# The flattenings of the layers of rotating planets and satellites deformed by a tidal potential

## Abstract

We consider the Clairaut theory of the equilibrium ellipsoidal figures for differentiated nonhomogeneous bodies in nonsynchronous rotation (Tisserand, Mécanique Céleste, t.II, Chaps. 13 and 14) adding to it a tidal deformation due to the presence of an external gravitational force. We assume that the body is a fluid formed by \(n\) homogeneous layers of ellipsoidal shape and we calculate the external polar flattenings \(\epsilon _k,\, \mu _k\) and the mean radius \(R_k\) of each layer or, equivalently, their semiaxes \(a_k\), \(b_k\), and \(c_k\). To first order in the flattenings, the general solution can be written as \(\epsilon _k=\mathcal {H}_k\epsilon _{h}\) and \(\mu _k=\mathcal {H}_k\mu _{h}\), where \(\mathcal {H}_k\) is a characteristic coefficient for each layer that depends only on the internal structure of the body and \(\epsilon _{h}\) and \(\mu _{h}\) are the flattenings of the equivalent homogeneous problem. For the continuous case, we study the Clairaut differential equation for the flattening profile using the Radau transformation to find the boundary conditions when the tidal potential is added. Finally, the theory is applied to several examples: (i) a body composed of two homogeneous layers, (ii) bodies with simple polynomial density distribution laws, and (iii) bodies following a polytropic pressure-density law.

### Keywords

Polar flattenings Tidal potential Rotation Differentiated bodies Clairaut equation Ellipsoidal figure of equilibrium Exoplanets Polytropes## 1 Introduction

Several theories of tidal evolution, since the theory developed by Darwin appeared in the nineteenth century (Darwin 1880), are based on the figure of equilibrium of an inviscid tidally deformed body (e.g., Ferraz-Mello et al. 2008; Ferraz-Mello 2013). The addition of viscosity to the model is made at a later stage, but the way it is introduced is not unique and can vary when different tidal theories are considered. Frequently, the adopted figure is a Jeans prolate spheroid or, if the rotation is important, a Roche triaxial ellipsoid (Chandrasekhar 1969). It is worth recalling that ellipsoidal figures are excellent first approximations but not exact figures of equilibrium (Poincaré 1902; Lyapounov 1925, 1927). In addition, Maclaurin, Jacobi, Roche, and Jeans ellipsoids are valid only for homogeneous bodies. Real celestial objects, however, are quite far from being homogeneous. This causes significant deviations which need to be taken into account in astronomical applications.

The nonhomogeneous problem, when one considers only deformation by rotation, has been extensively studied. The problem of one body formed by \(n\) rotating homogeneous spheroidal layers as well as its extension to the continuous case was studied by Clairaut (1743) [revisited by Tisserand (1891) and Wavre (1932)]. Their works were based on the hypotheses of small deformations (linear theory for the polar flattenings) and constant angular velocity inside a body. The general case of homogeneous layers rotating at different angular velocities (nonlinear theory) was studied by Montalvo et al. (1983) and Esteban and Vazquez (2001) [see Borisov et al. (2009) for a detailed review] and was generalized to the continuous inviscid case by Bizyaev et al. (2015).

The case of uniformly rotating layers has been studied by several authors. Kong et al. (2010) discussed the particular case of a body formed by two homogeneous layers with the same angular velocity. Hubbard (2013), with a recursive numerical form of the potential of an N-layer rotating planet in hydrostatic equilibrium, showed a solution for the spheroidal shapes of the interfaces of the layers.

Regarding cases where the tidal forces acting on a body are taken into account along with the rotation, the literature is much less extensive. Usually a spin-orbit synchronism is assumed, so that a rotating-body solution can be used (e.g., Van Hoolst et al. 2008). Tricarico (2014), assuming synchronism, found a recursive analytic solution for the shape of a body formed by an arbitrary number of layers. For this, he developed the potentials of homogeneous ellipsoids in terms of the polar and equatorial shape eccentricities. However, the results do not include tidally deformed bodies whose rotation is nonsynchronous, such as, for instance, the Earth, solar-type stars hosting close-in planets, and hot Jupiters in highly eccentric orbits.

In this work, we generalize the linear Clairaut theory, adding a tidal potential due to the presence of an external body, *without the synchronism hypothesis*. The paper is organized as follows. In Sect. 2, we present the \(2n\) classical equations of equilibrium. The resolution of the system of equilibrium equations is shown in Sect. 3. In Sect. 4, we study Clairaut’s equation for the continuous problem and its solution. In Sect. 5, we calculate the potential at a point in space due to the deformed body and calculate a generalized Love number for differentiated nonhomogeneous bodies. In Sect. 6, we apply the theory to a body composed of two homogeneous layers, while bodies with continuous density laws are studied in Sect. 7. Finally, in the Sect. 8, we present our conclusions.

## 2 Equilibrium equation of a fluid in rotation

## 3 Flattenings of the layers

It is important to note that in the case of a synchronous satellite, when the approximation \(\epsilon _J \simeq 3\epsilon _M\) is adopted^{1}, the system (12) is equivalent to that found by Tricarico (2014), where the square of the polar and equatorial “*eccentricities*” used there are related to the polar flattenings through \(e_{pi}^2\approx 2\epsilon _i\) and \(e_{qi}^2\approx 2\epsilon _i-2\mu _i\).

The calculations done are valid only for small flattenings, i.e., they assume that the perturbation due to the tide and the rotation are small enough so as not to deform the body too much (in the second order, the figure ceases to be an ellipsoid).

## 4 Extension to the continuous case

^{2}), we assume that the number of layers tends to infinity, so that the increments \(\Delta R_k=R_k-R_{k-1}\) are infinitesimal quantities. When \(\Delta R_k\rightarrow 0\), Eq. (15) becomes

### 4.1 Boundary conditions. Radau transformation

*Radau’s parameter*(Bullen 1975). Defining \(\eta (x=1)=\eta _n\) and using relationship (22) and transformation (24), the boundary conditions of (20) are

## 5 Potential of the tidally deformed body

^{3}The total potential is the sum of the potentials of all ellipsoids:

## 6 Two-layer core–shell model

If \(\lambda =1\) or \(\xi =1\), then the constants are \(\mathcal {H}_1=\mathcal {H}_2\) solutions for a homogeneous body.

When the core is denser than the mantle, \(\mathcal {H}_2\geqslant \mathcal {H}_1\), and the flattenings of the nucleus are smaller than the flattenings of the surface (where \(\epsilon _1=\mathcal {H}_1(\epsilon _J+\epsilon _M) \leqslant \epsilon _2=\mathcal {H}_2(\epsilon _J+\epsilon _M)\) and \(\mu _1=\mathcal {H}_1\epsilon _M\leqslant \mu _2=\mathcal {H}_2\epsilon _M\)).

Since \(\mathcal {H}_2\leqslant 1\), the maximum surface flattening is given by the homogeneous solution. In the presence of a core, the surface is always less flattened than it is in the homogeneous case.

While \(\mathcal {H}_1\) may take all possible values between 0 and 1, \(\mathcal {H}_2\) is always larger than the critical limit 0.4, corresponding to the degenerate limit case in which the whole mass would tend to concentrate in the center and be surrounded by a zero-density shell (case of Huygens–Roche). Therefore, the flattenings of the outer surface can never be less than 40 % of the homogeneous reference values. This is the same result given by Eq. (21) for the continuous case.

### 6.1 Fluid Love number

Figure 4 shows the possible value of \(k_f\) as a function of the core size \(\xi \) and of the relative density of the shell \(\lambda \). If we obtain \(k_f\), for example by determining \(\mathcal {H}_n\) by direct observation of the surface flattenings, then Eq. (41) defines a continuous curve of possible values for the size of the nucleus \(\xi \) and the relative density of the shell \(\lambda \) under the hypothesis of two homogeneous layers. Moreover, as can be seen in this figure, a maximum value for these physical parameters can be predicted.

## 7 Application to different density distribution laws

In this section, we present some applications of the theory developed in this paper to bodies with continuous density distributions. For this we use two examples of density distributions: polynomial and polytropic density laws.

In both cases, Clairaut’s equation is solved numerically after introduction of the variable defined by Eq. (24). The flattening profile \(\mathcal {H}(x)\) and the Love number are then obtained through the inverse transformation.

### 7.1 Polynomial density functions

The resulting flattening profiles \(\mathcal {H}(x)\) are shown in Fig. 5b. In all cases, the flattening profile \(\mathcal {H}(x)\) is an increasing monotonic function, and for all \(x\) the values of \(\mathcal {H}(x)\) increase when the power \(\alpha \) increases.

^{4}

### 7.2 Polytropic pressure–density laws

*Lane–Emden equation*(Chandrasekhar 1939)

It is worth mentioning that several real cases exist that correspond to polytropes. For example, when convection is established in the interior of a star, the resulting configuration is a polytrope; when the gas is degenerate, the corresponding equations of state have the same form as the polytropic equation of state, and so forth (Collins 1989). We also mention recent results by Leconte et al. (2011) showing that the density profile of hot Jupiters is well approximated by a polytrope.

The resulting flattening profiles \(\mathcal {H}(x)\) are shown in Fig. 7b. In all cases, the flattening profile \(\mathcal {H}(x)\) is an increasing monotonic function, and for all \(x\) the values of \(\mathcal {H}(x)\) decrease when the polytropic index \(n\) increases.

## 8 Conclusions

In this paper, we extended the classical results on nonhomogeneous rotating figures of equilibrium to the case in which a body is also under the action of a tidal potential owing to the presence of an external body, without the restrictive hypothesis of spin-orbit synchronization. The only assumptions in this paper are that the body is formed by \(n\) homogeneous ellipsoidal layers in equilibrium and that there are small enough tidal and rotational deformations with symmetry axes perpendicular to each other (remember that, in the second order, the figure ceases to be an ellipsoid). We calculated the \(2n\) equilibrium equations for small flattenings and found that the two polar flattenings \(\epsilon _k\) and \(\mu _k\) were linearly related, both being proportional to the homogeneous reference values with a factor of proportionality \(\mathcal {H}_k\) that is the same in both cases. The deformations propagate toward the interior of the body in the same way, depending, in the first approximation, only on the density profile, not on the origin of the two considered deformations. Then the problem of finding the \(2n\) flattenings corresponds to finding the \(n\) coefficients \(\mathcal {H}_k\) with \(n\) equilibrium equations. An important consequence of this approach is that the flattening profile \(\mathcal {H}_k\) is the same regardless of whether the rotation of the body is synchronous or nonsynchronous, and the results for \(\mathcal {H}_k\) are equivalent to those found by Tricarico (2014).

We also studied the continuous case as the limit for a very large number of layers of infinitesimal thickness, which leads to Clairaut’s differential equation for the function \(\mathcal {H}(x)\) (i.e., the same equation for both flattenings). This result was expected because the coefficients of the Clairaut equation only depend on the internal distribution of matter \(\rho (x)\). Therefore, the differential equation that generates the functional form of the profile flattening \(\mathcal {H}(x)\) does not change when we change the nature of the deformation, provided that it is small. For densities decreasing monotonically with the radius, we found that, at the surface, \(\mathcal {H}_n\) takes values larger than 0.4 [see Eq. (21)] and takes the limit value 1 in the homogeneous case. This means that the surface flattenings of a differentiated body are always smaller than the flattenings of the corresponding homogeneous ellipsoids but always larger than 40 % of them.

In a realistic case where the core is denser than the shell, the flattening of the nucleus is smaller than the flattening of the surface. This is a result classically known to Tisserand (1891) and discussed in recent papers by Zharkov and Trubitsyn (1978), Hubbard (2013), and Tricarico (2014).

In the presence of a core, the surface is always less flattened than the homogeneous reference flattening but larger than 40 % of the latter value.

The fluid Love number \(k_f<1.5\) defines a continuous curve of possible values for the size of the nucleus \(\xi \) and the relative density of the shell \(\lambda \) and predicts their maximum value.

In all cases, the function \(\mathcal {H}(x)\) is an increasing monotonic function.

For all \(x\), the values of \(\mathcal {H}(x)\) increase from 0.530 to 1 when the power \(\alpha \) increases from 0 to \(\infty \), in contrast to the polytropic densities, where the values of \(\mathcal {H}(x)\) decrease from 1 to 0.4 when the polytropic index \(n\) increases from 0 to the limit case \(n=5\).

The fluid Love number \(k_f\) varies between 0.326 and 1.5 in the same range of the power \(\alpha \) for polynomial densities. For the polytropic laws, the fluid Love number \(k_f\) varies between 1.5 and 0 when the polytropic index \(n\) increases.

For polynomial laws, the values of \(C/m_TR^2\) increase from 0.24 to 0.4 when the power \(\alpha \) increases, and for the polytropic laws, the values of \(C/m_TR^2\) decreases from 0.4 to 0 when the polytropic index \(n\) increases.

## Footnotes

- 1.
The exact relation is \(\epsilon _J = 3 \epsilon _M\frac{a^3}{r^3} \frac{M}{M+m_T}\). The approximation is valid only if the mass of the deformed body and the orbital eccentricity are small, that is \(r \simeq a\) and \(m_T << M\).

- 2.
See Appendix C in the online supplement for more details.

- 3.
For the details of the calculation of \(\delta V_2^{k}\), see Eq. (A.13) in Appendix A (in the online supplement).

- 4.
An elementary calculation allows one to find the relationship \(\frac{C}{m_TR^2}\approx \frac{2}{3}\frac{\int _0^1 \widehat{\rho }z^4 \hbox {d}z}{\int _0^1 \widehat{\rho }z^2\hbox {d}z}=\frac{2}{5}\times \frac{3+\alpha }{5+\alpha }\).

## Notes

### Acknowledgments

The authors wish to thank one anonymous referee for comments and suggestions that helped to improve the manuscript. This investigation was supported by the National Council for Scientific and Technological Development (CNPq), Grants 141684/2013-5 and 306146/2010-0, and by St. Petersburg University, Grant 6.37.341.2015.

## Supplementary material

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