TIDAL FRICTION AND TIDAL LAGGING. APPLICABILITY LIMITATIONS OF A POPULAR FORMULA FOR THE TIDAL TORQUE

A bona fide theory of bodily tides implies (1) decomposition of the tide into Fourier harmonic modes and (2) endowment of each separate Fourier mode with a phase delay and a magnitude decrease of its own. The first part of this program, Fourier decomposition, was accomplished in full by Kaula (1964), though a partial sum of the Fourier series was developed yet by Darwin (1879). The second part of this program, the quest for an adequate frequency dependence of the phase lags and dynamical Love numbers, is now in progress. While earlier attempts seldom went beyond the Maxwell model, more realistic rheologies are now coming into use. A rheology combining the Andrade model at higher frequencies with the Maxwell model at the lowest frequencies was investigated by Efroimsky (2012a, 2012b).¹ The necessity for such a combined model originates from the fact that different physical mechanisms of friction dominate tidal dissipation at different frequencies. Several other rheological laws were probed by Henning et al. (2009) and Nimmo et al. (2012).

Some authors tried to sidestep a Fourier decomposition by building simpler toy models which would preserve qualitative features of the consistent tidal theory and, ideally, would yield some reasonable quantitative estimates. Two radically simplistic ad hoc tools known as the constant geometric lag model (MacDonald 1964; Goldreich 1966; Murray & Dermott 1999) and the constant time lag model (Singer 1968; Mignard 1979, 1980, 1981; Heller et al. 2011; Hut 1981) are often resorted to and are applied to rocky moons and planets and gas giants and stars alike. Historically, both these approaches were introduced for the ease of analytical treatment rather than on sound physical principles.²

The first of the approaches mentioned, the constant geometric lag model, will be addressed in this paper. Our goal is to demonstrate that the model should be discarded, both for physical and mathematical reasons. On the one hand, the model is not well grounded in physical reality because it assumes a constant tidal response independent of the rotation frequency everywhere except at the 1:1 resonance where it singularly changes sign. On the other hand, the model is genuinely contradictory in its mathematics. The source of the inherent conflict is the popular formula for the tidal torque (and its analogue for the tidal potential) through which the model is implemented. It turns out that these formulae tacitly imply constancy (frequency independence) of the time lag, a circumstance prohibiting the additional imposition of the constant-geometric-lag (= frequency-independent quality factor) condition.

Prior to executing the plan, we shall provide a comprehensive, review-style introduction into the methods of incorporation of friction into the tidal theory. The review will then enable us to recognise the aforementioned inconsistency in the constant geometric lag model.

In a subsequent publication (Makarov & Efroimsky 2013), we shall explore the constant time lag model. This is an approach implying that all the tidal strain modes should experience the same temporal delay relative to the appropriate modes comprising the tidal stress. While the method is often assumed³ to work in the purely viscous limit (which, hypothetically, may be the case of stars and gaseous planets; see Hut 1981 and Eggleton et al. 1998), there is no justification for using it for terrestrial-type bodies such as the Moon, Phobos, or any exoplanet with a rocky or partially molten mantle. In light of the current rheological knowledge, the tidal response is very different, and its frequency dependence experiences especially steep variations in the vicinity of spin-orbit resonances. In Makarov & Efroimsky (2013), we shall demonstrate that illegitimate application of the constant time lag model to telluric objects leads to nonexistent phenomena like pseudosynchronous rotation—not to mention that it squeezes the tidal-evolution timescales (Efroimsky & Lainey 2007) and alters the probabilities of capture into resonances (Makarov et al. 2012).

Let a body of radius R experience tides from a perturber of mass M*₁ placed at ${\boldsymbol r}^{*} = (r^*, \phi ^*, \lambda ^*)$, with r* ⩾ R.

At a point $\boldsymbol R= (R,\phi ,\lambda)$ on the perturbed body's surface, the potential W due to the perturber is expanded over the Legendre polynomials P_l(cos γ) as⁶

$$\begin{eqnarray} W(\boldsymbol R, \boldsymbol r^{ *})&=&\sum _{{\it {l}}=2}^{\infty } W_{\it {l}}(\boldsymbol R, \boldsymbol r^{ *}) = - \frac{GM^*_{1}}{r^{ *}} \sum _{{\it {l}}=2}^{\infty } \left(\frac{R}{r^{ *}} \right)^{\textstyle {^{\it {l}}}} P_{\it {l}}(\cos \gamma) \nonumber \\ &=&{-} \frac{G M^*_{1}}{r^{ *}}\sum _{{\it {l}}=2}^{\infty }\left(\frac{R}{r^{ *}}\right)^{\textstyle {^{\it {l}}}}\sum _{m=0}^{\it l} \frac{({\it l}-m)!}{({\it l}+m)!}(2-\delta _{0m}) \nonumber\\ &&\times P_{{\it {l}}m}(\sin \phi)P_{{\it {l}}m}(\sin \phi ^*) \cos m(\lambda -\lambda ^*) , \end{eqnarray} \tag{ 5 }$$

where δ_ij is the Kronecker delta symbol, G is Newton's gravity constant, while γ denotes the angular separation between the vectors ${\boldsymbol r}^{*}$ and R pointing from the center of the perturbed body. The longitudes λ, λ* are reckoned from a fixed meridian on the perturbed body, the latitudes ϕ, ϕ* being reckoned from the equator. The integers l and m are called the degree and order, accordingly. The associated Legendre functions P_lm(x) are referred to as the associated Legendre polynomials when their argument is the sine or cosine of some angle.

The ${\emph {l}}$th term $W_{\it {l}}(\boldsymbol R, \boldsymbol r^{ *})$ of the perturber's potential introduces a distortion into the perturbed body's shape, assumed to be linear. Then the ensuing ${\emph {l}}$th amendment U_l to the perturbed body's potential will also be linear in W_l. Since $U_{\it {l}}(\boldsymbol r)$ falls off outside the body as $r^{-(\it {l}+1)}$, the overall change in the exterior potential of the perturbed body will be

$$\begin{eqnarray} &&U(\boldsymbol r) = \sum _{{\it l}=2}^{\infty } U_{\it {l}}(\boldsymbol r) = \sum _{{\it l}=2}^{\infty } k_{\it l}\left(\frac{R}{r} \right)^{{\it l}+1}W_{\it {l}}(\boldsymbol R,\boldsymbol r^{*}) , \end{eqnarray} \tag{ 6 }$$

where k_l are the static Love numbers, R is the mean equatorial radius of the perturbed body, while $\boldsymbol R=(R, \phi , \lambda)$ and $\boldsymbol r=(r, \phi , \lambda)$ are a surface point and an exterior point above it, respectively, so that r ⩾ R.

Combining Equation (6) with Equation (5), we arrive at a useful formula for the amendment to the potential of the tidally disturbed body:

$$\begin{eqnarray} U(\boldsymbol r) &=& {-} {GM^*_{1}} \sum _{{\it {l}}=2}^{\infty }k_{\it l} \frac{R^{ \textstyle {^{2\it {l}+1}}}}{r^{ \textstyle {^{\it {l}+1}}}{r^{*}}^{ \textstyle {^{\it {l}+1}}}}\sum _{m=0}^{\it l}\frac{({\it l} - m)! }{({\it l} + m)!}(2-\delta _{0m}) \nonumber\\ &&{\times} P_{{\it {l}}m}(\sin \phi)P_{{ \it {l}}m}(\sin \phi ^*)\cos m(\lambda -\lambda ^*). \end{eqnarray} \tag{ 7 }$$

This is how the tidally generated change in the perturbed body's potential is "felt" at a point r. The change is expressed as a function of the spherical coordinates $\boldsymbol r=(r, \phi , \lambda)$ of this point and the spherical coordinates $\boldsymbol r^{ *}=(r^*, \phi ^*, \lambda ^*)$ of the tide-raising body. The formula may be employed when we have two exterior bodies: if one such body, a perturber of mass M*₁ located at r*, produces tides on the perturbed body, then the other exterior body, located at r, will experience a potential perturbation (7) due to these tides.

By changing variables from the spherical coordinates $\boldsymbol r^{ *}=(r^*, \phi ^*, \lambda ^*)$ and $\boldsymbol r=(r, \phi , \lambda)$ to the Keplerian coordinates $\boldsymbol r^{ *}=(a^*, e^*, {\it i}^*, \Omega ^*, \omega ^*, {\cal M}^*)$ and $\boldsymbol r=(a, e, {\it i}, \Omega , \omega , {\cal M})$, one obtains a formula equivalent to Equation (7):

$$\begin{eqnarray} U(\boldsymbol r) &=& {-} \sum _{{\it l}=2}^{\infty }k_{\it l}\left(\frac{R}{a} \right)^{\textstyle {^{{\it l}+1}}}\frac{G M^*_{1}}{a^*}\left(\frac{R}{a^*} \right)^{\textstyle {^{\it l}}}\sum _{m=0}^{\it l}\frac{({\it l} - m)!}{({\it l} + m)!} (2-\delta _{0m}) \nonumber\\ &&\times \sum _{p=0}^{\it l}F_{{\it l}mp}({\it i}^*)\sum _{q=-\infty }^{\infty }G_{{\it l}pq} (e^*) \sum _{h=0}^{\it l}F_{lmh}({\it i})\sum _{j=-\infty }^{\infty } G_{lhj}(e) \nonumber\\ &&\times \cos ((v_{{\it l}mpq}^*-m\theta ^*)- (v_{{\it l}mhj}-m\theta )) _{\textstyle {_{\textstyle ,}}} \end{eqnarray} \tag{ 8 }$$

where

$$\begin{eqnarray} &&v_{{\it l}mpq}^*\equiv ({\it l}-2p) \omega ^* + ({\it l}-2p+q){\cal M}^* + m \Omega ^* , \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} &&v_{{\it l}mhj} \equiv ({\it l}-2h) \omega + ({\it l}-2h+j){\cal M} + m \Omega , \end{eqnarray} \tag{ 10 }$$

q and j being arbitrary integers, p and h being arbitrary non-negative integers, F_lmp(i) being the inclination functions, with G_lpq(e) being the eccentricity polynomials coinciding with the Hansen coefficients $X^{\textstyle {^{(-l-1), (l-2p)}}}_{\textstyle {_{(l-2p+q)}}}(e)$. Also, keep in mind that θ* is the same as θ, which is the sidereal angle of the tidally perturbed body. Following Kaula (1964), we append an asterisk to θ, when it shows up in expressions corresponding to the tide-raising body. In expression (8), the terms −mθ* and −mθ cancel one another, wherefore their presence may seem redundant. We keep them, though, for they will no longer cancel when lagging comes into play.

Decomposition (8) was pioneered by Kaula (1961, 1964). However, its partial sum, with |l|, |q|, |j| ⩽ 2, was derived by Darwin (1879). Darwin's work in modern notations is discussed by Ferraz-Mello et al. (2008).⁷

For our further developments, it would be important to emphasize that formulae (7) and (8) are equivalent to one another because the latter is obtained from the former simply by a change of variables.

Derived for a static tide, formulae (7) and (8) extend trivially to an elastic dynamical setting where the tide adjusts instantaneously to the changing position $\boldsymbol r^{ *}(t)$ of the perturber.

The key point is that, to get $U(\boldsymbol r)$ at the point r at time t, we insert into Equation (7) or (8) the perturber's position $\boldsymbol r^{ *}(t)$ taken at that same time t, and not at an earlier time. Formulae (7) and (8) stay equivalent to one another, and remain unchanged, except that the distances, the sidereal angle, and the angular coordinates acquire a simultaneous time dependence:⁸ r becomes r(t), r* becomes r*(t); while $\theta, \theta^*, \phi, \phi^*, \lambda, \lambda^*, \omega, \omega^*, \Omega, \Omega^*, \mathcal{M}$, and $\mathcal{M}^*$ become $\theta(t), \theta^*(t), \phi(t), \phi^*(t), \lambda(t), \lambda^*(t), \omega(t), \omega^*(t), \Omega(t), \Omega^*(t), \mathcal{M}(t)$, and $\mathcal{M}^*(t)$.

Thus, to obtain $U(\boldsymbol r(t))$, we take the values of all variables at time t, leaving no place for any lagging. This is possible only for an absolutely elastic, i.e., frictionless perturbed body.

Let us now write down the modes over which the tidal disturbance of the body gets expanded. We begin with expression (5) for the perturbing potential at a fixed point $\boldsymbol R=(R, \phi , \lambda)$ on the surface of the perturbed body. Using the technique developed by Kaula (1961, 1964), we change the coordinates of the tide-raising body from $\boldsymbol r^{ *}=(r^*, \phi ^*, \lambda ^*)$ to $\boldsymbol r^{ *}=(a^*, e^*, {\it i}^*, \Omega ^*, \omega ^*, {\cal M}^*)$. However, the location on the body's surface, where the disturbance is observed, is still parameterized with its spherical coordinates $\boldsymbol R=(R, \phi , \lambda)$:

$$\begin{eqnarray} W(\boldsymbol R, \boldsymbol r^{ *}) &=& {-}\frac{G M^*}{a^*} \sum _{{\it l}=2}^{\infty } \left(\frac{R}{a^*} \right)^{\textstyle {^{\it l}}}\sum _{m=0}^{\it l} \frac{({\it l} - m)!}{({\it l} + m)!} (2- \delta _{0m})\nonumber\\ &&\times P_{{\it {l}}m}(\sin \phi) \sum _{p=0}^{\it l} F_{{\it l}mp}({\it i}^*) \sum _{q= - \infty }^{\infty } G_{{\it l}pq} (e^*) \nonumber\\ &&\times \left\lbrace \begin{array}{@{} c @{}}\cos \\ \sin \end{array} \right\rbrace ^{{\it l} - m \mbox{even}}_{{\it l} - m \mbox{odd}} (v_{{\it l}mpq}^*-m(\lambda +\theta ^*)). \end{eqnarray} \tag{ 11 }$$

Since R is the radius of the tidally perturbed body, and since the latitude ϕ and the longitude λ are reckoned from the equator and a fixed meridian, correspondingly, then Equation (11) is just another expression for the perturbing potential at the fixed point (R, ϕ, λ) of the body's surface. In Equation (11), the expression in round brackets can be reshaped as

$$\begin{eqnarray} &&v_{{\it l}mpq}^*-m(\lambda +\theta ^*) = \omega _{{\it l}mpq} (t - t_0) - m \lambda+v_{{\it l}mpq}^*(t_0)-m\theta^*(t_0), \nonumber\\ \end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray} &&\omega _{lmpq} \equiv ({\it l}-2p) \dot{\omega }^* + ({\it l}- 2p+q) n^* + m (\dot{\Omega }^* - \dot{\theta }^*). \end{eqnarray} \tag{ 13 }$$

Here $n^* \equiv {\bf {\dot{\cal {M}}}}^{ *}$ is the mean motion of the perturber, while t₀ is the time of perigee passage wherefrom the mean anomaly ${\cal {M}}^{ *}$ of the perturber is reckoned.

We see from Equation (12) that the quantities ω_lmpq given by Equation (13) are the Fourier modes over which the tidal perturbation (11) is expanded. While these modes can be positive or negative, the physical forcing frequencies,

$$\begin{eqnarray} &&\chi _{lmpq} \equiv | \omega _{lmpq} |, \end{eqnarray} \tag{ 14 }$$

at which the stress oscillates, are definitely positive.

Having developed formulae (8)–(11), Kaula (1961, 1964) never stipulated⁹ that the Fourier modes of the tide are given by Equation (13). Possibly, he was not interested in the frequency dependence of the phase or time lags.

In Section 6 of his book, Lambeck (1980) explained some aspects of Kaula's theory. While Lambeck's Equation (6.1.13b) indicates that Lambeck could have been aware of how the Fourier modes look, he too never wrote down the formula for the modes explicitly. Perhaps, like Kaula, Lambeck had no interest in the frequency dependence of lags—he just introduced a time lag Δt, which in his developments was implicitly regarded frequency independent.

While in the review by Efroimsky & Williams (2009) and in Efroimsky (2012a, 2012b) the expression for ω_lmpq was written down explicitly, its derivation was omitted.

Therefore, in the literature of which we are aware, the formula for the Fourier modes either was implied tacitly or was employed with no proof. This was our motivation to derive it here in such detail.

To conclude, at the point $\boldsymbol R=(R, \phi , \lambda)$ of the surface of the perturbed body, the perturbing potential is expressed through the tidal Fourier modes as

$$\begin{eqnarray} \nonumber W(\boldsymbol R, \boldsymbol r^{ *}) &=& {-} \frac{G M^*}{a^*} \sum _{{\it l}=2}^{\infty } \left(\frac{R}{a^*} \right)^{\textstyle {^{\it l}}}\sum _{m=0}^{\it l} \frac{({\it l} - m)!}{({\it l} + m)!} (2-\delta _{0m}) \nonumber\\ &&{\times} P_{{\it {l}}m}(\sin \phi) \sum _{p=0}^{\it l} F_{{\it l}mp}({\it i}^*) \sum _{q= - \infty }^{\infty } G_{{\it l}pq} (e^*) \nonumber\\ &&{\times} \left\lbrace \begin{array}{@{} c @{}}\cos \\ \sin \end{array} \right\rbrace ^{{\it l} - m \ \mbox{even}}_{{\it l} - m \ \mbox{odd}} (\omega _{lmpq} (t - t_0) - m \lambda\nonumber\\ &&+v^*_{\it lmpq}(t_0)+m\theta^* (t_0)), \end{eqnarray} \tag{ 15 }$$

the tidal modes ω_lmpq being given by Equation (13). In an idealized situation, when the extended body is frictionless and its response is instantaneous, we can employ the static formula (6), as explained in Section 4.2. Combining formula with expression (15), we see that the additional tidal potential generated by a perfectly elastic body at the point $\boldsymbol r=(r, \phi , \lambda)$ right above R will now read as

$$\begin{eqnarray} \nonumber U(\boldsymbol r, \boldsymbol r^{ *}) &=& {-} \frac{G M^*}{a^*} \sum _{{\it l}=2}^{\infty } \left(\frac{R}{r} \right)^{\textstyle {^{l+1}}} \left(\frac{R}{a^*} \right)^{\textstyle {^{\it l}}}\sum _{m=0}^{\it l} \frac{({\it l} - m)!}{({\it l} + m)!} (2- \delta _{0m} ) \nonumber\\ &&{\times} P_{{\it {l}}m}(\sin \phi) \sum _{p=0}^{\it l} F_{{\it l}mp}({\it i}^*) \sum _{q= - \infty }^{\infty } G_{{\it l}pq} (e^*) k_l \nonumber\\ &&{\times} \left\lbrace \begin{array}{@{} c @{}}\cos \\ \sin \end{array} \right\rbrace ^{{\it l} - m \ \mbox{even}}_{{\it l} - m \ \mbox{odd}} (\omega _{lmpq} (t - t_0) - m \lambda\nonumber\\ &&+v^*_{\it lmpq}(t_0)+m\theta^* (t_0) ). \end{eqnarray} \tag{ 16 }$$

It is due to the lack of friction that the lmpq term of Equation (16) is in phase with the lmpq term of Equation (15). Below we shall see that inclusion of friction into the picture renders a phase shift between these terms. It is also in anticipation of the discussion of friction that we placed the Love numbers inside the ∑_mpq sum in expression (16). Mode independent in the perfectly elastic case, the Love numbers may acquire dependence upon the Fourier modes ω_lmpq because friction may mitigate amplitudes of distortion differently at different frequencies.

Naively, dissipation and the ensuing lagging can be included into the picture by assuming that the exterior body located at point r at time t is subject not to the tidal potential created simultaneously by the perturber residing at $\boldsymbol r^{ *}(t)$, but to the potential generated by the perturber lagging in time on its orbit¹⁰ by some Δt. Loosely speaking, the no-asterisk exterior body located at $\boldsymbol r(t)$ "feels" the tide given by Equation (7) or (8), as if the tide were generated by the asterisk perturber located on its orbit, not at $\boldsymbol r^{ *}(t)$ but at $\boldsymbol r^{ *}(t - \Delta t)$, as seen in the frame corotating with the tidally deformed body.

Mathematically, this implies that the tidal potential $U(\boldsymbol r)$ at t must be calculated via Equation (7) or (8), by insertion of the no-asterisk coordinates taken at t, and the coordinates with an asterisk taken at t − Δt. This naive strategy also implies that the Love numbers k_l keep their static values, though this detail is seldom spelled out.

With this approach implemented, our formulae (7) and (8) will acquire the following form:

$$\begin{eqnarray} \nonumber U(\boldsymbol r(t)) &=& {-}{G M^*_{1}}\sum _{{\it {l}}=2}^{\infty }k_{\it l} \frac{R^{ \textstyle {^{2\it {l}+1}}}}{r(t)^{ \textstyle {^{\it {l}+1}}}{r^{ *}(t-\Delta t)}^{ \textstyle {^{\it {l}+1}}}}\sum _{m=0}^{\it l}\frac{\left(l - m\right)! }{({\it l} + m)!} \nonumber\\ &&\times (2-\delta _{0m})P_{{\it {l}}m}(\sin \phi (t))P_{{ \it {l}}m} (\sin \phi ^*(t -\Delta t))\nonumber\\ &&\times \cos m[\lambda (t)-\lambda ^*(t-\Delta t)] \quad \quad \end{eqnarray} \tag{ 17 }$$

and

$$\begin{eqnarray} U(\boldsymbol r) &=& {-} \sum _{l=2}^{\infty } k_{\it l} \left(\frac{R}{a} \right)^{\textstyle {^{{\it l}+1}}}\frac{G M^*_{1}}{a^*} \left(\frac{R}{a^*} \right)^{\textstyle {^l}}\sum _{m=0}^{\it l} \frac{({\it l} - m)!}{({\it l} + m)!} \nonumber\\ &&{\times} (2 - \delta _{0m})\sum _{p=0}^{l}F_{lmp}({\it i}^*)\sum _{q=-\infty }^{\infty }G_{lpq} (e^*) \sum _{h=0}^{\it l}F_{lmh}({\it i}) \nonumber\\ &&{\times} \sum _{j=-\infty }^{\infty } G_{lhj}(e) \cos ([v_{{\it l}mpq}^*(t-\Delta t) - m\theta ^*(t-\Delta t) ] \nonumber\\ && {-} [v_{{\it l}mhj}(t)-m\theta (t)]) _{\textstyle {_{\textstyle.}}} \end{eqnarray} \tag{ 18 }$$

Just like their static precursors (7) and (8), our dynamical formulae (17) and (18) remain equivalent to one another. They still reflect a mere switch from the spherical to the Keplerian coordinates,¹¹ except that now r is taken at time t, while r* is taken at t − Δt.

The longitude reckoned from a fixed meridian on the perturbed body is expressed through the true anomaly ν, the periapse ω, and the node Ω as

$$\begin{eqnarray*} &&\nonumber \lambda = - \theta + \Omega + \omega + \nu + O({\it i}^2). \end{eqnarray*}$$

Neglecting the nodal and the apsidal precession, and for a small obliquity, this results in

$$\begin{eqnarray*} &&\nonumber \dot{\lambda } \approx - \dot{\theta } + \dot{\nu } + O({\it i}^2). \end{eqnarray*}$$

In Equation (17), the argument of cosine may now be written, in a linear approximation over Δt, as

$$\begin{eqnarray} m [ \lambda (t) - \lambda ^*(t-\Delta t) ] &=& m [ \lambda (t) - \lambda ^*(t) + \dot{\lambda }^* \Delta t ] \nonumber\\ &=& m (\lambda - \lambda ^*) - m (\dot{\theta } - \dot{\nu }) \Delta t + O({\it i}^2). \nonumber\\ \end{eqnarray} \tag{ 19 }$$

In Equation (18), the argument of cosine may be shaped, in a linear approximation over Δt, as

$$\begin{eqnarray} && [v_{{\it l}mpq}^*(t-\Delta t)- m \theta ^*(t-\Delta t)]- [v_{{\it l}mhj}(t)-m \theta (t)] \nonumber\\ &&\quad =[v_{{\it l}mpq}^*(t)- m \theta ^*(t)]-[v_{{\it l}mhj}(t)-m \theta (t)] - \epsilon _{lmpq}, \qquad \end{eqnarray} \tag{ 20 }$$

where

$$\begin{eqnarray} \nonumber \epsilon _{lmpq}&\equiv & [\dot{v}_{{\it l}mpq}^*(t)- m \dot{\theta }^*(t)] \Delta t = [({\it l}-2p) \dot{\omega }^* \nonumber\\ && + ({\it l}- 2p+q) n^* + m (\dot{\Omega }^* - \dot{\theta }^*)] \Delta t = \omega _{lmpq} \Delta t \qquad \end{eqnarray} \tag{ 21 }$$

is the phase lag corresponding to the mode ω_lmpq.

From here we observe that the method chosen above of taking the tidal friction into account fixes the phase lags in a very specific way: through shifting the perturber back on its orbit by a fixed time Δt, we set the phase lags (21) to be proportional to this Δt. It should be emphasized once again that the shift is performed in a frame corotating with the perturbed body. In a frame comoving but not corotating with it, the shift will thus be accompanied by rotation of the perturbed body back by $\dot{\theta}\Delta t$, hence the term ${-}\dot{\theta}\Delta t$ in the expression (21) for the phase lag.

Our formulae (17) and (18) can be written in another, equivalent form:

$$\begin{eqnarray} U(\boldsymbol r, \boldsymbol r^{ *}) &=& {-} \frac{G M^*}{a^*} \sum _{{\it l}=2}^{\infty } \left(\frac{R}{r} \right)^{\textstyle {^{l+1}}} \left(\frac{R}{a^*} \right)^{\textstyle {^{\it l}}}\sum _{m=0}^{\it l} \frac{({\it l} - m)!}{({\it l} + m)!} (2 - \delta _{0m}) \nonumber \\ &&{\times} P_{{\it {l}}m}(\sin \phi) \sum _{p=0}^{\it l} F_{{\it l}mp}({\it i}^*) \sum _{q= - \infty }^{\infty } G_{{\it l}pq} (e^*) k_l \nonumber\\ &&{\times} \left\lbrace \begin{array}{@{} c @{}}\cos \\ \sin \end{array} \right\rbrace ^{{\it l} - m \ \mbox{even}}_{{\it l} - m \ \mbox{odd}} (\omega _{lmpq} (t - \Delta t - t_0) - m \lambda\nonumber\\ &&+v _{lmpq}^*(t_0)-m\theta^*(t_0)). \end{eqnarray} \tag{ 22 }$$

This form is analogous to Equation (16), except for the time lag Δt, the same for each Fourier mode.

Expression (22) is equivalent to expression (17) and is obtained from it by a switch from the spherical coordinates $\boldsymbol r^{ *}=(r^*, \phi ^*, \lambda ^*)$ to the Kepler elements $\boldsymbol r^{ *}=(a^*, e^*, {\it i}^*, \Omega ^*, \omega ^*, {\cal M}^*)$, with the variables $\boldsymbol r=(r, \phi , \lambda)$ kept unchanged. From the equivalence of expressions (22) and (17), we observe that, after the same time lag is added into all terms of Equation (22), all the terms shifted thus add up to the equilibrium bulge geometrically displaced in such a way as if it were a static bulge generated by the perturber at a slightly different time. No other rheology can make this claim because, more generally, the lag in each term in the expansion would correspond to its own increment in t. This tells us that by setting Δt to be frequency independent we impose a highly restrictive rheological rule, obedience to which cannot be expected of realistic mantles.

A more profound problem with this approach lies in the fact that it is illegitimate to introduce lags, keeping at the same time the Love numbers unchanged. Mitigation of the magnitude and lagging of the phase are inseparably connected, though the link becomes apparent only within a consistent approach based on the Fourier expansion of the tide and on employment of one or another rheological law. That law will then define both lagging in phase and reduction in magnitude.

The above expression (21) for phase lags contains in itself an obvious hint on how a general-type rheology should be built into the tidal theory—to that end, one simply has to endow each mode ω_lmpq with a time lag Δt_l(ω_lmpq) of its own. Another adjustment is the mode dependence of the Love numbers: k_l = k_l(ω_lmpq). Expression (18) will now become

$$\begin{eqnarray} U(\boldsymbol r) &=& {-} \sum _{l=2}^{\infty } \left(\frac{R}{a} \right)^{\textstyle {^{{\it l}+1}}}\frac{G M^*_{1}}{a^*} \left(\frac{R}{a^*} \right)^{\textstyle {^l}}\sum _{m=0}^{\it l} \frac{({\it l} - m)!}{({\it l} + m)!} (2- \delta _{0m}) \nonumber\\ &&\times \sum _{p=0}^{l}F_{lmp}({\it i}^*)\sum _{q=-\infty }^{\infty }G_{lpq} (e^*) \sum _{h=0}^{\it l}F_{lmh}({\it i})\sum _{j=-\infty }^{\infty } G_{lhj}(e) \nonumber\\ &&\times k_{\it l}(\omega _{lmpq}) \cos ([v_{{\it l}mpq}^*(t)- m\theta ^*(t)] \nonumber\\ && - [v_{{\it l}mhj}(t)-m\theta (t) ] - \epsilon _{l}(\omega _{lmpq})) _{\textstyle {_{\textstyle ,}}} \end{eqnarray} \tag{ 23 }$$

where

$$\begin{eqnarray} \epsilon _{l}(\omega _{lmpq}) & \equiv & \omega _{lmpq} \Delta t_{l}(\omega _{lmpq}) = [ ({\it l}-2p) \dot{\omega }^* + ({\it l}- 2p+q) n^* \nonumber\\ &&+ m (\dot{\Omega }^* - \dot{\theta }^*)] \Delta t_{l}(\omega _{lmpq}). \end{eqnarray} \tag{ 24 }$$

In Equations (23) and (24), we prefer to denote the phase and time lags not as _lmpq and Δt_lmpq, but as _l(ω_lmpq) and Δt_l(ω_lmpq). Indeed, their dependence on the indices mpq is solely due to the argument ω_lmpq. However, we cannot strip or Δt of the subscript l, because the functional form of the frequency dependence of the phase or time lag is different for different l's. The tidal friction is not the same as the seismic friction, and the degree l affects the tidal damping rate.¹² Hence in formulae (23) and (24) we have _l and Δt_l, and not just or Δt.

We would also write down the expression for the tidal potential in terms of the Keplerian elements of the perturber and the spherical coordinates of the point where this potential is observed:

$$\begin{eqnarray} && U(\boldsymbol r, \boldsymbol r^{ *}) \nonumber\\ &&= {-} \frac{G M^*}{a^*} \sum _{{\it l}=2}^{\infty } \left(\frac{R}{r} \right)^{\textstyle {^{l+1}}} \left(\frac{R}{a^*} \right)^{\textstyle {^{\it l}}}\sum _{m=0}^{\it l} \frac{({\it l} - m)!}{({\it l} + m)!} (2 - \delta _{0m}) \nonumber\\ &&\quad {\times} P_{{\it {l}}m}(\sin \phi) \sum _{p=0}^{\it l} F_{{\it l}mp}({\it i}^*) \sum _{q= - \infty }^{\infty } G_{{\it l}pq} (e^*) k_l(\omega _{lmpq}) \nonumber\\ &&\quad {\times} \left\lbrace \begin{array}{@{} c @{}}\cos \\ \sin \end{array} \right\rbrace ^{{\it l} - m \ \mbox{even}}_{{\it l} - m \ \mbox{odd}} (\omega _{lmpq} [ t - \Delta t_l(\omega _{lmpq}) - t_0 ] - m \lambda\nonumber\\ &&+v^*_{lmpq}(t_0)-m\theta^* (t_0)). \end{eqnarray} \tag{ 25 }$$

This form is analogous to Equations (16) and (22), except for two details. First, each mode ω_lmpq now has a time lag Δt_l = Δt_l(ω_lmpq) of its own. Likewise, the dynamical Love number at each Fourier mode is a function of this mode: k_l = k_l(ω_lmpq).

The reason why we wrote down $U(\boldsymbol r, \boldsymbol r^{ *})$ in the above form is that it immediately furnishes an expression for the geometric lag angle of an arbitrary lmpq bulge:

$$\begin{eqnarray} &&\delta _{lmpq} = \frac{\omega _{lmpq}}{m} \Delta t_l(\omega _{lmpq}). \end{eqnarray} \tag{ 26 }$$

For example, the geometric lag angle of the principal, semidiurnal bulge is $\delta _{2200}= ({\textstyle \omega _{2200}}/{\textstyle 2}) \Delta t_2 = (n-\dot{\theta }) \Delta t_2$, where the time lag is taken at the appropriate, semidiurnal mode: Δt = Δt₂(ω₂₂₀₀).

It is customary to introduce the convention that the phase and time lags and the Love numbers are functions not of the tidal mode ω_lmpq but of the positively defined frequency χ_lmpq ≡ |ω_lmpq|. Simplifying some calculations, this convention makes it necessary to introduce, by hand, sign factors into the terms of the Fourier expansions of the tidal force and torque (Efroimsky 2012b).

In practical applications, most important is the special case when the exterior body located at r coincides with the tide-raising perturber located at r*. In this situation, the perturber is acting upon itself through the tide it creates on the perturbed body. Keeping the phase lags intact, and identifying $\boldsymbol r(t)$ with $\boldsymbol r^{ *}(t)$, we may be tempted to misassume that the expression with no asterisk, standing in the argument of the cosine in Equation (23), compensates the expression with an asterisk, so the phase lag becomes all that is left. However, this would only furnish us with the secular term of the tidal potential $U(\boldsymbol r)$ wherewith the perturber acts upon itself through the medium of the tidally perturbed body. This term is proportional to cos _lmpq. As p, q and h, j are independent pairs of integers, there will also be contributions with {p, q} ≠ {h, j}. These are the oscillating components of the tidal potential $U(\boldsymbol r)$. Although their time average is nil, they do contribute to the heat production, while the appropriate oscillating components of the tidal torque may influence free librations. The topic was first addressed in Efroimsky (2012b), and was revisited by Makarov et al. (2012) who explored whether the oscillating part of the torque can influence capture into spin–orbit resonances. It has turned out that, naturally, the oscillating part of the torque alters the outcome of a particular realization of the capture scenario, but leaves the statistics unchanged.

Let us calculate the tidal torque, using Equation (17). By employing this expression, we automatically set the rheology to be Δt = const.

Consider an exterior body of mass M₁ located at r, which is subject to the additional tidal potential $U(\boldsymbol r)$ of the tidally perturbed body. Then its energy in this potential will be $M_1 U(\boldsymbol r)$. When the position of the exterior body is rendered in spherical coordinates, the polar component of the torque acting on it can be conveniently expressed as: $T_z = - M_{1} {\partial U(\boldsymbol r)}/{\partial \lambda }$. The polar torque wherewith the exterior body acts back on the tidally perturbed body is the negative of T_z:

$$\begin{eqnarray} &&{\cal {T}}_z (\boldsymbol r) = M_{1} \frac{\partial U(\boldsymbol r)}{\partial \lambda }. \end{eqnarray} \tag{ 27 }$$

Here polar means orthogonal to the perturbed body's equator. For small obliquities (and, therefore, small latitudes), insertion of Equation (17) into Equation (27) yields

$$\begin{eqnarray} {\cal {T}}_z (\boldsymbol r(t)) &=& {G M_{1} M^*_{1}}\sum _{{\it {l}}=2}^{\infty }k_{\it l} \frac{R^{ \textstyle {^{2\it {l}+1}}}}{r(t)^{ \textstyle {^{\it {l}+1}}}{r^{ *}(t-\Delta t)}^{ \textstyle {^{\it {l}+1}}}}\sum _{m=0}^{\it l} m \frac{\left(l - m\right)! }{({\it l} + m)!} \nonumber\\ &&\times (2-\delta _{0m})P_{lm}(0)P_{lm}(0) \sin (m (\lambda -\lambda ^*) \nonumber\\ && - m (\dot{\theta }-\dot{\nu }) \Delta t ) + O(i^2), \end{eqnarray} \tag{ 28 }$$

where we made use of Equation (19).

Since the integer m is now entering the above expression as a multiplier, the term with m = 0 becomes nil. As P₂₁(0) = 0, the term with m = 1 also vanishes. Hence, the term with l = m = 2 is leading. Neglecting the smaller terms, we thus obtain

$$\begin{eqnarray} {\cal {T}}_z(\boldsymbol r(t)) & \approx & \frac{\textstyle 3}{\textstyle 2} {G M_{1} M^*_{1}} k_{2} \frac{R^5}{r(t)^{ \textstyle {^{3}}}{r^{ *}(t-\Delta t)}^{ \textstyle {^{3}}}} \sin (2 (\lambda -\lambda ^*) \nonumber\\ && {-} 2 (\dot{\theta }-\dot{\nu }) \Delta t ). \end{eqnarray} \tag{ 29 }$$

Consider the special case when the tide-raising perturber (with the asterisk) coincides with the other external body (with no asterisk). The perturber creates tides on the perturbed body, and then interacts with the tides it itself has created. Hence the perturber becomes subject to a tidal torque T it exerted upon itself, through the medium of the tidal bulge it creates on the perturbed body. Evidently, a torque ${\boldsymbol {\cal T}} = - \boldsymbol T$, of the same magnitude but opposite direction, will be acting upon the perturbed body. We arrive at this torque by setting λ = λ* and M₁ = M*₁ in the above expression:¹³

$$\begin{eqnarray} &&{\cal {T}}_z(\boldsymbol r) \approx \frac{\textstyle 3}{\textstyle 2} {G M_{1}^{ 2}} k_{2} \frac{R^5}{r^{ \textstyle {^{6}}} } \sin (2 (\dot{\nu }-\dot{\theta }) \Delta t). \end{eqnarray} \tag{ 30 }$$

Naturally, for the model with a frequency-independent Δt, the quantity

$$\begin{eqnarray} &&\epsilon _g \equiv (\dot{\nu }-\dot{\theta }) \Delta t \end{eqnarray} \tag{ 31 }$$

is the geometric lag, i.e., the angular separation between the planetocentric directions toward the perturber and the bulge. Accordingly, within the said model, the quantity

$$\begin{eqnarray} &&\chi = 2 | \dot{\nu }-\dot{\theta } | \end{eqnarray} \tag{ 32 }$$

acts as an instantaneous tidal frequency.

The quantity

$$\begin{eqnarray} &&\epsilon _{ph} \equiv 2 (\dot{\nu }-\dot{\theta }) \Delta t = 2 \epsilon _g \end{eqnarray} \tag{ 33 }$$

is commonly interpreted as an instantaneous phase lag, so the torque in expression (30) may be written down as

$$\begin{eqnarray} &&{\cal {T}}_z(\boldsymbol r) \approx \frac{\textstyle 3}{\textstyle 2} {G M_{1}^{ 2}} k_{2} \frac{R^5}{r^{ \textstyle {^{6}}}} \sin \epsilon _{ph} = \frac{\textstyle 3}{\textstyle 2} {G M_{1}^{ 2}} k_{2} \frac{R^5}{r^{ \textstyle {^{6}}}} \sin 2\epsilon _{g} , \quad \quad \end{eqnarray} \tag{ 34 }$$

which is exactly the expression (1) we are concerned with. In Section 2, we mentioned several popular papers and books, including Murray & Dermott (1999, Equation (4.159)),¹⁴ where this formula is employed. Now that we have derived this formula accurately, we see that its validity hinges on the time lag being frequency independent.

While it is common (MacDonald 1964; Goldreich 1966; Kaula 1968; Murray & Dermott 1999) to treat

$$\begin{eqnarray} &&Q \equiv 1/ | \sin (2 (\dot{\nu }-\dot{\theta }) \Delta t ) | = 1/\sin | \epsilon _{ph} | \end{eqnarray} \tag{ 35 }$$

as an instantaneous quality factor, the validity of this interpretation of Equation (35) remains questionable. For a nonzero eccentricity, the instantaneous tidal frequency (32) is varying in time. So it is not readily apparent whether the instantaneous Q is connected to the damping rate in the manner the proper quality factor introduced at a certain frequency links to the damping rate at that frequency. MacDonald (1964) and Goldreich (1966) tried to sidestep this difficulty by assuming that Q is a frequency independent constant. However, at finite eccentricities this assumption does not work, as it is incompatible with the constant Δt assumption (the latter assumption being a necessary prerequisite to using formulae (31)–(35), as we saw above). For more on this, see Williams & Efroimsky (2012).

When starting out with expression (18), it is convenient to use the formula

$$\begin{eqnarray} &&{\cal {T}}_z(\boldsymbol r) = - M_{1} \frac{\partial U(\boldsymbol r)}{\partial \theta } , \end{eqnarray} \tag{ 36 }$$

instead of Equation (27). Technically, we should first differentiate U with respect to the sidereal angle θ, and then set θ = θ*. We should also set the orbital elements with asterisk equal to their counterparts with no asterisk, with the understanding that the tide-raising perturber is the same as the other exterior body which "feels" the tides on the perturbed body. The development will furnish us with the polar component of the torque with which the perturber acts upon the tidally deformed body. For exploration of dynamics in a low-obliquity configuration, this component is sufficient.

Insertion of the Fourier series (18) into Equation (36) yields a Fourier series for the polar component of the torque, which is presented in Efroimsky (2012b). Here we shall not repeat this long formula, but shall only make an important comment on it. Insofar as Δt stays frequency independent (i.e., has the same value for all phase lags (21) entering the expansion for the torque), the resulting series for the torque stays fully equivalent to Equation (28), with λ and λ* set equal to one another in the latter formula.

This equivalence is ensured by expression (18) for the potential U, which is equivalent to expression (17) whence formula (28) originated, and by our agreement to keep Δt the same for all phase lags.

As soon as we abandon the assumption that the phase lags (21) contain the same fixed Δt, i.e., as soon as we switch from the lags (21) to those rendered by Equation (24), we acquire an opportunity to describe a tidal torque acting on a perturbed body of an arbitrary rheology. Indeed, as the time lags can have an arbitrary mode-dependence, this also relates to the phase lags. Above that, we now permit the Love numbers to be mode dependent.

To derive the tidal torque, we now combine formula (36) not with expansion (18) but with expansion (23) where the time lags are, generally, all different. The rheological emancipation, though, comes at a cost: the Fourier decomposition for the torque, obtained through Equation (23), with mode-dependent $\Delta t_l(\omega _{\textstyle {_{lmpq}}})$ and $k_l(\omega _{\textstyle {_{lmpq}}})$, will no longer be equivalent to the concise and elegant formula (28). The customary and widely used leading-order approximation of Equation (28), given by Equation (30) or by Equation (34), will not work for an arbitrary rheology.

If, for example, we choose to set the factors $({\textstyle k_l(\omega _{\textstyle {_{lmpq}}})}/{\textstyle Q_l(\omega _{\textstyle {_{lmpq}}})}) \mbox{Sgn}(\omega _{\textstyle {_{lmpq}}}) = k_l \sin \epsilon _l$ to be mode independent, then, to calculate the torque, we shall have to plug the same value of k_lsin _l into all terms of the Fourier series for the torque (Equation (106) from Efroimsky 2012b). However, we shall not be able to employ the neat formula (34).