% ${{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ % ~~~~~~~}^{^{^{Submitted~to~``Astronomy~\&~Astrophysics"}}}}$\\ {\Large{The Theory of Bodily Tides.\\ The Models and the Physics. }}
{\Large{Michael Efroimsky}}
{{US Naval Observatory, Washington DC 20392 USA}}
% The Gerstenkorn-MacDonald-Kaula theory of bodily tides assumes constancy of the % geometric lag angle $\delta_1$, while the Singer-Mignard theory asserts constancy % of the time lag $\,\Delta t\,$. \noindent Any model of tides is based on a specific hypothesis of how the geometric lag $\delta_1$ depends on the tidal-flexure frequency $\chi$. % (For example, a circular-orbit equatorial satellite acts on the surface % with a frequency $\;\chi\,=\,2\,|\,\omega_p\,-\,n\,|\;$, quantities $\, % \omega_p\,$ and $\,n\,$ being the planet's spin rate and the satellite's % mean motion.) Some authors put $\delta_1 \sim\chi^0$, the others set $\delta_1\sim\chi^1$. The actual dependence % of the lag on the frequency is complicated and determined by the planet's rheology. A particular form of this dependence will % unambiguously fix the form of the frequency dependence of the tidal quality factor $\,Q\,$. Since at present we know the shape of the function $Q(\chi)$, we can reverse our line of reasoning and single out the appropriate frequency-dependence of the lag. The dependence is different from those employed hitherto, and makes a considerable alteration in the theory of tides. The second alteration is needed to evade some difficulties inherent in Kaula's model. Kaula's expansion is divergent for eccentricities higher than a certain value. It is also subject to ``Goldreich's admonition": due to nonlinearity, the quality factors introduced for higher harmonics are badly defined (Goldreich 1963). To avoid these predicaments, we abandon the expansion and introduce an instantaneous tidal frequency that bears a dependence upon the moon's true anomaly (concept prompted by the WKB approximation in quantum mechanics). We also introduce an overall quality factor, the only $\,Q\,$ factor emerging in the model. This factor is a function of the instantaneous tidal frequency and therefore it, too, depends on the true anomaly. The third alteration will be our departure from the popular belief that $\,Q\,$ is inversely proportional to the ``tangential lag" (the angle $\delta_1$ subtended at the planet's centre between the moon and the bulge). That this is correct only in the limit of a circular equatorial orbit was pointed out by Kaula (1968) -- see the caveat preceding his derivation of formula (4.5.19) for the damping rate. Generalisation of Kaula's formula to a finite eccentricity and inclination yields extra terms. These become leading on crossing the synchronous orbit, when the ``tangential lag" vanishes. We show, by some different method, that in the general case $\,Q\,$ is interconnected, in the standard way, not with the ``tangential lag", but with the overall lag $\,\delta\,\equiv\,\sqrt{(\mbox{{tangential~lag}})^{\textstyle{^2}}+\,( \mbox{radial~lag})^{\textstyle{^2}}\,}\,$. While the ``tangential lag" $\,\delta_1\,$ originates because the location of the bulge lags, the``radial" lag comes into play since the height of the bulge lags too. % Each of the three above alterations can % be employed independently from the others. It turns out that, for an arbitrary eccentricity and inclination, the total lag and the instantaneous $\chi$ and $Q$ are always interconnected via the same relations as in the trivial case of an equatorial circular orbit. This makes our model concise and self-explanatory. The model addresses only the land tides, and therefore is intended to be a tool for exploring the dynamics of the Martian satellites. It is applicable to the Earth-Moon system only for the aeons preceding the formation of oceans.
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 \begin{document}
 \title{
 %   ${{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 %   ~~~~~~~}^{^{^{Submitted~to~``Astronomy~\&~Astrophysics"}}}}$\\
 {\Large{\textbf{The Theory of Bodily Tides.\\
 The Models and the Physics.}
            }}}
 \author{
 {\Large{Michael Efroimsky}}\\
 {\small{US Naval Observatory, Washington DC 20392 USA}}\\
 {\small{e-mail: me @ usno.navy.mil~}}\\
 }
 \maketitle
 \begin{abstract}
 % The Gerstenkorn-MacDonald-Kaula theory of bodily tides assumes constancy of the
 % geometric lag angle $\delta_1$, while the Singer-Mignard theory asserts constancy
 % of the time lag $\,\Delta t\,$.
 \noindent
 Any model of tides is based on a specific hypothesis
 of how the geometric lag $\delta_1$ depends on the tidal-flexure frequency
 $\chi$.
 % (For example, a circular-orbit equatorial satellite acts on the surface
 % with a frequency $\;\chi\,=\,2\,|\,\omega_p\,-\,n\,|\;$, quantities $\,
 % \omega_p\,$ and $\,n\,$ being the planet's spin rate and the satellite's
 % mean motion.)
 Some authors put $\delta_1
 \sim\chi^0$, the others set
 $\delta_1\sim\chi^1$. The actual dependence
   % of the lag on the frequency is complicated and
 determined by the planet's rheology. A particular form of this
 dependence will
 % unambiguously
 fix the form of the frequency dependence of the tidal quality factor $\,Q\,$. Since
 at present we know the shape of the function $Q(\chi)$, we can reverse our line of
 reasoning and single out the appropriate frequency-dependence of the lag. The
 dependence is different from those employed hitherto, and makes a considerable
 alteration in the theory of tides.

 The second alteration is needed to evade some difficulties inherent in Kaula's model.
 Kaula's expansion is divergent for eccentricities higher than a certain value. It is also subject to ``Goldreich's
 admonition": due to nonlinearity, the quality factors introduced for higher harmonics
 are badly defined (Goldreich 1963). To avoid these
 predicaments, we abandon the expansion and introduce
 an instantaneous tidal frequency that bears a dependence upon the moon's true
 anomaly (concept prompted by the WKB approximation in quantum mechanics). We also
 introduce an overall quality factor, the only $\,Q\,$ factor emerging in the
 model. This factor is a function of the instantaneous tidal frequency
 and therefore it, too, depends on the true anomaly.

 The third alteration will be our departure from the popular belief that $\,Q\,$
 is inversely proportional to the ``tangential lag" (the angle $\delta_1$ subtended at the planet's
 centre between the moon and the bulge). That this is correct only in the
 limit of a circular equatorial orbit was pointed out by Kaula (1968) -- see the
 caveat preceding his derivation of formula (4.5.19) for the damping rate.
 Generalisation of Kaula's formula to a finite eccentricity and inclination yields extra
 terms. These become leading on crossing the synchronous orbit, when the ``tangential
 lag" vanishes. We show, by some different method, that in the general case $\,Q\,$ is
 interconnected, in the standard way, not with the ``tangential lag", but with the
 overall lag  $\,\delta\,\equiv\,\sqrt{(\mbox{{tangential~lag}})^{\textstyle{^2}}+\,(
 \mbox{radial~lag})^{\textstyle{^2}}\,}\,$. While the ``tangential lag" $\,\delta_1\,$
 originates because the location of the bulge lags, the``radial" lag comes into
 play since the height of the bulge lags too.

 % Each of the three above alterations can
 % be employed independently from the others.

 It turns out that, for an arbitrary eccentricity and inclination, the total lag and the
 instantaneous $\chi$ and $Q$ are always interconnected via the same relations as in the
 trivial case of an equatorial circular orbit. This makes our model concise and
 self-explanatory.

 The model addresses only the land tides, and therefore is intended to be a tool for
 exploring the dynamics of the Martian satellites. It is applicable to the Earth-Moon
 system only for the aeons preceding the formation of oceans.
 \end{abstract}

 \section{The Gerstenkorn-MacDonald-Kaula and\\
          Singer-Mignard theories of bodily tides}

 If a satellite is located at a planetocentric position $\erbold$, it generates a
 tidal bulge that either advances or retards the satellite motion, depending on the
 interrelation between the planetary spin rate $\omega_p$ and the tangential part of
 satellite's velocity $\vbold$ divided by $r\equiv|\erbold|$. It is convenient to
 imagine that the bulge emerges beneath a fictitious satellite located at
 \ba
 \erbold_f\,=\,\erbold\,+\,\fbold\;\;\;,
 \label{1}
 \label{401}
 \ea
 where the position lag $\,\fbold\,$ is given by
 \ba
 {\vec{\mbox{\it\textbf{f}}}}\;=\;\Delta t\;\left(\;\omegabold_p\times
 \erbold\;-\;\vbold\;\right)\;\;\;,
 \label{2}
 \label{402}
 \ea
 $\Delta t\,$ being the time lag between the real and fictitious
 tide-generating satellites. We shall also introduce the angular lag as
 \ba
 \delta\;\equiv\;\frac{|\fbold |}{r}\;=\;\frac{\Delta t}{r}\;
 |\omegabold_p\times\erbold\;-\;\vbold|\;\;\;.
 \label{3}
 \label{403}
 \ea
 \begin{center}
 \begin{figure}
 \includegraphics[width=7.8in]{Tidal_picture_corrected.eps}
 %  \includegraphics[width=5in]{Tidal_picture.jpg}
 \caption{The position lag $\,\efbold\,$ and the angular lag $\,\delta_1\,$
 of a satellite located below the synchronous orbit. (For a moon
 located above the synchronous orbit both lags will be pointing in an opposite
 direction.) Be mindful that $\,\delta\,$ defined through (\ref{3}) does not
 coincide with the subtended angle $\,\delta_1\,$, except for circular orbits.}
 \label{Tidal_picture}
 \end{figure}
 \end{center}
 For zero eccentricity, $\,\delta\,$ is simply the absolute value of the angle
 $\,\delta_1\,$ subtended at the planet's centre between the satellite and the
 tidal bulge, as on Fig.~\ref{Tidal_picture}. For non-circular orbits, our
 $\,\delta\,$ differs from the absolute value of the subtended angle
 $\,\delta_1\,$. We have deliberately arranged for this difference, to make
 our $\,\delta\,$ embrace both the ``tangential" and the ``radial" lagging,
 i.e., the lag in the position and also that in the height of the
 bulge.\footnote{~The
 necessity of including the ``radial" part has long been
 pointed out in the literature -- for example, in the book by Murray
 and Dermott (1999), pp. 170 - 171, this issue is discussed in regard to
 dissipation inside satellites. We consider this effect in regard to the
 dissipation within the planet.}

 The imaginary satellite is merely a way of illustrating the time lag
 between the tide-raising potential and the distortion of the body. This
 concept implies no new physics, and is but a
 convenient figure of speech employed to express the fact that at each
 instance of time the dynamical tide is modelled with a static tide where
 all the time-dependent variables are shifted back by $\,\Delta t\,$, i.e.,
 (a) the moon is rotated back by $\,\vbold\,\Delta t\,$, and (b) the
 attitude of the planet is rotated back by $\,\omegabold_p\,\Delta t\,$.
 From the viewpoint of a planet-based observer, this means that a dynamical
 response to a satellite located at $\,\erbold\,$ is modelled with a static
 response to a satellite located at $\,\erbold_f\,\equiv\,\erbold\,-\,
 \Delta t\,(\vbold\,-\,\omegabold_p\times\erbold)\,$.

 This delayed response implies a lag in the location of the bulge, and a
 lag in its height. As a result, the tidal dissipation consists of two
 inputs -- the ``tangential" and the ``radial" one. Our
 $\,\delta\,$ defined by (\ref{3}) is equal to $\,\sqrt{(\mbox{{tangential
 ~lag}})^{\textstyle{^2}}\,+\,(\mbox{radial~lag})^{\textstyle{^2}}\,}\,$.
 ~This, together with the interconnection between our $\,\delta\,$ and the
 overall quality factor, ensures that our model indeed includes both lags
 -- that in the position and that in the height of the tide.
 Leaving details for Section 3 below, we would emphasise already at this
 point that the well-known relation $\;1/Q\,=\,\tan 2\delta\;$ interconnects
 the quality factor not with the ``tangential lag" $\,\delta_1\,$ but exactly
 with the total lag that includes both the ``tangential" and ``radial" part.


 The earliest efforts aimed at modeling bodily tides were undertaken in the
 end of the XIX$^{th}$ century  by Darwin (1879, 1880). Furthering of this
 line of research by Gerstenkorn (1955), MacDonald (1964) and Kaula (1964)
 gave birth to what we now may call `the old theory of tides'. In their
 works, Gerstenkorn, MacDonald, and Kaula\footnote{~To be more exact,
 Kaula's starting point was an assertion of constancy of the tidal quality
 factor.} assumed the subtended angle to be a fixed constant. With the
 latter assumption regarded as a critical deficiency, their theory soon
 fell into disuse in favour of the approach offered by Singer and Mignard.
 Singer (1968) suggested that the subtended angle should be proportional
 to the principal frequency $\chi$ of the tide. This was equivalent to
 setting $\Delta t$ constant in (\ref{2} - \ref{3}). Singer applied this
 new theory to the Moon and to Phobos and Deimos. Sadly, Singer's
 interesting work is seldom cited. A detailed mathematical development of
 Singer's idea can be found in Mignard (1979, 1980) who, actually,
 completely avoided using the lag angle and operated only with the
 position and time lags. Later, Singer's assumption of a constant
 $\,\Delta t\,$ was employed also by Touma \& Wisdom (1994) and Peale \&
 Lee (2000).

 \section{The tidal frequency $\chi$ and the quality factor $Q$}

 \subsection{How to define the tidal frequency?}

 Singer (1968) approximated the principal frequency of tidal flexure,
 $\chi$, with the expression $2\,|n-\omega_p|\,$, where $\,n\,$
 and $\omega_p\equiv|\,\omegabold_p|$ stand for the satellite's mean
 motion and the planet's spin rate. Thus he restricted the applicability
 realm of his theory to the case of low inclinations and eccentricities.
 While it is clear that, for an equatorial circular orbit, the satellite
 velocity relative to the surface is
 \ba
 |\,\omegabold_p\times\erbold~-~\vbold\,|~=\;r\;|\,\omega_p\,-\,n\,|\;\;\;,
 \label{4}
 \label{404}
 \ea
 the tidal frequency is
 \ba
 \chi\;=\;2\;|\,\omega_p\,-\,n\,| \;\;\;\;,
 \label{5}
 \label{405}
 \ea
 and the angular lag is
 \ba
 \delta\;\equiv\;\frac{\Delta t}{r}\;|\,\omegabold_p\times\erbold\;-\;
 \vbold\,|\;=\;\frac{\Delta t}{2}\;\chi\;\;\;,
 \label{6}
 \label{406}
 \ea
 the question remains how can this machinery be extended to the generic case when
 neither $\,\inc\,$ nor $\,e\,$ is small. In particular, how can one define
 the tidal frequency (or, perhaps, the principal tidal frequency and the
 higher frequencies) when the satellite may appear over a
 different point on the surface after each revolution, thus making the
 entire notion of flexure cycle hard to define?

 \subsection{Goldreich's admonition:\\ a general difficulty stemming from nonlinearity}

 Generalisation of the above construction was offered already by Darwin
 (1908) and, later, by Jeffreys (1961) and Kaula (1964). These works'
 starting point was that each elementary volume of the planet is subject
 to a tide-raising potential, which in general is not periodic but can be
 expanded into a sum of periodic terms. They then employed the linear
 approximation introduced by Love, according to which the
 tidal perturbations of the potential yield linear response of the shape
 and linear variations of the stress. In extension of the linearity
 approximation, these authors also implied that the overall dissipation
 inside the planet may be represented as a sum of attenuation rates
 corresponding to each periodic disturbance:
 \ba
 % \nonumber
 \langle\,\dot{E}\;\rangle\;=\;\sum_{i}\;\langle\,
 \dot{E}(\chi_{\textstyle{_i}})\;\rangle
 \label{407}
 \ea
 where, at each frequency $\,\chi_i\,$,
 \ba
 \langle\,\dot{E}(\chi_{\textstyle{_i}})~\rangle~=~-~2~\chi_{\textstyle{_i}}~
 \frac{\,\langle\,E(\chi_{\textstyle{_i}})~\rangle\,}{Q(\chi_{\textstyle{_i}})
 }~=\,\;-\;\chi_{\textstyle{_i}}\;\frac{\,E_{_{peak}}(\chi_{\textstyle{_i}})
 \,}{Q(\chi_{\textstyle{_i}})}\;\;\;,
 \label{408}
 \ea
 $\langle\,.\,.\,.\,\rangle~$ standing for averaging over flexure cycle,
 $\,E(\chi_{\textstyle{_i}})\,$ denoting the energy of deformation at
 the frequency $\,\chi_{\textstyle{_i}}\,$, and $Q(\chi_{\textstyle{_i}})\,$
 being the quality factor of the material at this frequency. Introduced
 empirically as a means to figleaf our lack of knowledge of the attenuation
 process in its full complexity, the notion of $\,Q\,$ has proven to be practical
 due to its smooth and universal dependence upon the frequency and temperature.
 At the same time, this empirical treatment has its predicaments and
 limitations. Its major inborn defect was brought to light by Peter
 Goldreich who pointed out that the attenuation rate at a particular
 frequency depends not only upon the appropriate Fourier component of the
 stress, but also upon the {\emph{overall}} stress. This happens because
 for real minerals each quality factor $\,Q(\chi_{\textstyle{_i}})\,$
 bears dependence not only on the frequency $\,\chi_{\textstyle{_i}}\,$,
 but also on the $\,\chi_{\textstyle{_i}}\,$ component of the stress and,
 most importantly, also on the {\emph{overall}} stress. This, often-neglected,
 manifestation of nonlinearity may be tolerated only when the
 amplitudes of different harmonics of stress are comparable. However, when
 the amplitude of the principal mode is orders of magnitude higher than
 that of the harmonics (tides being the case), then the principal mode
 will, through this nonlinearity, make questionable our entire ability to
 decompose the overall attenuation into a sum over frequencies. Stated
 differently, the quality factors corresponding to the weak harmonics will
 no longer be well defined physical parameters.

 Here follows a quotation from Goldreich (1963):\\
 {\emph{``... Darwin and Jeffreys both wrote the tide-raising potential
 as the sum of periodic potentials. They then proceeded to consider the
 response of the planet to each of the potentials separately. At first
 glance this might seem proper since the tidal strains are very small
 and should add linearly. The stumbling block in this procedure, however,
is the amplitude dependence of the specific dissipation function. In
the case of the Earth, it has been shown by direct measurement that
$\,Q\,$ varies by an order of magnitude if we compare the tide of
frequency  $\,2\omega - 2n \,$ with the tides of frequencies $\,
2\omega - n \,$,  $\, 2\omega - 3n \,$, and $\,\frac{\textstyle
3}{\textstyle 2}n\,$. This is because these latter tides have
amplitudes which are smaller than the principle tide (of frequency
$\, 2\omega - 2n \,$ ) by a factor of eccentricity or about 0.05. It
may still appear that we can allow for this amplitude dependence of
Q merely by adopting an amplitude dependence for the phase lags of
the different tides. Unfortunately, this is really not sufficient
since a tide of small amplitude will have a phase lag which
increases when its peak is reinforcing the peak of the tide of the
major amplitude. This non-linear behaviour cannot be treated in
detail since very little is known about the response of the planets
to tidal forces, except for the Earth."}}

 On these grounds, Goldreich concluded the paragraph with an important
 warning that we {\emph{``use the language of linear tidal theory, but we
 must keep in mind that our numbers are really only parametric fits to a
 non-linear problem."}}

 In order to mark the line beyond which this caveat cannot be ignored, let
 us first of all recall that the linear approximation remains applicable
 insofar as the strains do not approach the nonlinearity threshold, which
 for most minerals is of order $\,10^{-6}\,$. On approach to that
 threshold, the quality factors may become dependent upon the strain
 magnitude. In other words, in an attempt to extend the expansion
 (\ref{407} - \ref{408}) to the nonlinear case, we shall have to introduce,
 instead of $\,Q(\chi_{\textstyle{_i}})\,$, some new functions
 $\,Q(\,\chi_{\textstyle{_i}}\,,\;E_{_{peak}}(\chi_{\textstyle{_i}})\,,
 \;E_{_{overall}}\,)\;$. (Another complication is that in the nonlinear
 regime new frequencies will be generated, but we shall not go there.)
 Now consider a superposition of two forcing stresses -- one at
 the frequency $\,\chi_{\textstyle{_1}}\,$ and another at
 $\,\chi_{\textstyle{_2}}\,$.
 Let the amplitude $\;E_{_{peak}}(\chi_{\textstyle{_1}})\;$ be close or
 above the nonlinearity threshold, and
 $\;E_{_{peak}}(\chi_{\textstyle{_2}})\;$ be by an order or two
 of magnitude smaller than $\;E_{_{peak}}(\chi_{\textstyle{_1}})\;$.
 To adapt the linear machinery (\ref{407} - \ref{408}) to the
 nonlinear situation, we have to write it as
 \ba
 \langle\,\dot{E}~\rangle~=~\langle\,\dot{E}_1~\rangle~+~\langle\,\dot{E}_2
 ~\rangle~=~-~\chi_{\textstyle{_1}}~\frac{~E_{_{peak}}(\chi_{\textstyle{_1}
 })~}{Q(\,\chi_{\textstyle{_1}}\,,~E_{_{peak}}(\chi_{\textstyle{_1}})\,)}~
 -~\chi_{\textstyle{_2}}~\frac{\,E_{_{peak}}(\chi_{\textstyle{_2}})\,}{Q(
 \chi_{\textstyle{_2}}\,,\;E_{_{peak}}(\chi_{\textstyle{_1}})\,,\;
 E_{_{peak}}(\chi_{\textstyle{_2}})\,)}\;\;\;,\;\;\;\;\;
 \label{409}
 \ea
 the second quality factor bearing a dependence not only upon the frequency
 $\,\chi{\textstyle{_2}}\,$ and the appropriate magnitude $\;E_{_{peak}}(
 \chi_{\textstyle{_2}})\;$, but also upon the magnitude of the
 {\emph{first}} mode, $\;E_{_{peak}}(\chi_{\textstyle{_1}})\;$, -- this
 happens because it is the first mode which makes a leading contribution
 into the overall stress. Even if (\ref{409}) can be validated as an
 extension of (\ref{407} - \ref{408}) to nonlinear regimes, we should
 remember that the second term in (\ref{409}) is much smaller than the
 first one (because we agreed that $\;E_{_{peak}}(\chi_{\textstyle{_2}})\,
 \ll\,E_{_{peak}}(\chi_{\textstyle{_1}})\;$). This results in two quandaries.
 The first one (not mentioned by Goldreich) is that a nonlinearity-caused
 non-smooth behaviour of $\;Q(\,\chi_{\textstyle{_1}}\,,\;E_{_{peak}}(
 \chi_{\textstyle{_1}})\,)\;$ will cause variations of the first term in
 (\ref{409}), which may exceed or be comparable to the entire second term.
 The second one (mentioned in the afore quoted passage from Goldreich) is
 the phenomenon of nonlinear superposition, i.e., the fact that the
 smaller-amplitude tidal harmonic has a higher dissipation rate (and,
 therefore, a larger phase lag) whenever the peak of this harmonic is
 reinforcing the peak of the principal mode. Under all these circumstances,
 fitting experimental data to (\ref{409}) will become a risky
 business. Specifically, it will become impossible to reliably measure the
 frequency dependence of the second quality factor; therefore the entire
 notion of the quality factor will, in regard to the second frequency, become
 badly defined.

 The admonition by Goldreich had been ignored until an alarm sounded. This happened when
 the JPL Lunar-ranging team applied the linear approach to determining the
 frequency-dependence of the Lunar quality factor, $\,Q(\chi)\,$. They obtained (or, as
 Peter Goldreich rightly said, {\emph{fitted}} their data to) the dependency $\,Q(\chi)\,
 \sim\,\chi^{\alpha}\,$ (Williams et al. 2001). The value of the exponential for the Moon
 turned out to be {\emph{negative}}: $\,\alpha\,=\,-\,0.07\;$, a result firmly tabooed by
 the condensed-matter physics for this range of frequencies (Karato 2007).

 One possible approach to explaining this result may be the following. Let us begin with
 a very crude estimate for the tidal strains. For the Moon, the tidal displacements are
 of order $\,0.1m\,$ (zero to peak).
  % , and $\,0.25m\,$, for the Earth.
 The most rough estimate for the
 strain can be obtained through dividing the displacement by the radius of the body. While
  %  both for the Earth and the Moon
 this ratio is almost twenty times less than the
 $\,10^{-6}\,$ nonlinearity threshold, we should keep in mind that in reality the
 distribution of the tidal strain is a steep function of the radius, with the strain
 getting its maximum near the centre of the body, as can be seen from equations (48.17)
 in Sokolnikoff (1956). The values of strain can vary in magnitude, over the radius, by
 about a factor of five.\footnote{~Accordingly, the tidal-energy radial distribution,
 too, is strongly inhomogeneous, with a maximum in the planet's centre -- see Fig. 3 in
 Peale \& Cassen (1978).} Another order of magnitude, at least, will come from the fact
 that the Moon is inhomogeneous and that its warmer layers are far more elastic than its
 rigid surface. As a result, deep in the Lunar interior the tidal strain will exceed the
 afore mentioned nonlinearity threshold. Hence the uncertainties in determination of
 $\,Q(\chi)\,$ by the Lunar-ranging team.

 We shall not dwell on this topic in quantitative detail, leaving it for a future work.
 % (Karato \& Efroimsky 2007, in preparation).
 Our only goal here has been to draw the readers' attention to the existing difficulty
 stemming from the shortcomings of the extension of (\ref{407} - \ref{408}) to nonlinear
 regimes.

 \subsection{Specific problems with the Kaula expansion}

 The effect of lags in the tidal bulge can be described by expanding the potential of
 the tidally distorted planet in zonal harmonics, each of which is then transformed
 into a sum of spherical ones, with the further intention of expressing each term via
 the orbital elements and mean anomalies of the tide-raising and tide-disturbed
 satellites. A comprehensive development in this direction carried out by Kaula (1964)
 was later employed and quoted in numerous papers and books -- see, for example,
 section VI in Lambeck (1980). However, neither Kaula nor Lambeck adumbrated the
 applicability realm of this approach, nor did they dwell on its drawbacks. Below we
 shall mention several limitations, which restrict the usability of this method.

 First of all, this theory is subject to Goldreich's admonition. Indeed, in
 the Kaula expansion each frequency has its own phase lag, which must be
 interconnected with the quality factor appropriate to this frequency. For
 the reason described in the previous subsection, this inconvenience may be
 circumvented only in the linear case. In the nonlinear situations (like,
 say, dissipation inside Phobos and, most likely, inside the Moon), this
 becomes a problem, because the quality factors of higher harmonics become
 badly defined when the magnitudes of these harmonics are much lower than
 that of the principal mode.\footnote{~Goldreich (1966) employed the Kaula
 expansion to explore the Lunar-orbit history. To circumvent his own
 admonition, Goldreich had to assume all the lags to be the same, and to be
 constant. This, however, was long before the actual low-frequency
 behaviour of $\,Q\,$ was established.}

 The second problem with Kaula's series is that
 % it will badly converge for near-polar orbits.\footnote{~To trace the origin of
 % the divergence at $\,\inc\rightarrow\pi/2$, consider the expansion of the
 % tidal potential over the spherical harmonics -- formula (1) in Kaula (1964)
 % or (6.3.1) in Lambeck (1980). Each term of that sum contains an inclination
 % function $\,F_{lmp}(\inc)\,$ whose general expression can be found, for
 % example, in Kaula (1966). Formula (3.62) from that book demonstrates that the
 % subseries corresponding to $\,s=0\,$ diverges for $\,\inc\,=\,\pi/2\,$.}
 % Not being an issue for low-inclination satellites, this becomes an obstacle
 % when one attempts to apply the Kaula theory to studies of high-inclination
 % captures. The theory becomes of no use when one has to check if a satellite
 % captured at a retrograde orbit can evolve into a prograde one by making a
 % transition through a polar orbit. (It is unclear if such events may really
 % have happened, but it is for sure that a check of this hypothesis by the
 % Kaula theory is impossible.)
 %
 % A much worse complication emerges when one tries to apply the Kaula expansion
 % to high-eccentricity orbits. It is easy to see that in the limit of $\;e\,
 % \rightarrow\,1\;$ the expansion suffers the same vicissitudes\footnote{~The
 % Kaula expansion for the potential contains the eccentricity functions
 % $\,G_{lpq}(e)\,$, some of which diverge at $\;e\,\rightarrow\,1\;$. See table
 % 6.1 in Lambeck (1980) or the general formula in Kaula (1966.)} as for $\;\inc\,
 % \rightarrow\,\pi/2\;$. However, this is not the end of the story. The situation
 % is much worse because
 the derivation of the spherical-harmonics expansion --
 formula (1) in Kaula (1964) or formula (6.3.1) in Lambeck (1980) -- contains a
 transition from the true anomaly $\,\nu\,$ to the mean anomaly $\,M\,$,
 transition based on the trigonometric expansion of $\,\nu-M\,$ as a Fourier
 sine series in $\,M\,$. This expansion turns out to be convergent only for
 $\;e\,<\,0.6627434\,$. (Murray and Dermott 2000) As a result, the entire
 spherical-harmonics machinery remains legitimate only below this value of
 $\;e\,$.
% When developing his theory of tides, Kaula did not know of this
% booby-trap, because an explanation of it appeared only in Hagihara (1970).
 One negative consequence of this mishap is that the theory of Kaula (1964)
 cannot be applied to check the possibility of tidal capture hypothesised by
 Singer (1965) -- exploration of this mechanism makes it necessary to study
 tidal effects at initial eccentricities close to unity.

 \subsection{Quest for a new approach}

 In our opinion, the right way out of the afore described predicament lies
 not along the paths of mathematical rigour but through a careful choice
 of the particular physical meaning wherewith we want to endow the term
 ``tidal frequency."

 In the theory of tides, the notion of principal frequency is employed to
 make use of the attenuation theory borrowed from acoustics. There the
 basic fact is that, {\emph{for a sample of material}} (an important
 clause to reject later), the time-averaged rate of energy losses produced
 by small-amplitude alternating stresses can be expanded over the
 frequencies $\;\chi_{\textstyle{_i}}\;$ involved:
 \ba
 % \nonumber
 \langle\,\dot{E}\;\rangle\;=\;\sum_{i}\;\langle\,\dot{E}(\chi_{
 \textstyle{_i}})\;\rangle\;\;\;,
 \label{sum}
 \label{410}
 \ea
 where
 \ba
 \dot{E}(\chi_{\textstyle{_i}})\;=\;-\;\chi_{\textstyle{_i}}\;\frac{\,
 E_{_{peak}}(\chi_{\textstyle{_i}})\,}{Q(\chi_{\textstyle{_i}})}\;\;\;.
 \label{term}
 \label{411}
 \ea
 As the average dissipation rate $\,\langle\,
 \dot{E}\,\rangle\,$ is, by definition, the energy damped over a cycle,
 divided by the period $\,2\pi/\chi_{\textstyle{_i}}\,$, then one can
 shape the expression for the quality factor into the form:
 \ba
 \Delta E_{_{cycle}}(\chi_{\textstyle{_i}})\;=\;-\;2\;\pi\;\frac{E_{_{peak}}(\chi_i)
 }{Q(\chi_i)}
 \label{8}
 \label{412}
 \ea
 The advantage of definition (\ref{412}) is that it operates with only one
 cycle of flexure. (Even a half-cycle would be sufficient.) In no way does
 this definition refer to the afore emphasised clause about a certain
 sample of material. Rather, (\ref{412}) may be applied to a wave train and
 even to a solitary wave travelling through a medium; in this situation no
 fragment of the medium experiences a repeated pattern of strain. The
 solitary wave travels through this fragment just once. Nevertheless,
 (\ref{412}) still enables us to talk about the attenuation rate in regard to
 this wave. Let us apply this reasoning to tides and, specifically, to the
 situation when the satellite appears, after each revolution, over a
 different point of the surface. It is now {\emph{unimportant}} that the
 stress in a particular fragment of the Earth material lacks periodicity,
 and that we cannot clearly define a flexure frequency for a particular
 fragment. As agreed, we shall apply (\ref{412}) not to a fixed fragment,
 but to the travelling tidal bulge considered as a solitary half-wave of
 a frequency equal to the velocity of the satellite relative to the ground
 beneath it, divided by $\,r\,$:
 \ba
 \sigma~=~\,\frac{1}{r}~\,|\,\omegabold_p\times\erbold\;-\;\vbold\,|\;\;\;,
 \label{9}
 \label{413}
 \ea
 % the angular brackets symbolising averaging over one cycle of the
 % satellite orbit. (There is no need to average over a revolution of the
 % planet, because it is assumed to be uniform.)
 Had the satellite's pull created only one elevation on the planet's
 surface, we would employ (\ref{term}) to write $\,\dot{E}=-\sigma E_{
 peak}/Q(\sigma)\,$ or, equivalently, $\,\Delta E_{_{cycle}}(\sigma)=-
 2\pi E_{_{peak}}/Q(\sigma)\,$. However, since the moon's gravity
 produces two uplifts -- one on the facing side and another on the opposite
 side of the planet -- then the actual dissipated energy will be doubled:
 $\,\dot{E}=-2\sigma E_{peak}/Q(\sigma)\,$ or, the same, $\,
 \Delta E_{_{cycle}}(2\sigma)=-2\pi E_{_{peak}}/Q(\sigma)\,$.
 This can be shaped into the form similar to (\ref{term})
 \ba
 \dot{E}\,=\,-\,\chi\;\frac{E_{_{peak}}}{Q(\chi)}\;\;\;
 \label{10}
 \label{414}
 \ea
 or into the form like (\ref{412}):
 \ba
 \Delta
 E_{_{cycle}}(\chi)\;=\;-\;2\;\pi\;\frac{E_{_{peak}}}{Q(\chi)}
 \label{11}
 \label{415}
 \ea
 where
 \ba
 \chi~=~2~\sigma~=~\,\,\frac{2}{r}~\,|\,\omegabold_p\times\erbold~-~\vbold
 \,|\;\;\;,
 \label{12}
 \label{416}
 \ea
 is effectively playing the role of the tidal frequency. This is its
 {\textbf{definition}}: in our
 %  heuristic
 model, the tidal frequency is the quantity $\,\chi\,$, insertion whereof in
 (\ref{term}) or (\ref{412}) gives the correct dissipation rate. It remains to be
 said that in (\ref{415} - \ref{416}) we slightly cheated by substituting
 $\,Q(\sigma)\,=\,Q(\chi/2)\,$ with $\,Q(\chi)\,$. Formally, this trick can be
 justified as a convenient redefinition of the $\,Q\,$ factor.\footnote{~In
 reality, this formal justification is, of course, redundant, because the
 dependence $\,Q(\chi)\,$ is very slow, and noticeable changes of $\,Q\,$ take
 place over frequencies ranges that span, at least, an order of magnitude. In
 realistic seismological measurements, an increase of the frequency by some
 factor of two yields a variation in the value of $\,Q\,$, that will barely
 make it over the error bar.}

 The so-defined frequency can be expressed via the planet's spin rate $\,\omega_p\,$,
 the true anomaly $\,\nu\,$ of the satellite, and the orbital elements. Denoting the
 radial component of the moon's velocity with $v_r$, and its planetocentric distance
 with $\,r\,$, we write (see the Appendix for details):
 \ba
 |\,\omegabold_p\times\erbold~-~\vbold|\,=\,r\,\sqrt{(\dot{\nu}\,-\,
 \omega_p\,\cos\inc)^2\,+\,\omega_p^2\,\sin^2\inc\;\cos^2(\omega\,+\,\nu)\,
 +\,v_r^2\,r^{-2}\;}\;\;\;,
 \label{13}
 \label{417}
 \ea
 whence it can be derived (once again, see the Appendix) that
 \ba
 \nonumber
 \chi\,=\,2\;\sqrt{(\dot{\nu}\,-\,\omega_p\,\cos\inc)^2\,+\,\omega_p^2\,
 \sin^2\inc\;\cos^2(\omega\,+\,\nu)\,+\,v_r^2\,r^{-2}\;}\;\;\;\;
 \ea
 \ba
 =\;2\;\;\sqrt{\;\left[\,n\;\frac{\left(1+\,e\,\cos\nu
 \right)^2}{\left(1-e^2\right)^{3/2}}\,-\,\omega_p\,\cos\inc
 \right]^{{{\,2}}}+\,\omega_p^2\,\sin^2\inc\;
 \cos^2(\omega+\nu)\,+\,n^2\,e^2\,\frac{\left(1+
 e\,\cos\nu\right)^2}{\left(1-e^2\right)^{3}}\;\sin^2\nu\;\,}\;\;.\;\;
 % 2\;\sqrt{(n\,-\,\omega_p\,\cos\inc)^2\,+\,\frac{\textstyle 1}{
 % \textstyle 2}\;\,\omega_p^2\;\sin^2\inc\;+\;\frac{5}{2}\,n^2\,e^2\,~}
 % \;+\;O(\sin^4\inc)\;+\;O(e^2\,\sin^2\inc)\;+\;O(e^4)~~.\;\;\;\;
 \label{14}
 \label{418}
 \ea
 Here $\,n\,,\;e\,,\;\inc\,,\;\omega\,$ stand for the moon's mean motion,
 eccentricity, inclination, and the argument of the pericentre, while
 $\,\omega_p\,$ is the rotation rate of the planet.\\
 ~\\
 Let us draw conclusions:\\
 \\
 {\underline{\bf{1.}}} ~~The simple approximation (\ref{4} - \ref{6}) can
 be used only at a relatively late stage of the satellite's life -- when
 its orbit is well circularised and damped down toward the
 equator.\\
 \\
 {\underline{\bf{2.}}} ~~In the generic case of inclined and eccentric
 orbits, (\ref{4} - \ref{5}) must be substituted with (\ref{417} -
 \ref{418}), while the expression (\ref{6}) for the angular lag will
 preserve its form:
 \ba
 \delta\;\equiv\;\Delta t\,\;\frac{1}{r}\;\,|\,\omegabold_p\times\erbold
 \;-\;\vbold\,|\;=\;\frac{\Delta t}{2}\;\chi\;\;\;,
 \label{15}
 \label{419}
 \ea
 $\chi$ being now given by (\ref{418}).
 % So from now on, $\,\delta\,$ will stand for the {\emph{average}} lag.
 The generalised formalism (\ref{418} - \ref{419}) will render the satellite
 orbit evolution right after the capture, when
 the eccentricity (and, possibly, the inclination) is high.\\
 \\
 {\underline{\bf{3.}}} ~~~In our model, the above definition for the angular
 lag $\,\delta\,$ has been {\emph{deliberately}} chosen to differ, for
 non-circular orbits, from the angle $\,\delta_1\,$ subtended at the planet's
 centre between the satellite and the tidal bulge, as on Fig. 1. Had we not
 arranged for this difference, our model would be taking into account only the
 ``tangential" part of the tidal dissipation (one associated with the
 tangential motion of the satellite).\footnote{~That such a truncated model
 would be not only incomplete but also violently unphysical
 can be seen from the fact that, on each passage of a satellite through a
 spin-orbit resonance, the truncated model would cause the torque to
 formally approach infinity and then to suddenly reverse to the infinity
 of an opposite sign. Indeed, since $\,\delta\,$ is proportional to
 $\chi\,$ to a negative power, then in the absence of the ``radial"
 dissipation $\,\chi\,$ would be passing through zero on each crossing
 of the synchronous orbit. To spare the model from such vicissitudes,
 we must define the lag as in (\ref{419}).} Our definition (\ref{419})
 ensures that both the ``tangential" and ``radial" parts of the tidal
 dissipation contribute to the lag $\,\delta\,$ and, thereby, to the
 overall quality factor $\,Q\,$ interconnected with $\,\delta\,$ through
 (\ref{420}). So our model contains a radial-velocity-squared term
 in (\ref{417} - \ref{418}), a term that prevents $\,\chi\,$ from passing
 through zero at resonance crossings.\\
 ~\\
 {\underline{\bf{4.}}} ~~~Our model departs from the traditional approach
 on one more crucial point. Taking into consideration the Goldreich admonition,
 we refrain from decomposing the overall tide into modes and from attributing
 a separate quality factor to each of these. Instead, we introduce only one
 tidal frequency that bears an explicit dependence upon the satellite's true
 anomaly, and corresponds to an appropriate overall quality factor.\\

 The concept of an instantaneous frequency as a function of the true anomaly
 is an evident analogue to the evolving phase in the WKB approximation in
 quantum mechanics. This mathematical analogy, though, should not be taken
 too literally, because in quantum mechanics this method is used to solve a
 linear Schr{\"o}dinger equation, while here we employ it in a highly
 nonlinear and dissipative context.

 % The physical meaning of our model is that, on the one hand, we permit
 % both $\,\delta\,$ and $\,\Delta t\,$ to depend upon the tidal frequency,
 % while on the other hand we assume both these quantities to remain
% constant within one cycle of the satellite's revolution about the planet.
 % In other words, we keep and study the ``slow" time-dependence of $\,
 % \delta\,$ and $\,\Delta t\,$ (the dependence caused by the evolution of
 % the tidal frequency through aeons), but average out their ``fast"
 % time-dependence.

 ~\\

 \section{The quality factor $Q$ and the geometric lag angle $\delta $.}

~\\

 ${\left.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 ~~~~~~~~~~~\right.}^{-~Mr.~Wallace,~are~you~OK?}$~~\\
 ${\left.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 ~~~~~~~~~~~~~~~\,\right.}^{-~No,~I~am~very~far~from~being~OK...}$\\
 ${\left.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\right.}^{``Pulp~Fiction"}$\\
% \\



 During tidal flexure, the energy attenuation through friction is, as ever,
 accompanied by a phase shift between the action and the response. The tidal
 quality factor is interconnected with the phase lag $\,\epsilon$ and
 the angular lag $\delta\,$ via
 % \pagebreak
 \ba
 Q^{-1}\;=\;\tan \epsilon\;=\;\tan 2\delta
 \label{16}
 \label{420}
 \ea
 or, for small lag angles,
  \ba
 Q^{-1}\;\approx\;\epsilon\;=\;2\;\delta\;\;\;.~~
 \label{17}
 \label{421}
 \ea
 The doubling of the lag is a nontrivial issue. Many authors erroneously state that
 $\,Q^{-1}\,$ is equal simply to the tangent of the lag, with the factor of two
 omitted.\footnote{~Rainey \& Aharonson (2006) assume that $\,Q^{-1}\,$ is equal to
 the tangent of the geometric lag. As a result, they arrive to a value of $\,Q\,$
 that is about twice larger than those obtained by the other teams, for example by
 Bills et al (2005). As pointed out by Valery Lainey, in the work by Bills et al
 (2005) one letter, $\,\gamma\,$, is used to denote two different angles. In their
 paper, prior to equation (24), $\,\gamma\,$ signifies the {\emph{geometric}} lag
 (in our notations, $\,\delta_1\,$). Further, in their equations (24) and (25), Bills
 et al employ the notation $\,\gamma\,$ to denote the {\emph{phase}} lag (in our
 notations, $\,\epsilon\,$, which happens to be equal to $\,2\,\delta_1\,$). With
 this crucial caveat, Bills' equation $\,Q\,=\,1/\tan\gamma\,$ is correct. This
 mess in notations has not prevented Bills et al (2005) from arriving to a
 reasonable value of the Martian quality factor, $\,85.58\pm0.37\,$. (A more recent
 study by Lainey et al (2006) has given a comparable value of $\,79.91 \pm 0.69\,$.)}
 In Appendix 2.1, we offer a simple illustrative calculation, which explains whence this
 factor of two stems.

 As an aside, we would mention that Zschau (1978) contested the entire possibility
 of expressing the quality factor via only one lag. His point stemmed from the
 fact that attenuation of land tides, in distinction from earthquakes, takes place
 both due to rigidity and self-gravity of the body. In Appendix A.3 below we
 explain why a typical terrestrial planet or small body will be exempt from Zschau's
 objection, so that formulae (\ref{420} - \ref{421}) will hold for them.

 The popular oversight interconnects the quality factor with the ``tangential lag,"
 i.e., with the geometric angle $\,\delta_1\,$ depicted on Fig. 1. In our terms, this
 would read as $\,Q^{-1}\,=\,\tan 2\delta_1\,$ or $\,Q^{-1}\,\approx\,2\,\delta_1\,$.
 While correct in the limit of a circular equatorial orbit (see Appendix 2.1 below),
 in the general case this relation is plainly wrong. Nonetheless it was employed by
 many. Kaula (1968) explicitly acknowledged that his proof of this relation is valid
 only in the said limit.\footnote{~On page 201 of his book, Kaula (1968) calculates
 the dependence of $\,\dot{E}\,$ upon the tangential lag (which he calls $\,\delta\,$,
 and which we term $\,\delta_1\,$ -- mind the difference in notations). Kaula rightly
 prefaces his derivation with the words ``If the eccentricity and inclination of the
 moon's orbit are neglected", because his formula (4.5.19) is correct in this
 approximation only. Generalisation of Kaula's formula yields extra terms dependent
 upon the inclination and eccentricity. These terms become leading on approach to or
 crossing of the synchronous orbit.} To treat the case of an elongated inclined orbit,
 one can amend Kaula's formula with the terms he dropped. In Appendices A.2.2 - A.2.3
 below we shall offer an easier development, from which it will be seen that,
 for an eccentric inclined orbit, the ``tangential lag" $\,\delta_1\,$ should be
 substituted with the total lag $\,\delta\,=\,\sqrt{(\mbox{tangential~lag})^2\,+\,
 (\mbox{radial~lag})^2}\,$ defined through (\ref{403}).
 This will explain why in our formulae (\ref{420} - \ref{421}) we use the total lag $\,
 \delta\,$, not the subtended angle $\,\delta_1\,$.

 Taken the above elucidation, formulae (\ref{420} - \ref{421}) look reasonable: the higher
 the quality factor, the lower the damping rate and, accordingly, the smaller the lag. What
 look very far from being OK are the frequency dependencies ensuing from the assertions of
 $\,\delta\,$ being either constant or linear in frequency: the Gerstenkorn-MacDonald-Kaula
 theory implies that $\,Q\,\sim\,\chi^{0}\,$, while the Singer-Mignard theory yields
 $\,Q\,\sim\,\chi^{-1}\,$, neither option being in agreement with the geophysical data.

 \section{Dissipation in the mantle.}

 \subsection{Generalities}

 Back in the 60s and 70s of the past century, when the science of low-frequency
 seismological measurements was yet under development, it was widely thought that
 at long time scales the quality factor of the mantle is proportional to the inverse
 frequency.
 This fallacy proliferated into planetary astronomy where it was received most
 warmly, because the law $\,Q\,\sim\,1/\chi\,$ turned out to be the only model for
 which the linear decomposition of the tide gives a set of bulges displaced from
 the direction to the satellite by the same angle. Any other frequency dependence
 $\,Q(\chi)\,$ entails superposition of bulges corresponding to the separate
 frequencies, each bulge being displaced by its own angle. This is the reason why
 the scaling law $\,Q\,\sim\,1/\chi\,$, long disproved and abandoned in geophysics
 (at least, for the frequency band of our concern), still remains a pet model in
 celestial mechanics of the Solar system.

 Over the past twenty years, a considerable progress has been achieved in the
 low-frequency seismological measurements, both in the lab and in the field.
 Due to an impressive collective effort undertaken by
 several teams, it is now a firmly established fact that {\it{for
 frequencies down to about $\sim 1$ yr$^{-1}$ the quality factor of the mantle
 is proportional to the frequency to the power of a {\textbf{positive}} fraction}}
 $\,\alpha\,$. This dependence holds for all rocks within a remarkably broad band of
 frequencies: from several MHz down to about $\;1\;$yr$^{-1}\,$.

 At timescales longer than $\,1$ yr, all the way to the Maxwell time (about $\,100
 $~yr), attenuation in the mantle is defined by viscosity, so that the
 quality factor is, for all minerals, well approximated with $\,{\eta}\chi/M\,$,
 where $\eta$ and $M$ are the sheer viscosity and the sheer elastic modulus of the
 mineral. Although the values of both the viscosity coefficients and elastic moduli
 greatly vary for different minerals and are sensitive to the temperature, the
 overall quality factor of the mantle still scales linear in frequency.

 There still is no consensus in the seismological community in regard to the time
 scales exceeding the Maxwell time. One viewpoint (incompatible with the Maxwell
 model) is that the linear law $\,Q\,\sim\,\chi\,$ extends all the way down to the
 zero-frequency limit (Karato 2007). An alternative point of view (prompted by the
 Maxwell model) is that at scales longer than the Maxwell time we return to the
 inverse-frequency law $\,Q\,\sim\,1/\chi\,$.\\
 ~\\
 All in all, we have:\\
 \ba
 \mbox{For}\;\;\;\;\;\;10^7\;\;\mbox{Hz}\;\;>\,\;\chi\;\;>\;\;1\;yr^{-1}\;:\;\;\;Q\,\sim\,
 \chi^{\alpha}\;,\,\;\mbox{with}\;\,\alpha\,=\,0.2\,-\,0.4\;\;\;(\,0.2\;
 \mbox{for partial melts})\;.\;\;\;\,
 \label{one}
 \label{422}
 \ea
 \ba
 \mbox{For}\;\;\;\;\;~~\,1\,\;yr^{-1}\,>\,\chi\,>\,10^{-2}\;yr^{-1}\;:\;\;\;\;\;Q\,
 \sim\,\chi\;\;\;.\;\;\;\;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\,~~~~~~~~~~~~~
 ~~~~~~~~~
 \label{two}
 \label{423}
 \ea
  \ba
 \mbox{For}\;\;\;\;\;10^{-2}\;yr^{-1}\,>\,\chi\;:\;\;\;\mbox{arguably, ~it ~is ~still~~}
 Q\,\sim\,\chi\;.\;\;\;\;\;(\,\mbox{Or maybe~~~}Q\;\sim\;1/\chi~\,?\,)~~~~~~~~~~~~\;
 \label{three}
 \label{424}
 \ea
 Fortunately, in the study of tides {\emph{in planets}} one never has to approach the Maxwell-time scales, so the controversy remaining in (\ref{three}) bears no relevance to our subject. In
 this paper we shall address tidal dissipation only in planets, leaving satellites for
 later. We shall not address the unique case of the Pluto-Charon resonance, nor shall
 we address the binary asteroids locked in the same resonance. Thus we shall avoid the
 frequency band addressed in (\ref{two}). We shall be interested solely in the frequency
 range described in (\ref{one}).  It is important to emphasise that the positive-power
 scaling law (\ref{one}) is well proven not only for samples in the lab but also for
 vast seismological basins and, therefore, is universal. Hence, this law may be extended
 to the tidal friction.

 Below we provide an extremely squeezed review of the published data whence the scaling
 law (\ref{one}) was derived by the geophysicists. The list of sources will be incomplete,
 but a full picture can be restored through the further references contained in the works
 to be quoted below. For a detailed review on the topic, see Chapter 11 of the book by
 Karato (2007) that contains a systematic introduction into the theory of and experiments
 on attenuation in the mantle.

 \subsection{Circumstantial evidence: attenuation in minerals.\\
  Laboratory measurements and some theory}

 Even before the subtleties of solid-state mechanics with or without melt are brought up,
 the positive sign of the power $\,\alpha\,$ in the dependence $\,Q\,\sim\,\chi^{\alpha}
 \,$ may be anticipated on qualitative physical grounds. For a damped oscillator obeying
 $\;\ddot{z}\,+\,2\,\beta\,\dot{z}\,+\,\chi^2\,z\,=\,0\;\,$, the quality factor is equal
 to $\,\chi/(2\beta)\,$, i.e., $\,Q\,\sim\,\chi\,$.

 Solid-state phenomena causing attenuation in the mantle may be divided into
 three groups: the point-defect mechanisms, the dislocation mechanisms, and
 the grain-boundary ones.

 Among the point-defect mechanisms, most important is the transient diffusional
 creep, i.e., plastic flow of vacancies, and therefore of atoms, from one grain
 boundary to another. The flow is called into being by the fact that vacancies
 (as well as the other point defects) have different energies at grain boundaries
 of different orientation relative to the applied sheer stress. This anelasticity
 mechanism is wont to obey the power law $\,Q\,\sim\,\chi^{\alpha}\,$ with
 $\,\alpha\,\approx\,0.5\,$.

 Anelasticity caused by dislocation mechanisms is governed by the viscosity law
 $\,Q\,\sim\,\chi\,$ valid for sufficiently low frequencies (or sufficiently high
 temperatures), i.e., when the viscous motion of dislocations is not restrained by
 the elastic restoring stress.\footnote{~At higher frequencies or/and lower
 temperatures, the restoring force ``pins" the defects. This leads to the
 law $\,Q\,\sim\,(\textstyle{1\,+\,\tau^2\chi^2})\tau^{-1}\chi^{-1}\,$,
 parameter $\,\tau\,$ being the relaxation time whose values considerably
 vary among different mechanisms belonging to this group. As the mantle is
 warm and viscous, we may ignore this caveat.}

 The grain-boundary mechanisms, too, are governed by the law $\,Q\sim\chi^{\alpha}$,
 though with a lower exponent: $\,\alpha\approx 0.2 -0.3$. This behaviour gradually
 changes to the viscous mode ($\alpha=1$) at higher temperatures and/or at lower
 frequencies, i.e., when the elastic restoring stress reduces.

 We see that in all cases the quality factor of minerals should grow with frequency.
 Accordingly, laboratory measurements confirm that, within the geophysically
 interesting band of $\,\chi\,$, the quality factor behaves as $\,Q\,\sim\,
 \chi^{\alpha}\,$ with $\,\alpha\,=\,0.2\,-\,0.4\;$. Such measurements have been
 described in Karato \& Spetzler (1990) and Karato (1998). Similar results were
 reported in the works by the team of I. Jackson -- see, for example, the paper
 (Tan et al 1997) where numerous earlier publications by that group are also
 mentioned.

 To this we would add that in aggregates with partial melt the frequency
 dependence of $\,Q\,$ keeps the same form, with $\,\alpha\,$ leaning to
 $\,0.2\,$ -- see, for example, Fontaine et al (2005)
 % , Jackson et al (2004),
 and references therein.

 \subsection{Direct evidence: attenuation in the mantle.\\
 Measurements on seismological basins}

 As we are interested in the attenuation of tides, we should be prepared to face
 the possible existence of mechanisms that may show themselves over very large
 geological structures but not in small samples explored in the lab. No matter
 whether such mechanisms exist or not, we would find it safer to state that the
 positive-power
 scaling law $\,Q\,\sim\,\chi^{\alpha}\,$, even though well proven in the lab,
 must be propped up by a direct seismological evidence gathered over vast zones
 of the mantle. Fortunately, such data are available, and for the frequency range
 of our interest these data conform well with the lab results. The low-frequency
 measurements, performed by different teams over various basins of the Earth's
 upper mantle, agree on the pivotal fact: the seismological quality factor scales
 as the frequency to the power of a {\emph{positive}} fraction $\,\alpha\;$ -- see,
 for example, Mitchell (1995), Stachnik et al (2004), Shito et al (2004), and
 further references given in these sources.\footnote{~So far, Figure 11 in Flanagan
 \& Wiens (1998) is the only experimental account we know of, which only partially
 complies with the other teams' results. The figure contains two plots depicting
 the frequency dependencies of $\,1/Q_{sheer}\,$ and $\,1/Q_{compress}\,$. While
 the behaviour of both parameters remains conventional down to $\,10^{-1}\,$Hz,
 the sheer attenuation surprisingly goes down when the frequency decreases to
 $\,10^{-3}\,$Hz. Later, one of the Authors wrote to us that ``{\emph{Both P and
 S wave attenuation becomes greater at low frequencies. The trend towards lower
 attenuation at the lowest frequencies in Fig. 11 is not well substantiated.}}"
 (D. Wiens, private communication) Hence, the consensus on (\ref{one}) stays.}

 \subsection{Consequences for the tides\\}

 \subsubsection{Tidal dissipation vs seismic dissipation}

 One of the basic premises of our model will be an assumption that, for terrestrial
 planets, the frequency-dependence of the $\,Q\,$ factor of bodily tides is similar
 to the frequency-dependence (\ref{one} - \ref{two}) of the seismological $\,Q\,$
 factor. This premise is based on the fact that the tidal attenuation in the mantle
 is taking place, much like the seismic attenuation, mainly due to the mantle's
 rigidity. This is a nontrivial statement because, in distinction from earthquakes,
 the damping of tides is taking place both due to rigidity and self-gravity of the
 planet. Modelling the planet with a homogeneous sphere of density $\,\rho\,$,
 rigidity $\,\mu\,$, surface gravity $\,\mbox{g}\,$, and radius $\,R\,$, Goldreich
 (1963) managed to separate the rigidity-caused and self-gravity-caused inputs into
 the overall tidal attenuation. His expression for the tidal quality factor has the
 form
 \ba
 Q\;=\;Q_o\left(\;1\;+\;\frac{2}{19}\;\frac{\mbox{g}\;\rho\;R}{\mu}\;\right)\;\;\;,
 \label{Goldreich}
 \ea
 $Q_o\,$ being the value that the quality factor would assume were self-gravity absent
 (i.e., were damping due to rigidity only). To get an idea of how significant the
 self-gravity-produced input could be, let us plug there the mass and radius of Mars
 and the rigidity of the Martian mantle. For the Earth mantle, $\,\mu\,=\,65\,\div\,
 80\,$GPa. Judging by the absence of volcanic activity over the past hundred(s)
 of millions of years of Mars' history, the temperature of the Martian upper mantle is
 (to say the least) not higher than that of the terrestrial one. Therefore we may safely
 approximate the Martian $\,\mu\,$ with the upper limit for the rigidity of the
 terrestrial mantle: $\,\mu\,=\,8\,\times\,10^{10}\,$Pa. All in all, the relative
 contribution from self-gravity will look as
 \ba
 \frac{2}{19}\;\frac{\mbox{g}\;\rho\;R}{\mu}\;=\;
 % \frac{2}{19}\;\frac{\gamma\;\frac{\textstyle M}{\textstyle R^2}\;\frac{\textstyle
 % M}{\frac{\textstyle 4}{\textstyle 3}\;\textstyle{\pi\;R^3}}\;R}{\mu}\;=\;
 \frac{6}{76\,\pi}\;\frac{\gamma\;M^2}{\mu\;R^4}\;
 \approx\;\frac{1}{40}\;\frac{(\;6.7\,\times\,10^{\textstyle{^{-11}}}\,
 \mbox{m}^{\textstyle{^3}}\,\mbox{kg}^{\textstyle{^{-1}}}\,\mbox{s}^{\textstyle{^{-2}}}\;)
 \;\;(\;6.4\,\times\,10^{\textstyle{^{23}}}\,\mbox{kg}\;)^{\textstyle{^2}}}{(\;
 11^{\textstyle{^{10}}}\;\mbox{Pa}\;)\;\;(3.4\,\times\,10^{\textstyle{^6}}\,m)^{
 \textstyle{^4}}}\;\approx\;5.2\,\times\,10^{\textstyle{^{-2}}}\;\;,\;\;\;\;
 \label{estimate}
 \ea
 where $\gamma\,$ stands for the gravity constant. This rough estimate shows that
 self-gravitation contributes, at most, several percent into the overall count of
 energy losses due to tides. This is the reason why we extend to the tidal $\,Q\,$
 the frequency-dependence law measured for the seismic quality factor.

  The above estimate is conservative, because we approximated the Martian $\,\mu\,$
  with the terrestrial value of $\,\mu\,$. If the interior of Mars is cooler than
  that of the Earth, then its rigidity should be higher, and therefore
  the ratio (\ref{estimate}) may reduce to less than one percent.

 \subsubsection{Dissipation in the planet vs dissipation in the satellite}

 As already agreed above, in this work we are dealing solely with the
 satellite-generated tides on the planet. The input from the planet-caused tides on
 the satellite will be considered elsewhere. Hence, here it will not be a problem for
 us to encounter a configuration where a satellite shows the same face to its primary
 -- this kind of spin-orbit resonance will not imply a vanishing tidal frequency. Nor
 will it be a problem for us to have a satellite crossing a synchronous orbit -- the
 presence of the radial term in (\ref{13} - \ref{14}) will prevent the tidal frequency
 from becoming nil at the point of crossing. The same is true for the lag related to the
 frequency through formula (\ref{3}): although the ``tangential" lag $\,
 \delta_1\,$ vanishes at such a crossing, the ``radial" part of the lag remains finite.
 Hence, the overall $\,\delta\,$ stays finite.

 A very special situation is tidal relaxation toward the state where the planet
 shows the same side to its moon, like Pluto does to Charon.\footnote{~Such a complete
 locking is typical also for binary asteroids. Since at present most asteroids are
 presumed loosely connected, and since we do not expect the dependencies (\ref{one}
 - \ref{two}) to hold for such aggregates, our theory should not, without some
 alterations, be applied to binaries.} A gradual approach to this state makes the
 tidal frequency asymptotically go to zero. Mathematically, this situation still
 may be tackled by means of (\ref{one}) until the tidal frequency $\,\chi\,$
 decreases to $\;1\,$yr$^{-1}$, and then by means of (\ref{two}) while $\,\chi\,$
 remains above the inverse Maxwell time of the planet's material. Whether the latter
 is physically satisfactory remains an open issue of a generic nature that is not
 related to a specific theory of tides or to a particular frequency dependence of
 $\,Q\;$. The generic problem is whether we at all may use the concept of the
 quality factor on approach to the Maxwell time, or whether we should, beginning
 from some low $\,\chi\,$, employ a comprehensive hydrodynamical model. In the
 current work, we shall not address this question, leaving it for another paper.

 Thus, since we are talking only about dissipation inside the planet, and are not
 addressing the exceptional Pluto-Charon case, we may safely assume the tidal
 frequency to always exceed $\;1\,$yr$^{-1}$. Thence (\ref{one}) will render,
 for a typical satellite:
 \ba
 {\left.~~~~~~~~~\right.}
 Q\,\sim\,\chi^{\alpha}~~~,~~~~\mbox{with}~~\alpha~=~0.2~-~0.4~~~.
 \label{425}
 \ea
 Accordingly, (\ref{421}) will entail:
 \ba
 {\left.~~~~~~~~~\right.}
 \delta~\sim~\chi^{-\alpha}~~~,~~~~\mbox{with}~~\alpha~=~0.2~-~0.4~~~.
 \label{426}
 \ea

 \subsection{The frequency and the temperature}

 In the beginning of the preceding subsection we already mentioned that though the
 tidal $Q$ differs from the seismic one, both depend upon the frequency in the same
 way, because this dependence is determined by the same physical mechanisms. This
 pertains also to the temperature dependence, which for some fundamental reason
 combines into one function with the frequency dependence.
 % \footnote{~

 As explained, from the basic physical principles, by Karato (2007, 1998), the
 frequency and temperature dependencies of $\,Q\,$ are inseparably connected.
 Since the quality factor is dimensionless, it must retain this property despite
 the exponential frequency dependence. This may be achieved only if $\,Q\,$ is a
 function not of the frequency {\emph{per se}} but of a dimensionless product of
 the frequency by the typical time of defect displacement. This time exponentially
 depends upon the activation energy $\,A^*\,$, whence the resulting function reads
 as
 \ba
 Q\;\sim\;\left[\,\chi\;\exp(A^*/RT)\,\right]^{\alpha}\;\;\;.
 \label{427}
 \ea
 For most minerals of the upper mantle, $A^*$ lies within the limits of
 $360-540$ kJ mol$^{-1}$. For example, for dry olivine it is about $520$
 kJ mol$^{-1}$.
 % Most important is the value of $A^*$ appropriate to the layer wherein
 % most of the tidal dissipation takes place (i.e., where $k_2$ is maximal).

 Thus, through formulae (\ref{427}) and (\ref{421}), the cooling rate of
 the planet plays a role in the orbital evolution of satellites: the lower
 the temperature, the higher the quality factor and, thereby, the smaller
 the lag $\, \delta\,$. For the sake of a crude estimate, assume
 that most of the tidal attenuation is taking place in some layer, for which
 an average temperature $T$ and an average activation energy $A^*$ may be
 introduced.
 %  (\ref{427}) will entail:
 %  \ba
 %  {\cal E}\;=\;{\cal E}_o\;\exp\left[\,\frac{A^*}{R}\;
 %  \left(\,\frac{1}{T_o}\;-\;\frac{1}{T}\;\right)\;\right]\;\;\;,
 %  \label{43}
 %  \ea
 % $T_o\,$ being the present temperature of the layer.
 Then from (\ref{427}) we have: $\,\Delta Q/{Q}\,\approx\,\alpha\,A^*/RT\,$.
 For a reasonable choice of values $\,\alpha=0.3\,$ and $\,A^*=\,5.4\,\times
 \,10^5\,J/mol\,$, a drop of the temperature from $\,T_o=2000\,K\,$ down by
 $\,\Delta T=200\,K\,$ will result in $\,\Delta Q/Q\,\approx\,1\,$. So a
 $\,10\,\%\,$ decrease of the temperature can result in an about $\,100\,\%\,$
 growth of the quality factor.
 % A careful calculation for a multilayered planet
 % is more involved, but the scale of numbers should stay the same.

 Below we shall concentrate on the frequency dependence solely.

 \section{Formulae}

 The tidal potential perturbation at the surface of a planet due to a
 point satellite is given by
 \ba
 W(\gamma)\;=\;\sum_{n=2}^{\infty}\;W_n(\gamma\,,\;r)\;\;\;
 \label{23}
 \ea
 where
 \ba
 W_n(\gamma\,,\;r)\;=\;\frac{G\;m\;R^n}{~~~~r^{n+1}}\;P_n(\cos\gamma)\;\;\;,
 \label{24}
 \ea
 so that
 \ba
 W\;=\;\frac{G\;m}{2}\;\,\frac{R^2}{r^3}\;\left[\,\left(\,3\;\cos^2\gamma
 \;-\;1\,\right)\;+\;\frac{R}{r}\;\left(\,5\;\cos^3\gamma\;-3\;\cos\gamma
 \,\right)\;+\;\,.\;.\;.\;\right]\;\;\;,
 \label{25}
 \ea
 $R\,$ being the mean equatorial radius of the primary, and the angle $\;
 \gamma\;$ being reckoned from the planetocentric vector pointing at the
 satellite. Being small, the tidal perturbations of the potential yield a
 linear response of the planet's shape, deformations proportional to the
 Love numbers. These deformations, in their turn, amend the external
 potential of the planet with an addition, which would look as
 \ba
 U(\gamma)\;=\;\sum_{n=2}^{\infty}\;k_n\;\left(\,\frac{R}{r}\,\right)^{n
 +1}\;W_n(\gamma\,,\;r)\;\;\;,
 \label{26}
 \ea
 had the planet's response to the satellite's pull been synchronous. In
 reality the bulge lags, as if it emerges beneath a {\it{fictitious}}
 satellite lagging relative to the real one. If the real satellite is
 located at $\,\erbold\,\equiv\,\erbold(\nu)$, where $\;\nu\;$ is the
 true anomaly, then the fictitious satellite is at
 \ba
 \erbold_f\;=\;\erbold\;+\;\fbold\;\;\;,\;\;\;\;\;\;\fbold\;=\;
 \Delta t\;\left(\,{\omegabold}_p \times \erbold \,-\,\vbold \,\right)
 \label{27}
 \ea
 In these terms, the external potential of the planet will read:
 \ba
 U(\delta_1)\;=\;\sum_{n=2}^{\infty}\;k_n\;
 \left(\,\frac{R\;}{r_{\textstyle{_f}}}\,\right)^{\textstyle{^{n+1}}}
 W_n(\gamma\,,\;r)\;\;\;,
 \label{28}
 \ea
 the angle $\,\delta_1\,=\,\emph{arccos}\frac{\erbold\cdot\erbold_f}{|\erbold|\;
 |\erbold_f|}\;$ being reckoned from the bulge. (Mind that the lag
$\,\delta\,$ is, generally, different from $\,\delta_1\,$, and
coincides with it only in the limit of vanishing $\,e\,$.) To
calculate
 the back-reaction exerted by the bulge on its progenitor satellite,
 one should equate $\,\delta_1\,$ and $\,\gamma\,$. Then (\ref{28})
 will assume the form:
 \ba
 \nonumber
 U(\delta_1)\;=~\frac{GmR^5}{r_{\textstyle{_f}}^3\,\;r^3}\left[k_2P_2(\cos
 \delta_1)+k_3\frac{R^2}{r_{\textstyle{_f}}\,r}P_3(\cos\delta_1)+\,...\right]
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 \\
  \nonumber\\
 \nonumber\\
 \nonumber
  =
 \frac{GmR^5}{2\,r_{\textstyle{_f}}^3\,r^3}\left[k_2\left(3\,\cos^2\delta_1-
 1\right)+k_3\frac{R^2}{r_{\textstyle{_f}}\,r}\left(5\,\cos^3\delta_1-3\,
 \cos\delta_1\right)+...\right]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 \ea
 ~\\
 \ba
 =\;\frac{A_2}{r_{\textstyle{_f}}^5\;r^5}\;\left(\,3\;\left(\erbold_f
 \cdot\erbold\right)^2\;-\;\erbold_f^{\;2}\;\erbold^{\;2}\,\right)\;+
 \;\frac{A_3}{r_{\textstyle{f}}^7\;r^7}\;\left(\,5\;\left(\erbold_f
 \cdot\erbold\right)^2\;-\;3\;\erbold_f^{\;2}\;\erbold^{\;2}\,\right)
 \;+\;\;.\;\,.\;\,.~~
 \label{29}
 \ea
 where $\,r\,\equiv\,|\erbold|\,$ and $\,r_f\,\equiv\,|\erbold_f|\,$,
 while the constants are given by
 \ba
 A_2\;\equiv\;\frac{k_2\,G\,m\,R^5}{2}\;\;\;\;,\;\;\;\;\;\;A_3\;\equiv\;
 \frac{k_3\,G\,m\,R^7}{2}\;\;\;\;,\;\;\;\;\;\;\;.\,\;.\,\;.
 \label{30}
 \ea
 To (\ref{29}) we must add the potential due to the tidal deformation of the
 satellite by the planet. That input contributes primarily to the radial
 component of the tidal force exerted on the moon, and redounds to the decrease of eccentricity (MacDonald 1964). Here we omit this term, for our goal is to clarify the frequency dependence of the lag. In a subsequent work, devoted to orbit integration, we shall take that term into account.

 Another important comment worth mentioning is that in practical applications one often has to take into account the $\,k_3\,$ and sometimes even the $\,k_4\,$ terms of the expansion for the tidal potential. (Bills et al 2005) Once again, since in the current paper our goal is to explain the new model and not to perform whatever numerical integration, then we shall restrict ourselves, for brevity, to the leading term proportional to $\,k_2\,$.

 Our intention is take into consideration the frequency-dependence of the lag $\,\efcal\,$, but not of the parameter $\,k_2\,$. While the dependence $\,\efcal(\chi)\,$ will be derived through the interconnection of $\,\efcal\,$ with $\,\delta\,$ and therefore with $\,Q(\chi)\,$, the value of $\,k_2\,$ will be kept constant. This can be justified by means of the following formula derived by Darwin (1908) under for a Maxwell body (see also Correia \& Laskar 2003):
 \ba
 \nonumber
 k_2(\chi)\;=\;k_{\textstyle{_{fluid}}}\;\,\sqrt{\;\frac{1\;+\;\chi^2\;\eta^2/\mu^2}{1\;+\;
 \left(\,\chi^2\;\eta^2/\mu^2\,\right)\;\left(\,1\;+\;19\;\mu/(2\,g\,\rho\,R)\,\right)^2}\;}
 \label{k_2}\;\;\;.
 \ea
 $k_{\textstyle{_{fluid}}}\,$ being the fluid Love number. This is the value that $\,k_2\,$ would have assumed had the planet consisted of a perfect fluid with the same mass distribution as the Maxwell-body planet considered. Notations $\,\mu\,$, $\,\rho\,$, $\,\mbox{g}\,$, and $\,\mbox{g}\,$ stand for the rigidity, mean density, surface gravity, and the radius of the planet. The letter $\,\eta\,$ signifies the viscosity. For a terrestrial planet's mantle, the latter will be close to $\,10^{22}\,$kg/(m$\cdot$ s),
 give or take an order or two of magnitude. For the other parameters we may keep using the estimates from subsection 4.4.1. This will entail: $\,\chi^2\;\eta^2/\mu^2\,=\,(\,\chi\,\cdot\,10^{11}\,\mbox{s}\,)^2\,$,
 whence we see that in all realistic situations pertaining to terrestrial planets the frequency-dependence in Darwin's formula will cancel out. Thus we shall neglect the frequency-dependence of the Love number $\,k_2\,$ (but shall at the same time take into account the frequency-dependence of $\,Q\,$, for it will induce frequency-dependence of all three lags).

 This said, we can apply the standard rule of thumb according to which calculation of the tidal force consists of the following four steps:\\
 ~\\
 {\textbf{1.}} ~~Differentiate (\ref{29}) with respect to $~\erbold~$,
 keeping $~\erbold_f~$ constant.\\
 ~\\
 {\textbf{2.}} ~~{\emph{After the differentiation}},\footnote{
 ~At this point we are on the horns of dilemma. Should we first
 take the partial derivative $\;\left(\frac{\textstyle\partial}{\textstyle{
 \partial\erbold}}\right)_{_{\erbold_f}}\,$ and impose the constraint $\,
 \erbold_f\,=\,\erbold\,+\,\fbold\,$ afterwards, or should we perform these
 operations in the opposite order? The former option is conventional and,
 fortunately, correct, though its justification requires some effort.}
 insert the expression $\,\erbold_f\, =\,\erbold\,+\,\fbold\,$.\\
 ~\\
 {\textbf{3.}} ~~Substitute $\,\efbold\,$ with the product
 $\,\Delta t\;\left(\;\omegabold_p\times\erbold\;-\;\vbold\;\right)\;\,$.\\
 ~\\
 {\textbf{4.}} ~~Express everything in terms of the orbital elements.\\
 ~\\
 This will entail an unaveraged expression for the tidal force in terms of
 the satellite's orbital elements. It can then be plugged into
 the Gauss planetary equations. Averaging thereof
 will yield the equations describing the evolution of the secular parts of
 the orbital variables.

 Steps {\textbf{1}} and {\textbf{2}} lead us to the following formula for
 the force:
 \ba
 \nonumber
 \efcal\;=\;-\;\frac{3\;k_2\;G\;m^2\;R^5}{r^{7}}\;\left[\;\frac{\erbold}{r}\;-
 \;\frac{\vec{\mbox{\it\textbf{f}}}}{r}\;-\;2\;\frac{\erbold}{r}\;
 \frac{\,\erbold\cdot{\vec{\mbox{\it\textbf{f}} }}\,}{r^2}\;+\;O(\,{\efbold}^{\;\;{2}}/r^2)
 \right]\;+\;O(k_3Gm^2R^7/r^{9})\;\;\;\\
 \nonumber\\
 \approx\;-\;\frac{3\;k_2\;G\;m^2\;R^5}{r^{10}}\;\left[\;{\erbold}\;{r^2}\;-
 \;{\vec{\mbox{\it\textbf{f}}}}\;{r^2}\;-\;2\;{\erbold}\;\left(\,\erbold\cdot{\vec{\mbox{\it\textbf{f}}
}}\,\right)\;\right]\;\;\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;~~~~~~~~~~~~~~~~~~~~~~
 \label{31}
 \ea
 taken that $\;O(\,{\efbold}^{\;\;{2}}/r^2)\;=\;O(\,\delta^2)\,\ll\,1\;$, and assuming that the satellite is pulled by the bulge it itself has created, with no other moons involved.

 To perform step {\textbf{3}}, one should use connection
 (\ref{15}) between the position and angular lags:
 \ba
 \delta\;\equiv\,\;\frac{|\fbold \,|}{r}\,\;=\;\Delta t\;\;\frac{1}{r}\;|\;
 \omegabold_p\times\erbold\;-\;\vbold\;|\;\,=\;\frac{\Delta t}{2}\;\chi~~~
 \label{131}
 \label{32}
 \ea
 or, equivalently:
 % \footnote{~Equivalently -- in the framework of our physical model. See
 % the final paragraph of section 2.}
 \ba
 \fbold~=~{\bf{\hat{f}}}\;r\;\delta\;=\;r\;\frac{\Delta t}{2}\;\chi\;{\bf{
 \hat{f}}}\;\;\;,
 \label{132}
 \label{33}
 \ea
 where
 \ba
 {\bf{\hat{f}}}\;=\;\frac{~\omegabold_p \times \erbold\,-\,\vbold}{~|\,
 \omegabold_p\times\erbold\,-\,\vbold\,|}
 \label{133}
 \label{34}
 \ea
 is the unit vector pointing in the lag direction, and $\delta\,$ is a known
 function of the frequency $\chi\,$.

 Be mindful that all equations (\ref{31} - \ref{34}) are {\emph{exact}}, in that they
 do not make use of our (or any other) specific model of tides. It is of a crucial
 importance that the subtended angle (the ``tangential lag") $\,\delta_1\,$ does not
 appear in (\ref{31}). The expression for the force depends solely upon $\,\efbold\,$,
 which in its turn is interconnected with the total lag $\,\delta\,$ and the quantity
 $\,\chi(\nu)\,$ -- simply because $\,\delta\,$ is {\emph{defined}} as
 $\;\delta\,\equiv\,|\efbold|/r\,=\,r^{-1}|\omegabold\times\erbold-\vbold|\,\Delta t\;$
 and because $\,\chi(\nu)\,$ is {\emph{defined}} as $\;2r^{-1}|\omegabold\times\erbold-
 \vbold|\;$. We wish to emphasise that in the end of the day the actual physical force
 $\,\efcal\,$ depends not upon the ``tangential lag" $\,\delta_1\,$ but upon the total
 lag $\,\delta\,$.

 The realm of our particular model begins when, for the reasons explained in subsection 2.4, we interpret $\,\chi(\nu)\,$ as the instantaneous tidal frequency, in the spirit of the WKB approximation. This immediately brings into play the instantaneous quality factor $\,Q\,$ as a function of the
 instantaneous frequency: $\,Q\,=\,Q(\chi(\nu))\,$. Our knowledge of the scaling
 law (\ref{425} - \ref{426}) will then enable us to proceed within the framework of our model. Since, according to (\ref{426}), $\;\delta\,\sim\,\chi^{-\alpha}\;$, then (\ref{421}) necessitates for the quality factor: $\,Q\,\sim\,\chi^{\alpha}\,$. This power scaling law may be conveniently expressed as
 \ba
 \nonumber
 Q\,=\,{\cal E}^{\alpha}\,\chi^{\alpha}~~~,~~~
 \label{}
 \ea
 ${\cal E}^{\alpha}\,$ simply being the dimensional factor emerging in the relation $\,Q\,\sim\,\chi^{\alpha}\,$.  As mentioned in subsection 4.5,
 cooling of the planet should become a part of long-term orbital integration. It enters the integration via evolution of this factor $\,{\cal{E}}^{\alpha}\,$. If we suppose that most of the tidal attenuation is happening in some particular layer, for which an average activation energy $A^*$ and an average temperature $T$ may be defined, then (\ref{427}) will result in:
 \ba
 \nonumber
 {\cal E}\;=\;{\cal E}_o\;\exp\left[\,\frac{A^*}{R}\;
 \left(\,\frac{1}{T_o}\;-\;\frac{1}{T}\;\right)\;\right]\;\;\;,
 \label{}
 \ea
 $T_o\,$ being the temperature of this layer at some fiducial epoch. Then the physical meaning of the parameter $\,{\cal{E}}\,$ becomes evident: if the planet were made of a homogeneous medium, with a uniform temperature distribution, and if dissipation in this medium were caused by one particular physical mechanism, then $\,{\cal{E}}\,$ would be a relaxation time scale appropriate to this mechanism (say, the time of defect displacement). For an actual celestial body, $\,{\cal{E}}\,$ may be interpreted as a relaxation time averaged (in the sense of
 $\,Q\,=\,{\cal E}^{\alpha}\,\chi^{\alpha}\,$) over the body's layers and over the various attenuation mechanisms acting within these layers. This way, $\,{\cal{E}}\,$ is an integral parameter describing the overall tidal dissipation in an inhomogeneous body.

 As (\ref{one}) yields $\;\delta\,\approx\,1/(2\,Q)\,=\,(1/2)\,
 {\cal{E}}^{-\alpha}\,\chi^{-\alpha}\;$, then (\ref{131} - \ref{132}) give for the position lag:
 \ba
 {\vec{\mbox{\it\textbf{f}}}}~=~r\;\delta\;{\bf{\hat{f}}}\;=\;\frac{1}{2}\;
 r\;{\cal E}^{-\alpha}\;\chi^{-\alpha}\;{\bf{\hat{f}}}\;\;\;,
 \label{23}
 \ea
 and for the time lag:
 \ba
 {\Delta t}\;=\;{\cal E}^{-\alpha}\;\chi^{-(\alpha +1)}\;\;\;,
 \label{22}
 \ea
 ${\cal E}\,$ being the afore introduced integral parameter, and $\,\chi\,$ being a known function (\ref{418}) of the orbital variables. All in all, we have:
 \ba
 {\vec{\mbox{\it\textbf{f}}}}~=~\left({\cal{E}}{\chi}\right)^{-\alpha}a\;
 \frac{1\,-\,e^2}{1\,+\,e\;\cos\nu}\;\,\frac{~\omegabold_p \times\erbold\,-\,\vbold~}{~|
 \,\omegabold_p\times\erbold\,-\,\vbold\,|~}\;\;\;
 \label{24}
 \ea
 where
 \ba
 \chi\,\equiv\;2\;\;\sqrt{\;\left[\,n\;\frac{\left(1+\,e\,\cos\nu \right)^2}{
 \left(1-e^2\right)^{3/2}}\,-\,\omega_p\,\cos\inc\right]^{{{\,2}}}+\,
 \omega_p^2\,\sin^2\inc\;\cos^2(\omega+\nu)\,+\,n^2\,e^2\,\frac{\left(1+
 e\,\cos\nu\right)^2}{\left(1-e^2\right)^{3}}\;\sin^2\nu\;\,}\;\;.\;\;
 \label{137}
 \label{38}
 \ea
 The time lag is, according to (\ref{22}):
 \ba
 \nonumber
 \Delta t\;=\;2^{\textstyle{^{-\,(\alpha+1)}}}{\cal{E}}^{\textstyle{^{-\,\alpha}}}
 \left(\;\left[\,n\;\frac{\left(1\,+\,e\;\cos\nu\right)^2}{\left(1\,-\,e^2
 \right)^{3/2}}\;-\;\omega_p\;\cos\inc\right]^{{{\,2}}} \right.
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
  \nonumber\\
  \nonumber\\
 \left. +\;\,\omega_p^2\;\sin^2\inc\;\cos^2(\omega\,+\,\nu)\;+\; n^2\;e^2\;\frac{\left(1\,+\,e\;\cos\nu\right)^2
 }{\left(1\,-\,e^2\right)^{3}}\;\sin^2\nu\right)^{\textstyle{^{-\,(\alpha+1)/
 2}}}~~~.~~~~~~~~~~~~~
 \label{39}
 \ea
 Formulae (\ref{39}), (\ref{24}), and (\ref{31}) are sufficient to both compute the orbit evolution and trace the changes of the time lag.

 In neglect of the $\,O(e^2)\,$ terms, (\ref{24} - \ref{39}) simplify to
 \ba
 \fbold\;\approx\;\left({\cal{E}}{\chi}\right)^{\textstyle{^{-\alpha}}}\;
 \frac{a}{1\,+\,e\;\cos\nu}\;\,\frac{~\omegabold_p \times\erbold\,-\,\vbold~}{~|
 \,\omegabold_p\times\erbold\,-\,\vbold\,|}\;\;\;,
 \label{138}
 \label{40}
 \ea
 \ba
 \chi\,\approx\,2~\sqrt{~\,\left[\;n\,\left(1\,+\,2\,e\;\cos\nu\right)\,-\,
 \omega_p\,\cos\inc\,\right]^{\textstyle{^2}}\,+\;\omega_p^2\;\sin^2\inc\;
 \cos^2(\omega+\nu)}\;~~,\;\;\;\;
 \label{139}
 \label{41}
 \ea
 and
  \ba
 \Delta t\;\approx\;
 2^{\textstyle{^{-\,(\alpha+1)}}}{\cal{E}}^{\textstyle{^{-\,\alpha}}}\;\left(\,
 \left[\;n\,\left(1\,+\,2\,e\;\cos\nu\right)\,-\,\omega_p\,\cos\inc\,
 \right]^{\textstyle{^2}}\,+\;\omega_p^2\;\sin^2\inc\;\cos^2(\omega+\nu)\,
 \right)^{\textstyle{^{-\,(\alpha+1)/2}}}~.~
 \label{42}
 \ea
 In the MacDonald-Kaula-Gerstenkorn theory the geometric lag angle $\,
 \delta\,$ is constant (i.e., $\,\alpha\,=\,0\,$). The Singer-Mignard
 theory suggests that we treat the time lag $\,\Delta t\,$ as a constant
 (i.e., set $\,\alpha\,=\,-1\,$). As explained in the preceding sections,
 neither of these choices corresponds to the geophysical realities. The
 physics of attenuation dictates to us that for a realistic terrestrial
 planet, the exponential $\,\alpha\,$ lies within the range of $\,0.2 -
 0.4\,$, except in the case of near-synchronous satellites for which it
 increases to unity.

 To write down the planetary equations in the Euler-Gauss form, either the exact
 expression (\ref{24}) or its approximation (\ref{40}) should be plugged into
 (\ref{31}).\footnote{~Equivalently, one can simply plug (\ref{2}) into (\ref{31}),
 to get
  \ba
  \nonumber
 {\vec{\bf{\cal
 F}}}\;=\;-\;\frac{3\;k_2\;G\;m^2\;R^5}{r^{10}}\;\left[\;\erbold\;r\;^2\;+
 \;\left(\,2\,\erbold\,\left(\erbold\cdot\vbold\right)\,+\,r^2\,\left(
 \erbold\times\omegabold+\vbold\right)\,\right)\;\Delta t\;\right]\;+\;.\,
 .\,.\;\;\;,
 \label{}
 \ea
 and can insert it into the Euler-Gauss-type planetary equations. Formally, this
 will render the same unaveraged equations for the elements' evolution,
 as the equations derived by Mignard (1979, 1980). However, integration of our equations should be carried out with an important proviso that now $\,\Delta t\,$ is not a constant but is expressed by (\ref{22}), i.e., depends (through (\ref{39}) or (\ref{42})) upon the current values of the orbital elements and the true anomaly. This will cardinally alter the results of integration.} The equations can then be integrated. One possibility would be a straightforward numerical integration. An alternative option would be to perform analytical averaging over the true anomaly $\;\nu\;$, and only afterwards to integrate the averaged Euler-Gauss-type equations describing the evolution of the {\emph{secular parts}} of the satellite's orbital elements.

 A numerical simulation of the Mars-Phobos dynamics, in neglect of the primary's cooling and in neglect of the tides on the satellite, will be presented in our next publication (Efroimsky \& Lainey 2007b). Some preliminary results of that study have been reported in our talk Efroimsky \& Lainey (2007a). According to those computations, Phobos will fall on Mars in 45 Myr from now. This estimate is 50 percent longer than the one following from the tidal models used in the past. An overwhelming share of this difference comes from the actual frequency dependence of attenuation, and only a small part comes from our usage of the instantaneous tidal frequency instead of the Kaula expansion. This demonstrates that the currently accepted time scales of dynamical evolution should be reexamined using the actual frequency dependence of the lags.

 \section{Conclusions}

 In our model, we operate with the {\emph{overall}} quality factor $\,Q\,$ that
 describes both components of the tidal friction -- those associated with the
 ``tangential" and ``radial" lags. By the ``tangential" lag we
 understand the angle $\,\delta_1\,$ subtended at the planet's centre between the
 satellite and the bulge. It emerges because the {\emph{location}}
 of the bulge lags. The ``radial" lag is called into being by the fact that the
 {\emph{height}} of the bulge lags too.
 (This latter lag exists only for noncircular orbits, and vanishes after
 an orbit gets circularised.) Our model is based on the observation that the overall
 quality factor $\,Q\,$ is interconnected, via the standard relation (\ref{420}), not
 with the ``tangential lag" $\,\delta_1\,$, but with the {\emph{total}} lag $\,\delta\,$
 introduced as $\,\sqrt{(\mbox{{tangential~lag}})^{\textstyle{^2}}\,+\,(\mbox{radial~lag
 })^{\textstyle{^2}}\,}\,$.

 The model also employs an {\emph{instantaneous}} tidal frequency $\,\chi\,$, which is a
 function of the true anomaly $\,\nu\,$ of the satellite, -- a concept vaguely prompted
 by the WKB method in quantum mechanics. We {\emph{define}} this instantaneous tidal
 frequency $\,\chi\,$ as the quantity, insertion whereof in (\ref{term}) or (\ref{8})
 gives the correct dissipation rate. This definition yields the expression (\ref{14}) for
 $\,\chi\,$ through the satellite's orbital elements and the true anomaly. For a circular
 equatorial orbit, the right-hand side of that formula reduces to the well known expression
 $\,2|n\,-\,\omega_p|\,$.

 So defined total lag $\,\delta\,$, instantaneous frequency $\,\chi\,$, and the
 {\emph{overall}} quality factor $\,Q\,$ are always (i.e., for an arbitrary eccentricity
 and inclination of the moon) connected among themselves by the same relations (\ref{419}
 - \ref{420}) as in the trivial case of an equatorial circular orbit. This has enabled us
 to circumvent the difficulties of the standard, Kaula-expansion-based tidal theory by
 completely avoiding whatever decomposition of the tide. The instantaneous tidal frequency
 $\,\chi(\nu)\,$ and the instantaneous overall quality factor $\,Q(\chi)\,=\,Q(\,\chi(\nu
 )\,)\,$ are the only frequency and the only $\,Q\,$ factor entering the solution. This
 model is the only means of escape in nonlinear situations where the quality factors of
 the harmonics are badly defined (i.e., when the ``Goldreich's admonition" is in force --
 see subsection 2.2 above).

 Within our model we permit the overall quality factor and, therefore, both the overall
 angular lag $\,\delta\,$ and the time lag $\,\Delta t\,$ to depend upon the instantaneous
 tidal frequency $\,\chi\,$. While in the Gerstenkorn-MacDonald-Kaula theory the geometric
 lag is postulated to be constant, in the Singer-Mignard theory it is the time lag that is
 assumed constant. However, neither of these two choices conform to the geophysical data
 within the frequency band of our interest. As the overall angular lag $\,\delta\,$ is
 inversely proportional to the tidal $\,Q\,$ factor, the actual
 frequency-dependence of both $\,\delta\,$ and $\,\Delta t\,$ is unambiguously defined by
 the frequency-dependence of $\,Q\,$.

 The quality factor is wont, according to numerous studies,
 to obey the law $\,Q\,\sim\,\chi^{\alpha}\,$, where $\,\alpha\,$ lies within
 $\,0.2\,-\,0.4\,$.
 % For partial melts, $\,\alpha\,$ is a bit lower than for solids -- see
 % (\ref{one}) in Section 4.
 For this reason, the time lag $\,\Delta t\,$, employed in the
 Singer-Mignard theory as a fixed parameter, in reality is not a constant but a function
 (\ref{134}) of the frequency. Hence, through (\ref{39}), $\,\Delta t\,$ becomes a
 function of the current values of the orbital elements of the satellite, and so becomes
 $\delta$. This clause alters the integration of the planetary equations.

 In a subsequent publication we shall deal with the numerics, in application to the
 possible history of the Martian satellites (Efroimsky \& Lainey 2007b).

 % We leave to a companion paper the discussion of the startling
 % consequences of

  ~\\

 % \pagebreak

 {\underline{\textbf{\Large{Acknowledgments}}}}\\
 ~\\
 I wish to deeply thank Valery Lainey, George Kaplan, Victor Slabinski, and
 S. Fred Singer for our fruitful discussions on the theory of tides. My
 very special gratitude goes to Shun-ichiro Karato who so kindly consulted
 me on the theory and phenomenology of the quality factor.

   \pagebreak

% \noindent
{\underline{\textbf{\Large{Appendix}}}}\\
 ~\\
  ~\\
 % ~\\
 \noindent
 {{\textbf{\large{A.1.~~~The expression for $\,\frac{\textstyle 1}{
 \textstyle r}\,|\,\omegabold_p\times\erbold\;-\;\vbold\,|\,$}}}}\\
 ~\\
 % ~\\
 Our goal is to prove formula (\ref{13}). To that end, let us start out
 with the evident equality
 \ba
 \frac{\textstyle 1}{\textstyle r^2}\,|\,\omegabold_p\times\erbold\;-\;
 \vbold\,|^2\;=\;\frac{\textstyle 1}{\textstyle r^2}\,\left[\;\left(\,
 \omegabold_p\times\erbold\,\right)^2\;+\;\vbold^{~2}\;-\;2\,\left(\,
 \omegabold_p\times\erbold\,\right)\cdot\vbold\;\right]\;\;.\;\;\;\;
 \label{}
 \ea
 The satellite velocity $\,\vbold\,$ can be expanded into its radial and
 tangential components: $\,v_r\,=\,\dot{r}\,$ and $\,v_{_T}=r\,\dot{\nu}\,
 $, with $\,\nu\,$ standing for the true anomaly. Thence we get:
 \ba
 \frac{\textstyle 1}{\textstyle r^2}\,|\,\omegabold_p\times\erbold\;-\;
 \vbold\,|^2\;=\;\frac{\textstyle 1}{\textstyle r^2}\,\left[\;\left(\,
 \omegabold_p\times\erbold\,\right)^2\;+\;r^{2}\,\dot{\nu}^{\,2}\;+\;v_r^2
 \;-\;2\,\left(\,\omegabold_p\times\erbold\,\right)\cdot\vbold\;\right]\;\;
 \;
 \label{D}
 \ea
 where
  \ba
 v_r=\dot{r}=\frac{\partial M}{\partial t}\,\frac{\partial\nu}{\partial M}
 \,\frac{\partial}{\partial \nu}\left(\frac{a\,\left(1-e^2\right)}{1+e\;
 \cos\nu}\right)=n\,\;\frac{\left(1+\,e\;\cos\nu\right)^2}{\left(1-e^2
 \right)^{3/2}}\,\;\frac{a\,\left(1-e^2\right)\,e\;\sin\nu}{\left(1+e\;
 \cos\nu\,\right)^2}=\frac{n\;a\;e}{\sqrt{1\,-\,e^2}}\;\sin\nu\;\;.\;\;\;\;
 \;
 \label{O}
 \ea
 To continue, we need expressions for $~\left(\,\omegabold_p\times\erbold\,
 \right)^2~$ and $\,\;-\,2\,\left(\,\omegabold_p\times\erbold\,\right)\cdot
 \vbold\;.\;$ To derive these, we introduce a nonrotating planetocentric
 Cartesian coordinate system $\,(\hat{\bf x},\,\hat{\bf y},\,\hat{\bf z})\,
 $ with $\,\hat{\bf z}\,$ pointing northward along the planet
 spin axis, and with $\,\hat{\bf x}\,$, $\,\hat{\bf y}\,$ lying within
 the equatorial plane. (In our model, we neglect the planetary precession,
 nutation, Chandler wobble, or the polar wander, so this system may be
 referred to as quasi-inertial.) The satellite's longitude of the node may
 be reckoned, say, from the axis $\,\hat{\bf x}\,$.) In this coordinate
 system, the planet's angular spin rate is
 \ba
 \omegabold_p\,=\,\omega_p\;\hat{\bf z}\;\;\;,
 \label{A18}
 \ea
 while the satellite position
 \ba
 \erbold\;=\;x~\hat{\bf x}~+~y~\hat{\bf y}~+~z~\hat{\bf z}
 \label{A19}
 \ea
 has components:
 \ba
 x\;=\;a\;\frac{1\;-\;e^2}{1\;+\;e\;\cos \nu}\;\;\left[\;
 \cos \Omega\;\cos \left(\omega\;+\;\nu  \right)\;-\;\sin
 \Omega\;\sin \left( \omega\;+\;\nu \right)\;\cos \inc
 \;\right] \;\;,
 \label{A20}
 \ea
 \ba
 y\;=\;a\;\frac{1\;-\;e^2}{1\;+\;e\;\cos \nu}\;\; \left[ \;
 \sin \Omega\;\cos \left(\omega\;+\;\nu  \right)\;+\;\cos
 \Omega\;\sin \left( \omega\;+\;\nu \right)\;\cos \inc
 \; \right] \;\;,
 \label{A21}
 \ea
 \ba
 z\;=\;a\;\frac{1\;-\;e^2}{1\;+\;e\;\cos \nu}\;\;\sin \left( \omega\;+\;
 \nu \right)\;\sin\inc~~\;\;.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 \label{A22}
 \ea
 Formulae (\ref{A18} - \ref{A19}) entail:
 \ba
 \omegabold_p\,\times\,\erbold~=~-~\omega_p~y~\hat{\bf x}~+~\omega_p~x~
 \hat{\bf y}~~~,
 \label{A}
 \ea
 whence we get the desired expression for $\,\left(\,\omegabold_p\,\times\,
 \erbold\,\right)^2\,$:
 \ba
 \left(\,\omegabold_p\,\times\,\erbold\,\right)^2\;=\;\omega_p^2\;\left(\,
 x^2\,+\,y^2\,\right)\;=\;\omega_p^2\;\left(\,r^2\;-\;z^2\,\right)\;=\;
 \omega_p^2\;r^2\;\left(\,1\;-\;\sin^2\inc\;\sin^2(\omega\,+\,\nu)\,\right)
 \;\;.~~~
 \label{B}
 \ea
 The satellite velocity
 \ba
 \vbold\;=\;v_x~\hat{\bf x}~+~v_y~\hat{\bf y}~+~v_z~\hat{\bf z}
 \label{V}
 \ea
 has Cartesian components
 \ba
 \nonumber
 v_x\;=\;\frac{n\;a}{\sqrt{1\;-\;e^2}}\;\left[\;
 -\;\cos \Omega\;\sin (\omega\;+\;\nu)\;-\;\sin
 \Omega\;\cos(\omega\;+\;\nu)\;\cos \inc\;+ \right.\\
 \label{A23}\\
 \nonumber
 \left. e\;
 (\;-\;\cos \Omega\;\sin \omega\;-\;\sin \Omega\;\cos \omega\;\cos \inc )
 \;\right]\;\;,
 \ea
 \ba
 \nonumber
 v_y\;=\;\frac{n\;a}{\sqrt{1\;-\;e^2}}\;\left[\;
 -\;\sin \Omega\;\sin (\omega\;+\;\nu)\;+\;\cos
 \Omega\;\cos(\omega\;+\;\nu)\;\cos \inc\;+ \right.\\
 \label{A24}\\
 \nonumber
 \left. e\;
 (\;-\;\sin \Omega\;\sin \omega\;+\;\cos \Omega\;\cos \omega\;\cos \inc )
 \;\right]\;\;,
 \ea
 \ba
 v_z\;=\;\frac{n\;a}{\sqrt{1\;-\;e^2}}\;\sin \inc\;\left[\;\cos (\omega\;+
 \;\nu)\;+\;e\;\cos\omega\;\right]\;\;\;,~~~~~~~~~~~~~~~~~~~~~~~~
 \label{A25}
 \ea
 Together, (\ref{A}) and (\ref{A23} - \ref{A25}) will yield the needed
 expression for $\,\;-\,2~\,\left(\,\omegabold_p\times\erbold\,\right)\cdot
 \vbold\,\;$:
 \ba
 -~2~\left(\,\omegabold_p\times\erbold\,\right)\cdot\vbold~=~-~2\;\omega_p
 \;a^2\;n\;\sqrt{1\,-\,e^2\,}\;\cos\inc
 \label{C}
 \ea
 Finally, recall that the planetocentric distance $\,r\,$ to the satellite
 is
 \ba
 r\;=\;a\;\frac{1\,-\,e^2}{1\,+\,e\;\cos\nu}\;\;\;,
 \label{E}
 \ea
 wherefrom the following auxiliary formulae ensue:
 \ba
 \frac{-~2~\left(\,\omegabold_p\times\erbold\,\right)\cdot\vbold}{r^2}\;=\;
 -\;2\;n\;\omega_p\;\frac{\left(1\,+\,e\;\cos\nu\right)^{2}}{\left(1\,-\,
 e^2\right)^{3/2}}\;\cos\inc\;=\;-\;2\;\omega_p\;\dot{\nu}\;\cos\inc\;\;\;
 \label{F}
 \ea
 and
 \ba
 \frac{v_r^2}{r^2}\,=\,\left(\frac{n\;a\;e}{\sqrt{1\,-\,e^2}}\;\sin\nu
 \right)^2\,\left(\frac{a\,\left(1-e^2\right)}{1+e\;\cos\nu}\right)^{-2}\,
 =\,n^2\,e^2\,(1-e^2)^{\textstyle{^{-3}}}\sin^2\nu\;(1\,+\,e\,\cos\nu)^{
 \textstyle{^{2}}}\;\;\;.~~~
 \label{G}
 \ea
 Plugging of (\ref{O}), (\ref{B}), and (\ref{C} - \ref{E}) into (\ref{D})
 then leads to:
 \ba
 \nonumber
 \frac{\textstyle 1}{\textstyle r^2}\,|\,\omegabold_p\times\erbold\;-\;
 \vbold\,|^2\;=\;\frac{\textstyle 1}{\textstyle r^2}~\left[\,\omega_p^2\;
 r^2\;(\,1\;-\;\sin^2\inc\;\sin^2(\omega\,+\,\nu)\,)~+\;r^{2}\,\dot{\nu}^{
 \,2}\;+\;v_r^2\;-~2~\left(\,\omegabold_p\times\erbold\,\right)\cdot\vbold
 \;\right]
 \ea
 \ba
 =~\,\omega_p^2~\cos^2\inc~+~\omega_p^2~\sin^2\inc~\cos^2(\omega\,+\,\nu)~
 +~\dot{\nu}^{\,2}\;+\;\left(\frac{v_r}{r}\right)^2\;+\;\frac{-\;2~\left(\,
 \omegabold_p\times\erbold\,\right)\cdot\vbold}{r^2}\;~~,~~~~~~~~
 \label{H}
 \ea
 while the subsequent insertion of (\ref{F} - \ref{G}) will bring us to
\ba
 \nonumber
 \frac{1}{r^2}\,|\,\omegabold_p\times\erbold\,-\,\vbold\,|^2\,=\,\omega_p^2
 ~\cos^2\inc\,+\,\omega_p^2~\sin^2\inc~\cos^2(\omega+\nu)\,+\,\dot{\nu}^{\,
 2}\,+\,\left(\frac{v_r}{r}\right)^2\,-\,2\,\omega_p\,\dot{\nu}\;\cos\inc~~
 ~~
 \ea
 \ba
 ~~~~~~~~~=\;\left(\,\omega_p\;\cos\inc\;-\;
 \dot{\nu}\,\right)^2\;+\;\omega_p^2\;\sin^2\inc\;\cos^2(\omega\,+\,\nu)\;
 +\;\left(\frac{v_r}{r} \right)^2\;\;.
 \label{}
 \ea
 This proves formula (\ref{13}).

 One can introduce the mean principal frequency by averaging $\,\chi\,$
 over one revolution of the satellite. Applying the
 averaging rule $\;(2\pi)^{-1}\int_o^{2\pi}dM\,=\, (2\pi)^{-1}\,(1-e^2)^{
 3/2}\int_o^{2\pi}(1+\,e\,\cos \nu)^{-2} d\nu\,$ to the above expression
 for $\,\chi\,$, we arrive at
 \ba
 \langle\chi\rangle~=~2\;\sqrt{\,(n\;-\;\omega_p\;\cos\inc)^2\;+\;\frac{1}{
 2}\;\omega_p^2\;\sin^2\inc\;+\;\frac{5}{2}\;n^2\;e^2\;}\;+\;O(\sin^4\inc)
 \;+\;O(e^2\,\sin^2\inc)\;+\;O(e^4)\;~~.~~~~
 \label{}
 \ea

 % ~\\
 ~\\
 ~\\
 \noindent
 {\textbf{\large{A.2.~~~The quality factor and the angular lag. }}}\\
 ~\\
 The goal of this section is threefold. First, we wish to remind to the reader why, in the first
 approximation, the quality factor $\,Q\,$ is inversely proportional to the phase lag. Second,
 we shall remind why the phase lag is twice the
 geometric lag angle, as in formula (\ref{16}) above. Once again, while a comprehensive
 mathematical derivation of this fact can be found elsewhere (see the unnumbered formula
 between equations (29) and (30) on page 673 in Kaula (1964)), our goal here is to illustrate
 this, very counterintuitive result, by using the simplest setting. Third, we shall explain why
 in general (i.e., for nonequatorial noncircular orbits) the $\,Q\,$ factor is interconnected
 not just with the ``tangential lag" (the subtended geometrical angle $\,\delta_1\,$ on Fig. 1)
 but with the total lag $\,\delta\,$ defined as $\,\sqrt{(\mbox{{tangential~lag}})^{\textstyle{
 ^2}}+\,(\mbox{radial~lag})^{\textstyle{^2}}\,}\,$. (While the ``tangential" lag $\,\delta_1\,$
 emerges because the position of the bulge lags, the ``radial" lag comes into play due to the
 fact that also the height of the bulge lags.)\\
 ~\\
 ~\\
 \noindent
 {\textbf{{A.2.1~~The case of a circular equatorial orbit.}}}\\
  ~\\
 We shall begin with the simple case of an equatorial moon on a circular orbit. At
 each point of the planet, the tidal potential produced by this moon will read
  \ba
  W\;=\;W_o\;\cos \chi t\;\;\;,
  \label{A3}
  \label{469}
  \ea
 the instantaneous frequency being given by
 % in neglect of the $\,O(e^2)\,$ terms,
 % \ba
 % \chi\,\approx\,2~\sqrt{~\,\left[\;n\,\left(1\,+\,2\,e\;\cos\nu\right)\,-\,\omega_p\,\cos
 % \inc\,\right]^{\textstyle{^2}}\,+\;\omega_p^2\;\sin^2\inc\;\cos^2(\omega+\nu)}\;~~,\;\;\;\;
 % \label{}
 % \ea
 % or in the most rough approximation, with $\;O(e)\;$ and $\;O(\inc^2)\;$ neglected:
 \ba
 \chi\,=\,2~|n\;-\;\omega_p|~~~.~~~
 \label{A3}
 \label{470}
 \ea
 Let $\,\mbox{g}\,$ denote the free-fall acceleration. An element of the planet's volume
 lying beneath the satellite's trajectory will then experience a vertical elevation of
 \ba
 \zeta\;=\;\frac{W_o}{\mbox{g}}\;\cos (\chi t\;-\;2\delta_{_1})\;\;\;.
 \label{A4}
 \label{471}
 \ea
 Accordingly, the vertical velocity of this element of the planet's volume will amount to
  \ba
 u\;=\;\dot{\zeta}\;=\;-\;\chi\;\frac{W_o}{\mbox{g}}\;\sin (\chi t
 \;-\;2\;\delta_{_1})\;=\;-\;\chi\;\frac{W_o}{\mbox{g}}\;\left(\sin \chi
 t\;\cos 2\delta_{_1}\;-\;\cos \chi t\; \sin 2\delta_{_1}\right)\;\;.\;\;
 \label{A5}
 \label{472}
 \ea
 The expression for the velocity has such a simple form because in this case the instantaneous
 frequency $\,\chi\,$ is constant.\footnote{~Equally simple would be a hypothetical case
 of an inclined circular orbit about a slowly spinning primary ($\,\omega_p\,\ll\,n\,$).
  Here the frequency (\ref{418}) would, for an
 arbitrary inclination, be approximated simply with $\,n\,$, i.e, with a constant. Hence
 all derivations presented in this subsection would remain in force.}
 % In (\ref{A4} - \ref{A5}), the factor of two accompanies the geometric
 % lag for the same reason why it accompanies the frequency.
 The satellite generates two bulges -- on the facing and opposite sides of the planet -- so each
 point of the surface is uplifted twice through a cycle. This entails the factor of two in the
 expressions (\ref{470}) for the frequency. The phase in (\ref{A4}), too, is doubled,
 though the necessity of this is less evident.\footnote{~Let $\,x\,$ signify a position along
 the equatorial circumference of the planet. In the absence of lag, the radial elevation at a
 point $\,x\,$ would be:
 \ba
 \nonumber
 \zeta\;=\;\frac{W_o}{\mbox{g}}\;\cos
 k(x\;-\;v\;t)\;\;\;,\;\;\;\;\;\;v\,=\,R\,\sigma\;\;\;,
 \ea
 $v\,$ being the velocity of the satellite's projection on the ground,
 $\,R\,$ being the planet's radius, and
 $\sigma$ being simply $\,|n-\omega_p|\,$ because we are dealing with a
 circular equatorial orbit. The value of $\,k\,$ must satisfy
 \ba
 \nonumber
 k\;v\;=\;2\;\sigma\;\;\;,\;\mbox{i.e.,}\;\;\;\;k\;v\;=\;\chi\;\;\;,
 \ea
 to make sure that at each $\,x\,$ the ground elevates twice per
 an orbital cycle. The above two formulae yield:
 \ba
 \nonumber
 k\;R\;=\;2\;\;\;.
 \ea
 In the presence of lag, all above stays in force, except that the
 formula for radial elevation will read:
 \ba
 \nonumber
 \zeta\;=\;\frac{W_o}{\mbox{g}}\;\cos k(x\;-\;v\;t\;+\;D)\;\;\;,\;\;\;\;
 \mbox{where}\;\;\;\; D\;=\;R\;\delta_{_1}\;\;\;,
 \ea
 $D\,$ being the linear lag, and $\,\delta_{_1}\,$ being the angular
 one. Since $\,k\,v\,=\,2\,$, we get:
 \ba
 \nonumber
 \cos \left[\,k\;(x\;-\;v\;t\;+\;R\;\delta_{_1})\,\right]\;=\;\cos \left[\,k\;x
 \;-\;k\;v\;t\;+\;k\;R\;\delta\,\right]\;=\;\cos
 \left[\,k\;x\;-\;(k\;v\;t\;-\;2\;\delta_{_1})\,\right]\;\;\;,
 \ea
 so that, at some fixed point (say, at $\,x\,=\,0\,$) the elevation becomes:
 \ba
 \nonumber
 \zeta(t)\;=\;\frac{W_o}{\mbox{g}}\;\cos (k\;v\;t\;-\;2\;\delta_{_1})\;\;\;.
 \ea
 We see that, while the geometric lag is $\;\delta_{_1}\,$, the phase lag is
 double thereof.}

 The energy dissipated over a time cycle $\,T\,=\,2\pi/(\chi)\,$, per
 unit mass, will, in neglect of horizontal displacements,\footnote{~In Appendix
 A.2.3 below, it will be explained why the horizontal displacements may be neglected.} be
 \ba
 \nonumber
 \Delta E_{_{cycle}} &=& \int^{T}_{0}u\left(-\,\frac{\partial W}{
 \partial r}\right)dt=
 % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 % ~~~~~~~\\
 % \nonumber\\
 % \nonumber\\
 % \nonumber
 \,-\left(-\,\chi \frac{W_o}{\mbox{g}}\right)\,\frac{\partial W_o}{
 \partial r}\int^{t=T}_{t=0}\cos \chi t\,\left(\sin \chi t\,
 \cos 2\delta_{_1}\,-\,\cos \chi t\, \sin 2\delta_{_1}\right)dt\\
 \nonumber\\
 \nonumber\\
 % \nonumber
 &=&\,-\;\chi\;\frac{W_o}{\mbox{g}}\;\frac{\partial W_o}{\partial r}\;\sin 2\delta_{_1}\;
 \frac{1}{\chi}\;\int^{\chi t\,=\,2\pi}_{\chi t\,=\,0}\;\cos^2 \chi t\;\;d(\chi t)\;=\;-\;
 \frac{W_o}{\mbox{g}}\;\frac{\partial W_o}{\partial r}\;\pi\;\sin 2\delta_{_1}\;\;,\;\;\;
 ~~~~~~~~~~~~~~~~~~~~
 \label{A6}
 \label{}
 \ea
 while the peak energy stored in the system during the cycle will read:
 \ba
 \nonumber
 E_{_{peak}}&=&\int^{T/4}_{0} u \left(-\,\frac{\partial W}{
 \partial r}\right)dt =
 % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 % ~~~~~~~~\\
 % \nonumber\\
 % \nonumber\\
 % \nonumber
 \,-\left(-\,\chi\,\frac{W_o}{\mbox{g}}\right)\frac{\partial W_o
 }{\partial r}\int^{t=T/4}_{t=0}\cos \chi t\,\left(\sin
 \chi t\,\cos 2\delta_{_1}\,-\,\cos \chi t\,\sin 2\delta_{_1}\right)dt\\
 \nonumber\\
 \nonumber\\
 \nonumber
 &=&\;2\;\sigma\;\frac{W_o}{\mbox{g}}\;\frac{\partial W_o}{\partial r}\;\left[\;
 \frac{\cos 2\delta_{_1}}{\chi}\;\int^{\chi t\,=\,\pi/2}_{\chi t\,=\,0}
 \;\cos \chi t\;\sin \chi t\;\;d(\chi t)\;-\;\frac{\sin 2\delta_{_1}
 }{\chi}\;\int^{\chi t\,=\,\pi/2}_{\chi t\,=\,0}\;\cos^2 \chi t
 \;\;d(\chi t)\;\right]\\
 \nonumber\\
 \nonumber\\
 &=&\;\frac{W_o}{\mbox{g}}\;\frac{\partial W_o}{\partial r}\;\left[\;\frac{1}{2}
 \;\cos 2\delta_{_1}\;-\;\frac{\pi}{4}\;\sin 2\delta_{_1}\;\right]~~~~~~~~~~~~~~~~~~~
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 \label{A7}
 \label{}
 \ea\\
 whence
 \ba
 Q^{-1}\;=\;\frac{-\;\Delta E_{_{cycle}}}{2\,\pi\,E_{_{peak}}}\;=\;\frac{1}{2\,\pi}\;\,
 \frac{\pi\;\sin 2\delta_{_1}}{~\frac{\textstyle 1}{\textstyle 2}\;\cos 2\delta_{_1}\;-\;
 \frac{\textstyle\pi}{\textstyle 4}\;\sin 2 \delta_{_1}}\;\approx\;\tan 2 \delta_{_1}\;\;\;.
 \label{A8}
 \label{}
 \ea
 ~\\
 ~\\
{\textbf{{A.2.2~~The general case of
 %  \\  $\;{\left.\;\right.}^{\left.\;\right.}\;\,$~~~~~~
 a nonzero eccentricity and inclination.}}}\\
 ~\\
 In this subsection, we shall switch from the specific case of an equatorial circular
 orbit to the generic case of an inclined elongated one. If we want to keep using the
 ansatz (\ref{A3}), we must now accept that the tidal frequency $\,\chi\,$ is not a
 constant but a variable quantity $\,\chi(\nu)\,$ or, if we choose to parameterise
 the motion with time instead of the true anomaly, $\,\chi(t)\,$.

 Now we have the following expressions for the tidal potential and the (delayed by $\,
 \Delta t\,$) vertical elevation:
  \ba
  W\;=\;W_o\;\cos \left[\,\chi(t)\;t\,\right]\;\;\;,
  \label{}
  \label{}
  \ea
 \ba
 \zeta\;=\;\frac{W_o}{\mbox{g}}\;\cos\left\{\,\chi(t)\;\left[\,t\;-\;\Delta t\,
 \right]\,\right\}\;\;\;.
 \label{}
 \label{}
 \ea
 Back in (\ref{471}), the quantity $\,\chi\,\Delta t\,$ had an evident meaning of the doubled
 ``tangential lag" angle, so we preferred to write it as $\,2\delta_{\textstyle{_1}}\,$. Here,
 though, $\,\chi(t)\,\Delta t\,$ lacks such a straightforward meaning, and therefore we would
 prefer to deal with $\,\Delta t\,$.

  Generalising (\ref{A6}), we must keep in mind that $\,\chi\,$ is no longer constant:
 \ba
 \nonumber
 \Delta E_{_{cycle}} &=& \int^{t\,\chi(t)\,=\,2\pi}_{t\,\chi(t)\,=\,0}
 \left(-\,\frac{\partial W}{\partial r}\right)\;d\zeta=
 \,\frac{W_o}{\mbox{g}}\;\left(-\,\frac{\partial W_o}{\partial r}\right)\int^{t\,\chi(t)\,=
 \,2\pi}_{t\,\chi(t)\,=\,0}\;\cos\left[\chi(t)~t\right]~d\,\cos\left[\chi(t)\,t\,-\,\chi(t)
 \,\Delta t\,\right]
 \ea
 \ba
 \nonumber
 =\;-~\frac{W_o}{\mbox{g}}~\frac{\partial W_o}{\partial r}\int^{t\,\chi(t)\,=\,2\pi}_{t
 \,\chi(t)\,=\,0}\cos\left[\,\chi(t)\,t\,\right]~\frac{d\,\cos\left[\,\chi(t)\,t\,-\,
 \chi(t)\,\Delta t\,\right]}{d\left[\,\chi(t)\;t\,-\,
 \chi(t)\,\Delta t\,\right]}\;\,\frac{d\,\left[\,\chi(t)\,t\,-\,
 \chi(t)\,\Delta t\,\right]}{d\left[\,\chi(t)\;t\,\right]}\;\,d\left[\,\chi(t)\,t\,\right]
 \;\;\;\;\;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
 \nonumber\\
 \nonumber\\
 \nonumber
 =\;\frac{W_o}{\mbox{g}}\,\frac{\partial W_o}{\partial r}\int^{t\,\chi(t)\,=\,2\pi}_{t\,
 \chi(t)\,=\,0}\left(1\,-\,\frac{d\left[\,\chi(t)\,\Delta t\,\right]}{d\,t}\,\frac{d\,t}{d
 \left[\,\chi(t)\,t\,\right]}\right)\,\cos\left[\,\chi(t)\,t\,\right]~\,\sin\left[\,\chi(t
 )\,t\,-\,\chi(t)\,\Delta t\,\right]\,d\left[\,\chi(t)\;t\,\right]\;\;\;\;~~~~~~~~~~~~~~~~
 ~~~~~~~~~~
 \ea
 \ba
=~\frac{W_o}{\mbox{g}}~\frac{\partial W_o}{\partial r}~\int^{t\,\chi
 (t)\,=\,2\pi}_{t\,\chi(t)\,=\,0}\left[\,1\,-\,O(\dot{\chi}\Delta
 t/\chi)\,\right]\;
 \left(\,\cos\chi t~\cos\chi\Delta t\,-\,\sin\chi t~\sin
 \chi\Delta t\,\right)~\sin\chi t\; d\left(\chi t\right)\;\;.\;\;\;~~~~~~~~
 \label{VH}
 \ea
 Let us now recall that $\,\chi(t)\,$ is a slow function of time, i.e., that
 $\,\dot{\chi}/\chi^2\,$ can be treated as a small dimensionless parameter.\footnote{~It
 is straightforward from (\ref{418}) that $\,\dot{\chi}/\chi^2\,=\,O\left(\,e(n/\omega_p)^2
 \,\right)\,+\,O\left(\,(n/\omega_p)\,\sin^2\inc\,\right)\;$.} Also keep in mind that $\,
 \Delta t\,$, too, is small (i.e., that $\,\chi\,\Delta t\,$ too can be treated as a small
 dimensionless parameter). It will then be legitimate to neglect, in the square brackets,
 the term $\,O(\dot{\chi}\Delta t/\chi)\,=\,O\left(\,(\chi\,\Delta t)({\dot{\chi}}/{\chi^2
 })\,\right)\,$ whose order of smallness is the same as that of the product of these two
 small parameters. For the same reason, in the right-hand side of (\ref{VH}) one can
 approximate $\,\cos\chi\Delta t\,$ with unity and can treat $\,\sin\chi\Delta t\,$ as a
 constant. Thence we shall obtain an analogue to
 (\ref{A6}):
 \ba
 \Delta E_{_{cycle}}\;=\;-\;\frac{W_o}{\mbox{g}}\;\frac{\partial W_o}{\partial r}\;\pi\;
 \sin\left[\chi(t)\,\Delta t\right]\;\;.\;\;\;
 \label{I}
 \ea
 In the same approximation of $\,\chi(t)\,$ being a slow function, we can get an
 analogue to (\ref{A7}):
 \ba
 \Delta E_{_{peak}}\;=\;\frac{W_o}{\mbox{g}}\;\frac{\partial W_o}{\partial r}\;\left[\;
 \frac{1}{2}\;\cos \left[\chi(t)\,\Delta t\right]\;-\;\frac{\pi}{4}\;\sin \left[\chi(t)\,
 \Delta t\right]\;\right]~~~.~~~~~~~~~~~~~~~~~~~~~~~~
 \label{II}
 \ea
 Both in the latter and in the former formulae integration over the entire
 volume of the body is implied.\footnote{~Just as in the previous section, here we have in fact performed all calculations for an elementary volume, and did not bother, in (\ref{I}) or (\ref{II}), to integrate over the entire volume. We
 afford this omission, because this integration would not affect the $\,\delta_1-$dependent multipliers and therefore would yield the same functional form for the dependence $\,Q(\chi(t)\,\Delta )\,$. The reason for this is explained in Appendix A.3.} From (\ref{I}) and (\ref{II}), the following generalisation of (\ref{A8}) ensues:
 \ba
 Q^{-1}\;=\;\frac{-\;\Delta E_{_{cycle}}}{2\,\pi\,E_{_{peak}}}\;=\;\frac{1}{2\,\pi}\;\,
 \frac{\pi\;\sin \left[\chi(t)\,\Delta t\right]}{~\frac{\textstyle 1}{\textstyle 2}\;\cos
 \left[\chi(t)\,\Delta t\right]\;-\;\frac{\textstyle\pi}{\textstyle 4}\;\sin \left[\chi(t)
 \,\Delta t\right]}\;\approx\;\tan \left[\chi(t)\,\Delta t\right]\;\;\;.
 \label{L}
 \ea
 So in the general case the factor $\,Q\,$ is connected not to the subtended angle
 but to $\,\chi(t)\,\Delta t/2\,$.
 ~\\
 ~\\
 {\textbf{{A.2.3~~The final touch}}}\\

 In our above derivation of the interrelation between $\,Q\,$ and $\,\delta\,$, we
 greatly simplified the situation, taking into account only the vertical displacement
 of the planetary surface, in response to the moon's pull. Here we shall demonstrate
 that this approximation is legitimate, at least in the case when the planet is
 modelled with an incompressible (though not necessarily homogeneous) medium.
 Thence we shall see that the calculations offered in a simplified form in A.2.1 and
 A.2.2 can be expressed in the fancy language of continuum mechanics.

 As a starting point, recall that the tidal attenuation rate within the planet is
 well approximated with the work performed on it by the satellite:\footnote{~A small
 share of this work is continuously being spent for decelerating the Earth rotation.}

 The attenuation rate within a tidally distorted planet is:
 \ba
 \dot{E}\;=\;-\;\int\,\rho\;\Vbold\;\nabla W\;d^3x
 \label{A9}
 \ea
 where $\,\rho\,,\;\Vbold\,$ and $\,W\,$ are the density, velocity, and tidal potential
 inside the planet. To simplify this expression, we shall, following Goldreich
 (1963), employ the equality
 \ba
 \rho\,\Vbold\,\nabla W\,=\,\nabla\,\cdot\,(\rho\;\Vbold\;W)\;-\;W\;\Vbold\cdot
 \nabla\rho\;-\;W\;\nabla\,\cdot\,(\rho\;\Vbold)\,=\,\nabla\,\cdot\,(\rho\;
 \Vbold\;\nabla W)\;-\;W\;\Vbold\cdot\nabla\rho\;+\;W\;
 \frac{\partial\rho}{\partial t}\;\;.\;\;\;
 \label{}
 \ea
 For a homogeneous and incompressible primary, both the
 $\,\Vbold\;W\nabla \rho\,$ and $\,\partial\rho/\partial t\,$ terms are nil, whence
 \ba
 \dot{E}\;=\;-\;\int\rho\;W\;\Vbold\,\cdot\,{\vec{\bf{n}}}\;d^3x\;\;\;,
 \label{}
 \ea
 ${\vec{\bf{n}}}\,$ being the outward normal to the surface of the planet. We
 immediately see that, within the hydrodynamical model, it is only the radial
 elevation rate that matters.

 Now write the potential as $\;W\,=\,W_o\,\cos(\chi(t)\,t)\;$, where $\,\chi(t)\,$
 is a slow (in the sense explained in A.2.3) function of time. Since the response
 is delayed by $\, \Delta t\,$, the surface-inequality rate will evolve as $\;
 \Vbold{\vec{\bf{n}}}\,\sim\,\sin\left\{\chi(t)\,[\,t\,-\,\Delta t\,]\,\right\}\;$.
 All the rest is as in subsection A.2.2 above.\\
 ~\\
 ~\\
 \noindent
 {{\textbf{\large{A.3.~~~Our calculations in the light of Zschau's theory}}}}\\
 ~\\
 Amendments to the conventional theory of bodily tides were offered by
 Zschau (1978) whose  starting point was that attenuation of land tides, in distinction
 from earthquakes, takes place both due to rigidity and self-gravity of the body.

 For this reason, Zschau introduced not one but two different lags. The first was the
 ordinary lag $\,\delta\,$ depicting the delay between the bulge on the planet's surface
 and the position of the tide-raising satellite. (Zschau called this quantity $\,\alpha_H
 \,$.) The second was the (neglected in the literature thitherto) lag of the secondary
 potential emerging due to the planet's deformation. (Zschau called it $\,\alpha_K\,$.)
 These two lags are, generally, different from one another, and this difference reflects
 the fact that the phase delay between the tidal stress and strain varies over the depth
 of the planet and is, generally, different from that on the planet's surface.

 The traditional approach (in whose spirit our Appendix 2 above is written) implies that,
 as long the relation $\,Q^{-1}\,=\,\tan(2\delta)\,$ is proven for an elementary volume,
 it can be trivially applied also to the entire volume of the body (provided the body is
 assumed homogeneous). In the theory of Zschau, this generalisation is no longer trivial,
 {\emph{even if the body is homogeneous.}} This circumstance entails, for the overall
 quality factor of the planet, an expression more complicated than $\,Q^{-1}\,=\,\tan 2
 \delta\,$. According to Zschau (1978), the inverse quality factor should read as
 \ba
 Q^{-1}\,=\,\frac{1}{1\,+\,k\,-\,h}\;\left[\,(1\,+\,k)\,\tan 2\alpha_H\;-\;k\,
 \tan 2\alpha_K\,\right]\;\;\;,
 \label{Zschau}
 \ea
 $k\,$ and $\,h\,$ being some Love numbers. Zschau emphasises, though, that the second
 term in the square brackets becomes irrelevant when $\,k\,\ll\,1$, i.e., when the
 self-gravity may be neglected.

 Fortunately, it turns out that the self-gravity does not bring much into the overall
 dissipation process, because the mantle damps tidal displacements, much like the seismic
 waves, mainly due to its rigidity. If we model the body with a homogeneous sphere of
 density $\,\rho\,$, rigidity $\,\mu\,$, surface gravity $\,\mbox{g}\,$, and radius
 $\,R\,$, then the estimate presented in subsection 4.4.1 above tells us that
 self-gravitation contributes, at most, several percent into the overall energy
 dissipation due to tides.

 This fact entails two important consequences. One is that we may extend to the tidal
 $\,Q\,$ the frequency-dependence law measured for the seismic quality factor. The other
 consequence is that the extra term $\;-k\,\tan2\alpha_K\;$ in Zschau's expression for
 $\,Q^{-1}\,$ may be ignored (unless we are talking about some hypothetical terrestrial
 exoplanet several times larger than Mars). As soon as this term is omitted,
 \ba
 \nonumber
 Q^{-1}\;=\;\frac{1}{1\,-\,h}\;\tan 2\delta\;\;\;
 \ea
 where we are using the conventional notation $\,\delta\,$ instead of Zschau's $\,\alpha_H
 \,$. Effective renormalisation of $\,Q\,$ by a factor of $\,1/(1-h)\,$ is taking place
 due to gravitational prestressing, and is of a marginal importance to our study.
 We see that for the terrestrial planets (and, the more so, for small bodies) one
 still may use the conventional simple relation between the energy dissipated in one
 cycle divided by the peak energy and the tidal lag.


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\end{thebibliography}{}

\end{document}


 Variable stressing of a material unavoidably entails internal friction
 and, therefore, inelastic dissipation of the stress energy $\,E\,$. The
 dissipation rate in the mantle depends upon numerous factors -- the
 physical and chemical constituency of the minerals, the frequency of
 vibrations, the temperature, the pressure. Fortunately, certain features
 of this dependence are universal, because they can be traced down to
 basic physical mechanisms.

 As the planet goes through two cycles of flexure in one period of
 satellite's revolution, the actual frequency of the principal bulge is
 twice higher:



 \ba
 \fbold~=~r^{\textstyle{^{\alpha+1}}}\;{\cal B}\;\,
 \frac{\omegabold_p \times \erbold\,-\,\vbold~~~}{~|\,\omegabold_p\times
 \erbold\,-\,\vbold\,|^{\textstyle{^{\alpha+1}}}}
 \label{25}
 \ea
 The multiplier $\,r^{\textstyle{^{\alpha+1}}}\,$ enters the above
 expression to keep $\,\it f\,$ linear in $\,r\,$, so that $\,Q\,$ comes
 out $\,r$-independent in the formulae below. The constant $\,{\cal B}\,$
 may, for convenience, be expressed as $\,2^{\textstyle{^{-\alpha}}}\,{
 \cal E}\,$, where $\,{\cal E}\,$ is just another constant. Thence,
 \ba
 \fbold\;=\;r\;{\cal E}\;\chi^{{{-\alpha}}}\,\;\frac{~\omegabold_p\times
 \erbold\,-\,\vbold}{~|\,\omegabold_p \times \erbold\,-\,\vbold\,|}\;\;\;,
 \label{26}
 \ea
 the principal tidal frequency being given by
 \ba
 \chi\,=\,2\;\sqrt{(n\,-\,\omega_p\,\cos\inc)^2\,+\,\frac{\textstyle 1}{
 \textstyle 2}\;\,\omega_p^2\;\sin^2\inc\;+\;\frac{5}{2}\,n^2\,e^2\,~}\;+\;
 O(\sin^4\inc)\;+\;O(e^2\,\sin^2 \inc)\;+\;O(e^4)~~.\;\;\;\;
 \label{27}
 \ea
 In the leading order over the eccentricity and inclination,
 $\,\chi\,\approx\,2\,|\,\omega\,-\,n\,|\,$, with $\,n\,$ being the
 satellite's mean motion.

 Now, this is what we have for the geometric lag:
 \ba
 \delta\;\equiv\;\frac{f}{r}\;=\;{\cal E}\;\chi^{-\alpha}\;\;\;.
 \label{28}
 \ea
 which is exactly what is required by (\ref{13} - \ref{16}).

 In the case of $\,\alpha\,=\,0\,$, the above machinery simplifies to the
 MacDonald-Kaula-Gerstenkorn theory, with a constant geometric lag $\,
 \delta\,=\,{\cal B}\,=\,{\cal E}\,$, while for $\,\alpha\,=\,-\,1\,$ we
 arrive to the Singer-Mignard theory, with a constant time lag $\,\Delta t
 \,=\,{\cal B}\,=\,{\cal E}/2\,$. (For different values of $\,\alpha \,$,
 the constant $\,{\cal E}\,$ has different dimensions.)

 \noindent {{\textbf{\large{A.2.~~~Averaging of the expression for $\,
 \frac{\textstyle 1}{\textstyle r}\,|\,\omegabold_p\times\erbold\;-\;\vbold
 \,|\,$}}}}\\
 ~\\
 Our next step is to find
 \ba
 \sigma\,\equiv\,\langle\;\, \sqrt{(\dot{\nu}\,-\,\omega_p\,\cos\inc)^2\,
 +\,\omega_p^2\;\sin^2\inc\;\cos^2(\omega+\nu)\,+\,v_r^2\,r^{-2}\;\,}\;\;
 \rangle\;\;\;,
 % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
 % \nonumber\\
 % \nonumber\\
 % =\,\sqrt{(\dot{\nu}\,-\,\omega_p\,\cos\inc)^2\,+\,\omega_p^2\,\sin^2\inc
 % \;+\;\frac{1}{2}\,n^2\,e^2\,(1-e^2)^{\textstyle{^{-3/2}}}~}\;+\,
 % O(n^4e^4)\;\;.\;\;\;\;
 \label{sigma}
 \ea
 $\langle\;.\,.\,.\;\rangle\;$ symbolising the orbital average
 $\;(2\pi)^{-1}\int_o^{2\pi}dM\,=\,
 (2\pi)^{-1}\,(1-e^2)^{3/2}\int_o^{2\pi}(1+\,e\,\cos \nu)^{-2} d\nu\,$, and
 $\,v_r\,$ being the radial part of the satellite velocity, expressed by
 (\ref{O}). We now can decouple the expression under the square root in
 (\ref{sigma}) into a constant part, $\,A\,$, and an oscillating part, $\,B
 \,$:
 \ba
 (\dot{\nu}\,-\,\omega_p\,\cos\inc)^2\,+\,\omega_p^2\;\sin^2\inc\;\cos^2(
 \omega+\nu)\,+\,v_r^2\,r^{-2}\;=\;A\;+\;B\;\;\;,
 \label{}
 \ea
 where
 \ba
 A\;\equiv\;\,\langle\;\,\left[\;(\dot{\nu}\,-\,\omega_p\,\cos\inc)^2\,+\,
 \omega_p^2\;\sin^2\inc\;\cos^2(\omega+\nu)\,+\,v_r^2\,r^{-2}\;\right]\,\;
 \rangle
 \label{}
 \ea
 and
 \ba
 B\;\equiv\;\left[\,(\dot{\nu}\,-\,\omega_p\,\cos\inc)^2\,+\,\omega_p^2\;
 \sin^2\inc\;\cos^2(\omega+\nu)\,+\, v_r^2\,r^{-2}\;\right]\;-\;A
 \label{}
 \ea
 so that $\;\langle B\rangle\,=\,0\;$, $\;\langle A\rangle\,=\,A\;$, and
 also $\;\langle\sqrt{A\,}\rangle\,=\,\sqrt{A\,}\;$ because $\,A\,$ is a
 constant. After that, the evident equality
 \ba
 \nonumber
 \langle\;\sqrt{A\,+\;B\,}\;\rangle\;=\;\langle\;\,\sqrt{A\;}\;\,\sqrt{1\;
 +\;B/A\;}\,\;\rangle\;\,
 =\;\sqrt{A\;}\;\;\langle\;\left[\;1\;+\;\frac{1}{2}\;B/A\;-\;\frac{1}{4}\;
 \left(B/A\right)^2 \;+\;O\left(\;(B/A)^3\;\right)\;\right]\;\rangle ~~~\,
 \ea
 \ba
 \,\;
 =\;\sqrt{A\;}\;\,\left[\;1\;+\;\frac{1}{2}\;\;\langle\;{B}/{A}\;\rangle\;-
 \;\frac{1}{4}\;\langle\;\left(B/A\right)^2\;\rangle\;+\;O\left(\;(B/A)^3\;
 \right)\;\right]\;\;\;\;~~~~~~~~~~~~~~~~
 \label{}
 \ea
 can be employed in order to check if $\;\sigma\;$ defined by
 (\ref{sigma}) may be approximated simply with $\,A\,$.

From the auxiliary formulae
 \ba
 (\dot{\nu}\,-\omega_p\;\cos\inc)^2\;=\;(n\;-\;\omega_p\;\cos\inc)^2\;+\;
 (\dot{\nu}^2\,-\;n^2)\;-\;2\,(\dot{\nu}\,-\;n)\;\;\;,~~~~~~
 \label{}
 \ea
 \ba
 \langle\;\,\left(\,\dot{\nu}^2\,-\;n^2\,\right)\,\;\rangle\;=\;2\;n^2\;e^2
 \;+\;O(n^2e^4)\;\;\;,~~~~\langle\,\;\left(\,\dot{\nu}\;-\;n\,\right)\;\,
 \rangle\;=\;0\;\;\;,~~~~~~
 \label{}
 \ea
 \ba
 \omega_p^2\;\,\sin^2\inc\;\,\cos^2(\omega_p\,+\,\nu)\;=\;\frac{1}{2}\;
 \omega_p^2\;\sin^2\inc\;\,-\;\omega_p^2\;\sin^2\inc\;\,\cos 2(\omega_p\,+
 \;\nu)\;\;\;,
 \label{}
 \ea
 \ba
 \langle\;\frac{v_r^2}{r^2}\;\rangle=\,\frac{1}{2\pi}\,(1-e^2)^{3/2}
 \int_o^{2\pi}\frac{d\nu}{(1+\,e\,\cos\nu)^{2}}\,n^2e^2\,(1-e^2)^{
 \textstyle{^{-3}}}\,(1\,+\,e\,\cos\nu)^{\textstyle{^2}}\sin^2\nu\,=\,
 \frac{1}{2}\;\frac{n^2\,e^2}{\left(1\,-\,e^2\right)^{3/2}}\;\;.~~~
 \label{}
 \ea
 we see that
 \ba
 A\;=\;(n\;-\;\omega_p\;\cos\inc)^2\;+\;\frac{1}{2}\;\omega_p^2\;\sin^2\inc
 \;+\;\frac{5}{2}\;n^2\;e^2\;+\;O(e^4)\;\;\;,
 \label{}
 \ea
 while
 \ba
 B\;=\;\frac{1}{2}\;\omega_p^2\;\sin^2\inc\;\cos2\nu\;+\;O(e^2)\;\cos\nu\;
 +\;O(e^4)\;\;\;
 \label{}
 \ea
 whence
 \ba
 \langle\,B/A\,\rangle\;\;=\;O(e^4)\;\;\;,\;\;\;\;\;\;\langle\,(B/A)^2\,
 \rangle\;\;=\;O(\sin^4\inc)\;+\;O(e^2\;\sin^2\inc)\;+\;O(e^4)\;\;\;.
 \label{}
 \ea
 From the latter we can see that in all realistic situations (except,
 perhaps, near-polar orbits) $\;\sigma\;$ may well be approximated with $\,
 A\,$. Hence the tidal frequency, which is double of $\,\sigma\,$, may
 in this approximation be written down as
 \ba
 \chi~=~2\;\sqrt{\,(n\;-\;\omega_p\;\cos\inc)^2\;+\;\frac{1}{2}\;\omega_p^2
 \;\sin^2\inc\;+\;\frac{5}{2}\;n^2\;e^2\;}\;+\;O(\sin^4\inc)\;+\;O(e^2\,
 \sin^2\inc)\;+\;O(e^4)\;~~.~~~~
 \label{}
 \ea




  ~\\
 ~\\
 {\textbf{{A.2.4~~An extreme example: a radially moving tide-raising body.}}}\\
 ~\\
 As an illustrative thought-experiment, consider a radially moving body elevating a
 $\,\Delta t$-delayed tide on the planet. Let this body's planetocentric position
 evolve as $\,r_o\,+\,a\,\cos\chi t\,$. It is then straightforward from (\ref{25})
 that the potential it creates is
 \ba
 W\;=\;W_o\;\cos \chi t \;\;\;,
 \label{}
 \ea
 while the $\,\Delta t$-delayed elevation of the surface beneath the body looks as
  \ba
 \zeta\;=\;\frac{W_o}{\mbox{g}}\;\cos\left\{\,\chi(t)\;\left[\,t\;-\;\Delta t\,
 \right]\,\right\}\;\;\;.
 \label{}
 \label{}
 \ea

astro-ph0605521.extracted_bib
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