to determine ecosω for HATP13b are currently being made (Harrington & Hardy, priv. comm.).
6. Summary and conclusions
In this work, we presented new interior models of the transiting Hot Jupiter HATP13b with the aim of showing to what extent interior models of extrasolar giant planets can be constrained by using the tidal Love number k_{2} in addition to the known observables mass and radius. We also varied the envelope temperature and metallicity in order to demonstrate the uncertainties imposed on the inferred interior models.
One main result of our work is that based on the Love number k_{2} one cannot draw a conclusion on the precise core mass of the planet. Only a maximum possible core mass can be inferred which is given by adiabatic zeroenvelopemetallicity models. Taking into account the allowed k_{2} interval (0.116 < k_{2} < 0.425) derived from Batygin et al. (2009), we find a maximum value for HATP13b’s core mass of 87 M_{⊕}. Further radial velocity observations from Winn et al. (2010) gave a new value for the planet’s eccentricity, allowing us to construct a new k_{2} interval of 0.265 < k_{2} < 0.379. With this smaller interval we were able to constrain the core mass of HATP13b further down to 27 M_{⊕}.
We also showed the influence of the envelope’s temperature and metallicity. The coldest possible model defines a maximum possible value for k_{2} as this is the most homogeneous model. For HATP13b we find k_{2,max} = 0.379. This is well below the upper k_{2} limit from Batygin et al. (2009).
As our analysis showed, the temperature and metallicity of the planet’s envelope are important parameters whose knowledge would significantly increase the effectiveness of constraining the planet’s interior with the Love number k_{2}. We therefore calculated a model series based on a theoretical model atmosphere from Fortney et al. (2007). Models of this kind have T_{1 bar} = 2080 K and fall in our new k_{2} interval if they have an outer isothermal layer with 3 bar < P_{ad} < 44 bar. With this result we also estimated lower limits of the tidal Q value of HATP13b.
Based on our analysis and the applicability of the tidal fixed point theory we can rule out the majority of the calculated models. Assuming our new k_{2}interval, we note that hot models with T_{1 kbar} > 6500 K can be ruled out (compare Fig. 3). On the other hand, the very cold models with T_{1 kbar} < 4000 K are unlikely because they have a metallicity that is smaller than stellar. We favor models with T_{1 bar} = 2080 K, an isothermal outer layer of 3 − 44 bar, and an envelope metallicity of 1 < Z/Z_{∗} < 11 because they (i) are placed in our new k_{2} interval; (ii) are consistent with the model atmospheres from Fortney et al. (2007); and (iii) have an envelope metallicity that is above the stellar metallicity. Assuming these conditions really apply for HATP13b the planet would have a core mass of 27 M_{⊕} or less. Given the calculated tidal Q, if HATP13b turns out to have a small core and an envelope enrichment similar to Jupiter, it could represent a Jupiterlike extrasolar planet.
By comparing with the previous results from Batygin et al. (2009) we have seen that the core EOS affects the core mass and Love number, resulting into a difference of about 28% in the prediction of a maximum possible core mass of HATP13b.
As pointed out in Sect. 3.2, the eccentricity of the inner planet is a crucial parameter because it determines the Love number k_{2}. The value of the eccentricity is still uncertain. We analyzed the consequences of an e_{b} = 0.021 ± 0.009 (Bakos et al. 2009) and e_{b} = 0.0133 ± 0.0041 (Winn et al. 2010).
Finally, despite the uncertainty of the inner planet’s eccentricity it is still unclear whether the tidal fixed point theory from Mardling (2007) can really be applied to HATP13b. Further observations are necessary to prove the coplanarity and apsidal alignment configuration of the system. Also, TTV measurements could provide clues about the existence of other longperiod companions in the system.
In the recent months there have been several observational studies dedicated to the HATP13 system. In particular, the sudden occurence of a TTV signal reported by Pál et al. (2011) and their interpretation with a longperiod eccentric perturber is discussed by several authors. Using data from new transits, Nascimbeni et al. (2011) note that the data could also be explained with a sinusoidal TTV caused by a 5 M_{⊕} perturber locked in a 3:2 meanmotion resonance with HATP13b. In contrast, by analyzing a total of 22 transit light curves spanning four observational seasons, Fulton et al. (2011) find that the transit times are consistent with a linear ephemeris, with the exception of a single outlier from the Szabó et al. (2010) data whose origin remains unexplained. Southworth et al. (2012) support the hypothesis of a linear ephemeris and find no clear indication of the existence of TTVs.
Furthermore, Southworth et al. (2012) report new physical properties of the HATP13 system. Accordingly, they find that HATP13b is slightly more massive, but has a significantly larger radius of R_{p} = 1.487 ± 0.053 R_{J}. This leads to a mean density which is about 14% lower than the ρ_{min} considered in Sect. 4.2. We showed in Sect. 4.2. that zeroenvelopemetallicity models with a lower density have a slightly larger Love number and a smaller core which would lead to a decrease of the estimate of a maximum possible core mass.
The value of the eccentricity is actually a matter of debate and influences the results significantly (see Sect. 3.2).
With its high mass the companion may instead be classified as a brown dwarf. However, as the definition of planets and brown dwarfs is rather unclear (Baraffe et al. 2008), we call it a planet here.
Acknowledgments
We thank Konstantin Batygin, David Stevenson, Josh Winn and Stefanie Raetz for helpful discussions. U.K., R.R., and R.N. acknowledge support from the DFG SPP 1385 “The first ten million years of the Solar System”, N.N. from DFG RE882/111.
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Appendix A: Applicability of the tidal fixed point theory
The analysis in this work is based on the assumption that the tidal fixed point theory of Mardling (2007) can be applied in order to extract a k_{2} value from the inner planet’s eccentricity. For that purpose the planetary system has to fulfill the requirements mentioned in Sect. 3.1. Yet unproven and crucial to the evolution into the tidal fixed point is the coplanarity of the HATP13 system. In fact, if the orbits of planet b and c are mutually inclined, the system will not evolve to the tidal fixed point. Instead it will relax to a limit cycle in e_{b} − η parameter space where the eccentricity of the inner planet e_{b} and the angle between the periapse lines η oscillate (Mardling 2010). In that case the Love number k_{2} of the inner planet can not be determined unambiguously. In particular, Mardling (2010) shows that the analysis of Batygin et al. (2009) and hence the inferred k_{2}interval is only valid if the mutual inclination of planets b and c is less than about 10°.
A direct estimate of the mutual inclination Δi would be possible by observing planet c in transit and measuring the RossiterMcLaughlin effect. A transit of HATP13c was predicted to occur on April 28, 2010 (Winn et al. 2010). However, such a transit could not be observed. The multisite campaign by Szabó et al. (2010) revealed that a transit of HATP13c can be excluded with a significance level from 65% to 72%. Yet, a close to coplanar configuration cannot be ruled out. The transit probability of HATP13c is at most 8.5% for a mutual inclination of the two planets of 8° and decreases to zero for inclinations ≲4°^{3} (Beatty & Seager 2010). Another reason why the campaign of Szabó et al. (2010) could not detect a transit may be that they only observed for 5 days. The orbital parameters (eccentricity and period) of the outer planet are not well constrained due to the effect of the unknown stellar jitter, which translates into large uncertainties about the predicted transit times of planet c, making it necessary to observe over a wide range of nights – ~3 weeks (Payne & Ford 2011).
Support for the hypothesis of coplanarity of the HATP13 system comes from the study of the RossiterMcLaughlin effect to determine the stellar obliquity Ψ_{∗,b}. Winn et al. (2010) note that there is an indirect connection between Ψ_{∗,b} and Δi: a large Δi would lead to nodal precession of b′s orbit around c′s orbital axis, causing periodic variations in Ψ_{∗,b}. As a consequence, it would be unlikely to observe a low value of Ψ_{∗,b} unless Δi is also small. At any rate, only the skyprojected angle λ can be measured (not the true obliquity Ψ_{∗,b}) which makes it still impossible to draw firm conclusions about Δi. Winn et al. (2010) find a small λ = 1.9 ± 8.6 deg, suggesting that planet b′s orbital angular momentum vector is wellaligned with the stellar spin vector. This also increases the likelyhood that the orbits of planets b and c are coplanar.
Payne & Ford (2011) also suggest a method to determine the mutual inclination between planets b and c with the help of transit timing variations (TTV). In particular, they show that systems with Δi ~ 0° and Δi ~ 45° result in significantly different TTV profiles which can easily be distinguished. In fact, HATP13b has been subject to a TTV campaign. Pál et al. (2011) detected TTV in the order of 0.01 days which “suddenly” occured during their observations in December 2010 and January 2011 while in the first three years of observations of HATP13b there seemed to be a lack of TTV. If it is a periodic process, it should have a period of at least ~3 years, making it unlikely to be caused by HATP13c. Hence, Pál et al. (2011) conclude that the measured TTV may be the result of perturbations of another longperiod companion. Interestingly, Winn et al. (2010) also point to the possibility of an additional body d in the system as their models of two Keplerian orbits give an unacceptable fit to the RV data while a model assuming a third companion with a longer orbital period is successful. Furthermore, thinking about the origin of the HATP13 system, Mardling (2010) suggests that the system originally contained a third planet which was later scattered out of the system.
Given all these issues, to our current knowledge it seems impossible to say whether HATP13b is really subject to the tidal evolution into the tidal fixed point like envisioned by Mardling (2007). Further observations are necessary to pin down the mutual inclination of the planets and to prove or disprove the existence of another companion in the system.
All Tables
Lower limits for the tidal Q value of HATP13b for different eccentricity measurements and intrinsic temperatures.
All Figures
Fig. 1 Relation between the fixed point eccentricity e_{b} and the Love number k_{2}. The blue line shows the fourth order polynomial fit from Batygin et al. (2009). Different eccentricity measurements from Bakos et al. (2009) and Winn et al. (2010) are plotted in black and red, respectively, with solid lines showing the mean value of the measured eccentricity and dashed lines showing the error bars. The black dotted lines indicate the k_{2} interval inferred from Batygin et al. (2009) based on the eccentricity measurement from Bakos et al. (2009), in the figure also referred to as old k_{2} interval. The gray shaded area marks a region of k_{2} values not possible for models of HATP13b. The vertical black solid line (M_{core} = 0) shows the maximum possible k_{2} value we found based on our interior modeling (see Sect. 4 for details). Combining the information of the e_{b} − k_{2}relation from Batygin et al. (2009), the new eccentricity measurement from Winn et al. (2010), and our interior modeling, we find a new k_{2} interval ranging from 0.265 − 0.379 (rosy shaded). 

In the text 
Fig. 2 Love numbers k_{2} and core masses of twolayer models of HATP13b for different 1 kbar temperatures (color coded). For pressures <1 kbar the temperature profile is either adiabatic (colored solid) or isothermal (colored dashed). The metallicity increases along a line, reaching its maximum value for M_{core}/M_{p} = 0. The zero metallicity line for fully adiabatic models (brown solid) separates the region of all possible planetary models (white) from the prohibited area (light gray). Vertical black dashed lines indicate the allowed interval for k_{2}values derived by BBL. Based on that interval we find a maximum possible core mass of 0.32 M_{p} ≈ 87 M_{⊕}. Our new k_{2} interval is shown with vertical black dotdashed lines. It lowers the uncertainty of the planet’s core mass down to 0.1 M_{p} ≈ 27 M_{⊕}. The other black lines show our models that are based on our model atmosphere with a fully adiabatic envelope (solid black) and P_{ad} of 5 bar (dotted black), 10 bar (dashed black), 50 bar (dotdashed black), and 72 bar (black triangle). 

In the text 
Fig. 3 Love numbers k_{2} as a function of the envelope metallicity. Shown are the same models as in Fig. 2. The numbers close to some models give the core mass in M_{⊕}. The shaded areas demonstrate how models can be ruled out by taking into account the k_{2} intervals and assuming that the envelope metallicity is at least the stellar metallicity. Considering the information from the model atmosphere the favored models reduce to the green shaded area. 

In the text 
Fig. 4 P − T profiles near the atmosphere of acceptable and unacceptable models. Colors and line styles are the same as in Figs. 2 and 3. For lines with same color and style, models with Z = 0 and Z = Z_{max} are shown. Thick lines indicate acceptable models; compare these also to Fig. 3. The green shaded area marks the P − T regime for acceptable models. However, models in this regime are only acceptable when they also have a large enough amount of metals in their envelopes. 

In the text 
Fig. 5 Core mass (solid lines) and total mass of metals (dashed lines) as a function of envelope metallicity for different mean densities according to the measurement uncertainties in mass and radius. The numbers are the Love numbers k_{2} for zero and maximum possible metallicity in the envelope. Shown are only models compatible with our model atmosphere (T_{1 bar} = 2080 K) for P_{ad} = 1 bar and P_{ad} = 50 bar. 

In the text 
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