Tidal synchronization of close-in satellites and exoplanets. A rheophysical approach

Abstract

This paper presents a new theory of the dynamical tides of celestial bodies. It is founded on a Newtonian creep instead of the classical delaying approach of the standard viscoelastic theories and the results of the theory derive mainly from the solution of a non-homogeneous ordinary differential equation. Lags appear in the solution but as quantities determined from the solution of the equation and are not arbitrary external quantities plugged in an elastic model. The resulting lags of the tide components are increasing functions of their frequencies (as in Darwin’s theory), but not small quantities. The amplitudes of the tide components depend on the viscosity of the body and on their frequencies; they are not constants. The resulting stationary rotations (often called pseudo-synchronous) have an excess velocity roughly proportional to $6ne^2/(\chi ^2+\chi ^{-2})$ ($\chi$ is the mean-motion in units of one critical frequency—the relaxation factor—inversely proportional to the viscosity) instead of the exact $6ne^2$ of standard theories. The dissipation in the pseudo-synchronous solution is inversely proportional to $(\chi +\chi ^{-1})$; thus, in the inviscid limit, it is roughly proportional to the frequency (as in standard theories), but that behavior is inverted when the viscosity is high and the tide frequency larger than the critical frequency. For free rotating bodies, the dissipation is given by the same law, but now $\chi$ is the frequency of the semi-diurnal tide in units of the critical frequency. This approach fails, however, to reproduce the actual tidal lags on Earth. In this case, to reconcile theory and observations, we need to assume the existence of an elastic tide superposed to the creeping tide. The theory is applied to several Solar System and extrasolar bodies and currently available data are used to estimate the relaxation factor $\gamma$ (i.e. the critical frequency) of these bodies.

Appendix: Higher-order approximations

The eccentricity functions introduced in Sect. 3 are the Cayley expansions for the solutions of the Keplerian motion (see Cayley 1861; Fernandes 1996) corresponding to the entries $(r/a)^{-3} [\sin |\cos ] 2f$ in Cayley’s tables.

The approximations adopted in Sect. 3 may be enough for most purposes. They are certainly enough for low eccentricities. For high eccentricities, however, a different approach is necessary. Cayley expansions include in their derivation the Taylor expansion of the solution of Kepler’s equation, whose convergence radius is $e^*=0.6627434\ldots$ (see Wintner 1941). Some simple comparisons allow us to see that for eccentricities of the order of 0.5, or even less, the results are not accurate.

Since the equations appearing in this paper are written in terms of one of the families of Cayley expansions, the power series may be substituted by Eq. 16 in the form

$$\begin{aligned} E_{2,k}(e)=\frac{1}{2\pi \sqrt{1-e^2}}\int \limits _0^{2\pi }\frac{a}{r}\cos \big (2v+(k-2)\ell \big )\ dv. \end{aligned}$$

(63)

We may note that the function under the integral involves both the mean ($\ell$) and the true ($v$) anomalies, but can be easily handled numerically. The only difference with Eq. (16) is the preference in having the integration done over the true anomaly instead of being done over the mean anomaly (via the classical expression $r^2 dv=a^2\sqrt{1-e^2}d\ell$) to avoid having to solve the Kepler equation, which is operationally expensive.

The integral solves the problem of the accuracy in the determination of the Cayley coefficients (see Table 2), but not that of the Fourier series giving the variation of the elements. To obtain approximations valid in high eccentricities, it is necessary to use expansions not affected by the singularities of the Keplerian motion (see Ferraz-Mello and Sato 1989). However, one preliminary guess can be done comparing different approximations. Figure 8 shows the value of $|\dot{e}|$ for close-in companions in pseudo-synchronous stationary rotation, in arbitrary units, considering the sum of the eccentricity-dependent terms of Eq. 49 truncated at $N=$ 7, 20, 50 and 100, respectively. The abrupt changes of the curves with respect to the next one show a limiting eccentricity in each case. For instance the series truncated at $N=7$ (the usual cutoff of Cayley series) give wrong results for eccentricities above 0.4. It is important to stress that the calculations showed in Fig. 5, concerning the tidal evolution predicted by the standard Darwin theory are not affected by truncation effects since for that sake we have used Mignard’s formulation whose basic equations are given in closed form (Mignard 1979) without resorting to any expansions.

Fig. 8

$\log |de/dt|$ obtained with the series of Eq. (49) truncated at $N=$ 7, 20, 50 and 100. Arbitrary units

Table 2 Comparison of the calculated values of $E_{2,k}(e)$ for $e=0.4$