#
${{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}^{{}^{{}^{Published~{}in~{}the~{}Astrophysical~{}Journal\,,~{}Vol.~{}764\,,~{}id.~{}26\,~{}(2013)}}}}$
Tidal Friction and Tidal Lagging. Applicability Limitations of a Popular
Formula for the Tidal Torque

Michael Efroimsky
US Naval Observatory, Washington DC 20392
e-mail: michael.efroimsky @ usno.navy.mil

and

Valeri V. Makarov
US Naval Observatory, Washington DC 20392
e-mail: vvm @ usno.navy.mil

###### Abstract

Tidal torques play a key role in rotational dynamics of celestial bodies. They
govern these bodies’ tidal despinning, and also participate in the subtle
process of entrapment of these bodies into spin-orbit resonances. This makes
tidal torques directly relevant to the studies of habitability of planets and
their moons.

Our work begins with an explanation of how friction and lagging should be
built into the theory of bodily tides. Although much of this material can be
found in various publications, a short but self-consistent summary on the
topic has been lacking in the hitherto literature, and we are filling the gap.

After these preparations, we address a popular concise formula for the tidal
torque, which is often used in the literature, for planets or stars. We
explain why the derivation of this expression, offered in the paper by
Goldreich (1966; AJ 71, 1 - 7) and in the books by Kaula (1968, eqn. 4.5.29),
and Murray & Dermott (1999, eqn. 4.159), implicitly sets the time lag to be
frequency independent. Accordingly, the ensuing expression for the torque can
be applied only to bodies having a very special (and very hypothetical)
rheology which makes the time lag frequency independent, i.e, the same for all
Fourier modes in the spectrum of tide. This expression for the torque should
not be used for bodies of other rheologies. Specifically, the expression
cannot be combined with an extra assertion of the geometric lag being
constant, because at finite eccentricities the said assumption is incompatible
with the constant-time-lag condition.

##  1 Context, Motivation, and Plan

${\left.~{}~{}~{}~{}~{}~{}\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\right.}^{\mbox{\small\it
The mills of God grind slowly...}}$

Usually extremely weak, tidal interactions act upon celestial bodies for
extended spans of time (up to billions of years). Over æons, tides shape
celestial bodies’ spin modes, and govern the exchange of angular momentum. The
numerous and diverse manifestations of bodily tides range from the expected
fall of Phobos to synchronous locking of the Moon, to Mercury’s capture in the
3:2 spin-orbit resonance, to bloated hot-jupiter exoplanets in tight orbits
around their host stars, to dissipational coalescence of short-period binary
stars. This makes studies of tides essential to our understanding of the
dynamical properties and evolution of stellar systems.

While the slow work of tides is responsible for circularisation, obliquity
evolution, and synchronisation of planets and moons, the wide scope of these
dynamical phenomena is not always matched by sophistication or versatility of
the tidal models employed to describe them.

###  1.1 Requirements to a consistent theory,
and the history of simpler approaches

A bona fide theory of bodily tides implies (1) decomposition of the tide into
Fourier harmonic modes and (2) endowment of each separate Fourier mode with a
phase delay and a magnitude decrease of its own. The first part of this
programme, Fourier decomposition, was accomplished in full by Kaula (1964),
though a partial sum of the Fourier series was developed yet by Darwin (1879).
The second part of this programme, the quest for an adequate frequency
dependence of the phase lags and dynamical Love numbers, is now in progress.
While earlier attempts seldom went beyond the Maxwell model, more realistic
rheologies are now coming into use. A rheology combining the Andrade model at
higher frequencies with the Maxwell model at the lowest frequencies was
investigated by Efroimsky (2012a, 2012b).111 At lower frequencies, dissipation
is predominantly viscous, and the mantle is well approximated with the Maxwell
body. Its behaviour can be represented with a viscous damper and an elastic
spring connected in series. Experiencing the same force, these elements have
their elongations summed up. This illustrates how the total strain is
comprised by a sum of viscous and elastic contributions generated by the same
stress. At higher frequencies, the strain acquires the third component, one
intended to describe inelasticity. Inelasticity is produced by defect
unpinning, a process effective at frequencies higher than a certain threshold
(about 1 yr-1, in the case of Earth’s mantle, – see Karato & Spetzler 1990).
Hence, at frequencies above the threshold, dissipation is predominantly
inelastic, and the mantle behaves as the Andrade body. A combined rheological
model written down in Efroimsky (2012a, 2012b) embraces both frequency bands.
In Makarov et al. (2012) and Efroimsky (2012a), we mistermed the Andrade creep
as anelastic. It would be more accurate to call it inelastic, which means:
irrecoverable. (The term anelastic is applied to recoverable deformations,
like the Maxwell behaviour.) The necessity for such a combined model
originates from that fact that different physical mechanisms of friction
dominate tidal dissipation at different frequencies. Several other rheological
laws were probed by Henning et al. (2009) and Nimmo et al. (2012).

Some authors tried to sidestep a Fourier decomposition by building simpler toy
models which would preserve qualitative features of the consistent tidal
theory and, ideally, would yield some reasonable quantitative estimates. Two
radically simplistic ad hoc tools known as  the constant geometric lag model
(MacDonald 1964, Goldreich 1966, Murray & Dermott 1999) and the constant time
lag model  (Singer 1966; Mignard 1979, 1980, 1981; Heller et al. 2011; Hut
1981) are often resorted to, and are applied to rocky moons and planets and
gas giants and stars alike. Historically, both these approaches were
introduced for the ease of analytical treatment rather than on sound physical
principles. 222 Aside from its mathematical simplicity, the constant time lag
method is sometimes motivated by its analogy with the viscously damped
harmonic oscillator. This analogy, however, appeared in the literature a
posteriori, Alexander (1973) being the earliest work known to us where this
analogy was spelled out.

###  1.2 Plan

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

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

In the subsequent publication (Makarov & Efroimsky 2013), we shall explore the
constant time lag model. This is an approach implying that all the tidal
strain modes should experience the same temporal delay relative to the
appropriate modes comprising the tidal stress. While the method is often
assumed 333 It can be demonstrated that the purely viscous model implies a
frequency-independent time lag at sufficiently low frequencies only. Time lag
acquires frequency dependence at frequencies higher than
$\,G\rho^{2}R^{2}/\eta\,$, where $\,G$, $\,\rho$, $\,R$, and $\,\eta\,$ are
the Newton gravity constant, mean density, radius, and the mean viscosity of
the perturbed body. This circumstance lies outside the topic of this paper,
and we shall elaborate on it elsewhere. to work in the purely viscous limit
(which, hypothetically, may be the case of stars and gaseous planets – see Hut
1981 and Eggleton et al. 1998), there is no justification for using it for
terrestrial-type bodies such as the Moon, Phobos, or any exoplanet with a
rocky or partially molten mantle. In the light of the current rheological
knowledge, the tidal response is very different, and its frequency dependence
experiences especially steep variations in the vicinity of spin-orbit
resonances. In Makarov & Efroimsky (2013), we shall demonstrate that
illegitimate application of the constant time lag model to telluric objects
leads to nonexistent phenomena like pseudosynchronous rotation – not to
mention that it squeezes the tidal-evolution timescales (Efroimsky & Lainey
2007) and alters the probabilities of capture into resonances (Makarov,
Berghea, & Efroimsky 2012).

##  2 The constant-torque model

As we mentioned above, some authors tried to circumvent a consistent but
laborious treatment, by building simpler toy models. One such attempt was
undertaken by MacDonald (1964, eqn. 20) who assumed that the dynamical tide
mimics a static tide, except for being displaced by a geometric lag angle. The
same idea underlay the study, by Goldreich (1966), of a satellite on approach
to the 1:1 spin-orbit resonance. Thus, for mathematical convenience, both
authors set the geometric lag angle to be a frequency-independent constant.

The approach is known as the constant angular lag model. This name, however,
is inexact in the sense that, within the vicinity of the 1:1 spin-orbit
resonance, the lags and torque change their signs twice over a period: when
the bulge falls behind or advances (relative to the direction towards the
perturber), the sign is positive or negative, correspondingly. So in this
discussion the term constant  should be understood as frequency-independent :
both the instantaneous phase lag and the instantaneous torque
$\,\vec{\cal{T}}\,$ are independent of the tidal frequency $\,\chi\,$.
Consequently, the orbit-average (secular) tidal torque
$\,\langle\,\vec{\cal{T}}\,\rangle\,$ is also frequency-independent.

Sometimes this approach shows up in the literature under the name of MacDonald
torque (e.g., Touma & Wisdom 1994, section 2.7.1).

Unfortunately, the approach turns out to be inconsistent and should be
discarded. Physically, the constant angular lag model looks suspicious from
the beginning, because in the vicinity of the 1:1 spin-orbit resonance it
permits for abrupt switches of the torque, i.e., for situations where the
torque changes its sign, retaining the absolute value. Although the abrupt
switch can be substituted, by hand, with a continuous transition, this ad hoc
alteration still would not save the method, because it would not heal a more
fundamental defect. Mathematically, the derivation of the formula for the
tidal torque within the said model contains a subtle and often unappreciated
detail: this derivation implicitly sets the time lag $\,\Delta t\,$ to be
constant, as we shall demonstrate below. However, it can be shown that the
assertion of the time lag being frequency independent is incompatible with the
assertion of the geometric lag angle being frequency independent. In this way,
the discussed simplified approach is inherently contradictive.

Another defect of this approach is that it employs such entities as the
instantaneous phase lag and the instantaneous quality factor, the latter being
introduced as the inverse sine of the former. The so-defined instantaneous
quality factor is not guaranteed to be related to the energy damping rate in
the same manner as the regular (appropriate to a fixed frequency) quality
factor is related to the dissipation rate at that frequency (Williams &
Efroimsky 2012). Were the quality factor frequency-insensitive, this would not
be a problem. However, the latter option is excluded within the discussed
model,444 Rejection of the constant angular lag model does not, by itself,
prohibit setting the quality factor constant, at least over some limited
interval of frequencies. While realistic mantles never behave like this, such
a rheology, in principle, is not impossible. However, employment of this
rheology will not be available within a simplified model. Instead, one will
have to attribute the same value to $\,k_{l}/Q_{l}\,$ at all tidal modes, and
then will have to insert this value of $\,k_{l}/Q_{l}\,$ into all terms of the
Fourier expansion of the tidal torque (the Darwin-Kaula series). Each such
term will change its sign when the corresponding resonance gets transcended.
Up to the late 60s of the past century, there was a consensus in the
geophysical community that the seismic $\,Q\,$ of rocks should be
“substantially independent of frequency” (Knopoff 1964). This viewpoint was
later disproved by a large volume of experimental data obtained both in the
laboratory and in the field (see, e.g., Karato 2007 and references therein).
as being incompatible with the constant-$\Delta t\,$ assumption tacitly
present.

##  3 Goldreich (1966), Kaula (1968), Murray & Dermott (1999)

Numerous authors offer the following expression for the polar component of the
torque wherewith the tide-raising perturber acts on the tidally-perturbed
body:

$\displaystyle{\cal{T}}_{z}~{}=~{}\frac{3}{2}~{}G\,M_{1}^{\,2}~{}\frac{R^{5}}{r^{6}}~{}k_{2}\,\sin
2\epsilon_{g}~{}~{},$ (1)

$R$ being the radius of the disturbed body, $\,M_{1}\,$ standing for the mass
of the tide-raising perturber, $\,r\,$ denoting the instantaneous distance
between the bodies, $\,\epsilon_{g}\,$ standing for the angular lag, and the
obliquity being set nil. The subscript $\,z\,$ serves the purpose of
emphasising that the above formula furnishes the torque component orthogonal
to the equator of the tidally perturbed body.

Goldreich (1966) denotes the angular lag with $\,\Psi\,$, Kaula (1968, eqn.
4.5.29) calls it $\,\delta\,$, while Murray & Dermott (1999, eqn. 4.159) use
the letter $\,\epsilon\,$. We shall follow the latter notation, though
equipping it with the subscript $\,g\,$ which means: geometric .

Below we shall provide a detailed derivation of this formula, and shall see
that the angle standing in it is indeed the instantaneous geometric  tidal lag
angle

$\displaystyle\epsilon_{g}\,\equiv~{}(\dot{\nu}-\dot{\theta})\,\Delta t~{}\,,$
(2)

i.e., the instantaneous angular separation between the direction towards the
bulge and that towards the perturber. Here $\,\Delta t\,$ is the time lag,
$\,\nu\,$ is the true anomaly of the perturber, $\,\theta\,$ is the sidereal
angle of the perturbed body, and
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{\theta\;}}$ is its spin rate. It is
important to distinguish the instantaneous geometric lag $\,\epsilon_{g}\,$
from the instantaneous phase lag

$\displaystyle\epsilon_{ph}\,\equiv~{}2\,(\dot{\nu}-\dot{\theta})\,\Delta
t~{}=~{}2~{}\epsilon_{g}$ (3)

sometimes used in the literature (Efroimsky & Williams 2009, Williams &
Efroimsky 2012).

In Section 5 of Murray & Dermott (1999, eqns. 5.2 - 5.3), the authors rewrite
the expression for the torque, employing (in fact, implying) the above
expression of the lag through the true anomaly and spin rate:

$\displaystyle\sin
2\epsilon_{g}\,=\,\sin|2\epsilon_{g}|~{}\mbox{Sgn}\left(\epsilon_{g}\right)\,=~{}-~{}\sin|2(\dot{\theta}\,-\,\dot{\nu})\,\Delta
t|~{}\mbox{Sgn}(\,\dot{\theta}\,-\,\dot{\nu}\,)\,=~{}-~{}Q_{s}^{-1}~{}\mbox{Sgn}\left(\,\dot{\eta}\,-\,\dot{\varphi}\,\right)~{}\,.~{}$
(4)

The new angles showing up in this formula are $\,\eta\equiv\theta-{\cal{M}}\,$
and $\,\varphi\equiv\nu-{\cal{M}}\,$, with $\,{\cal{M}}\,$ denoting the mean
anomaly. These angles are depicted in Figure 5.1b in Ibid. Clearly, the time
derivatives $\,\dot{\eta}=\dot{\theta}-n\,$ and
$\,\dot{\varphi}=\dot{\nu}-n\,$ are the spin rate and true anomaly rate in a
frame which is centered on the tidally perturbed body 555 In Murray & Dermott
(1999), the role of a tidally perturbed body is played by the satellite, the
planet acting as its tide-raising perturber. In a different setting, perturbed
is the planet, the star or a satellite being the perturber. and is rotating
with the mean motion $\,n\,$. Interpreting the quantity
$\,Q_{s}\,=\,1/\sin|2(\dot{\theta}\,-\,\dot{\nu})\,\Delta t|$ as the
instantaneous quality factor, the authors assume that they can set it
frequency-independent (thus making the torque frequency-independent).

This approach contains two flaws. First, as explained in Williams & Efroimsky
(2012), it is not apparently evident whether the instantaneous quality factor
introduced as the inverse sine of the instantaneous phase lag has the physical
meaning usually instilled in a tidal dissipation factor at a certain
sinusoidal mode. Second, and most important, is that in reality it is
$\,\Delta t\,$ which gets implicitly set as frequency independent in the
derivation of (1). It then becomes impossible to assume that the geometric lag
$\,\epsilon_{g}\,$ also is frequency-independent – the two assumptions are
incompatible, as we shall see shortly. Consequently, setting the factor
$\,Q_{s}\,$ to be frequency independent is no longer an option. This makes the
entire constant geometric lag model or, to be exact, its implementation with
(1), inherently contradictive. Specifically, it is illegitimate to assert that
the tidal torque is proportional to Sgn$(\dot{\nu}-\dot{\theta})\,$.

##  4 Mathematical Introduction.

The tide-raising potential $\,W\,$ created by a perturber always changes the
shape and, as a result, the potential of the perturbed body. At the point
$\vec{R}$ of the perturbed body’s surface, the potential
$\,W(\mbox{{\boldmath$\vec{R}$}},\,\mbox{{\boldmath$\vec{r}$}}^{~{}*})\,$
created by the perturber residing at $\,\mbox{{\boldmath$\vec{r}$}}^{*}\,$ can
be expanded into a sum of terms
$\,W_{l}(\mbox{{\boldmath$\vec{R}$}},\,\mbox{{\boldmath$\vec{r}$}}^{~{}*})\,$
proportional to the Legendre polynomials $\,P_{\it l}(\cos\gamma)\;$. Here
$\,\gamma\,$ is the angle between the vectors
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,$ and $\vec{R}$ pointing from the
perturbed body’s centre towards the perturber and the point on the perturbed
body’s surface, where the potential $\,W\,$ is measured.

Tidal distortion of the body’s geometric form renders an addition
$\,U(\mbox{{\boldmath$\vec{r}$}})\,$ to the body’s potential in an exterior
point $\vec{r}$ . This addition turns out to be comprised of terms
$\,U_{l}(\mbox{{\boldmath$\vec{r}$}})\,$ each of which is proportional to the
term
$\,W_{l}(\mbox{{\boldmath$\vec{R}$}},\,\mbox{{\boldmath$\vec{r}$}}^{~{}*})\,$,
with the surface point $\vec{R}$ located exactly below (i.e., having the same
latitude and longitude as) the exterior point
$\,\mbox{{\boldmath$\vec{r}$}}^{~{}*}\,$.

The goal of this section is to provide a squeezed introduction into this
formalism and to explain how it should be generalised from a static
configuration setting to a dynamical setting.

###  4.1 Static tides

Let a body of radius $R$ experience tides from a perturber of mass
$\,M^{*}_{1}\,$ placed at
$\,{\mbox{{\boldmath$\vec{r}$}}}^{\;*}=(r^{*},\,\phi^{*},\,\lambda^{*})\,$,
with $\,r^{*}\geq R\,$.

At a point $\,\mbox{{\boldmath$\vec{R}$}}=(R,\phi,\lambda)\,$ on the perturbed
body’s surface, the potential $\,W\,$ due to the perturber is expanded over
the Legendre polynomials $\,P_{\it l}(\cos\gamma)\;$ as 666 Summation in
formula (5) goes over $\,l\geq 2\,$. The central term ($\,l=0\,$) is regarded
as the principal, Newtonian, part of the potential generated by the perturber,
and not as a part of the perturbation $\,W\,$ caused by the finite size of the
tidally perturbed body – indeed, the $\,l=0\,$ terms bears no dependence upon
$\vec{R}$ . The reason why the $\,l=1\,$ terms falls out is more subtle and is
related to the fact that we are developing our formalism in the frame of the
tidally perturbed body, not in an inertial frame. See, e.g., Efroimsky &
Williams (2009, eqns. 5 - 11).

$\displaystyle
W(\mbox{{\boldmath$\vec{R}$}}\,,\,\mbox{{\boldmath$\vec{r}$}}^{~{}*})$
$\displaystyle=$
$\displaystyle\sum_{{\it{l}}=2}^{\infty}~{}W_{\it{l}}(\mbox{{\boldmath$\vec{R}$}}\,,~{}\mbox{{\boldmath$\vec{r}$}}^{~{}*})~{}=~{}-~{}\frac{G\;M^{*}_{1}}{r^{\,*}}~{}\sum_{{\it{l}}=2}^{\infty}\,\left(\,\frac{R}{r^{~{}*}}\,\right)^{\textstyle{{}^{\it{l}}}}\,P_{\it{l}}(\cos\gamma)~{}~{}~{}~{}$
(5) $\displaystyle=$
$\displaystyle-\,\frac{G~{}M^{*}_{1}}{r^{\,*}}\sum_{{\it{l}}=2}^{\infty}\left(\frac{R}{r^{~{}*}}\right)^{\textstyle{{}^{\it{l}}}}\sum_{m=0}^{\it
l}\frac{({\it l}-m)!}{({\it
l}+m)!}(2-\delta_{0m})P_{{\it{l}}m}(\sin\phi)P_{{\it{l}}m}(\sin\phi^{*})~{}\cos
m(\lambda-\lambda^{*})~{}~{},\quad\,\quad$

where $\,\delta_{ij}\,$ is the Kronecker delta symbol, $\,G\,$ is Newton’s
gravity constant, while $\,\gamma\,$ denotes the angular separation between
the vectors $\,{\mbox{{\boldmath$\vec{r}$}}}^{\;*}\,$ and $\vec{R}$ pointing
from the centre of the perturbed body. The longitudes $\lambda,\,\lambda^{*}$
are reckoned from a fixed meridian on the perturbed body, the latitudes
$\phi,\,\phi^{*}$ being reckoned from the equator. The integers $\,l\,$ and
$\,m\,$ are called the degree and order, accordingly. The associated Legendre
functions $\,P_{lm}(x)\,$ are referred to as the associated Legendre
polynomials when their argument is sine or cosine of some angle.

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

$\displaystyle U(\mbox{{\boldmath$\vec{r}$}})~{}=~{}\sum_{{\it
l}=2}^{\infty}~{}U_{\it{l}}(\mbox{{\boldmath$\vec{r}$}})~{}=~{}\sum_{{\it
l}=2}^{\infty}~{}k_{\it l}\;\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\;W_{\it{l}}(\mbox{{\boldmath$\vec{R}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})~{}~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(6)

where $\,k_{l}\,$ are the static Love numbers, $\,R\,$ is the mean equatorial
radius of the perturbed body, while
$\,\mbox{{\boldmath$\vec{R}$}}=(R,\,\phi,\,\lambda)\,$ and
$\,\mbox{{\boldmath$\vec{r}$}}=(r,\,\phi,\,\lambda)\,$ are a surface point and
an exterior point above it, respectively, so that $\,r\geq R\,$.

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

$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})\;=\;\,-\,{G\;M^{*}_{1}}\sum_{{\it{l}}=2}^{\infty}k_{\it
l}\;\frac{R^{\textstyle{{}^{2\it{l}+1}}}}{r^{\textstyle{{}^{\it{l}+1}}}{r^{\;*}}^{\textstyle{{}^{\it{l}+1}}}}\sum_{m=0}^{\it
l}\frac{({\it l}-m)!}{({\it
l}+m)!}(2-\delta_{0m})P_{{\it{l}}m}(\sin\phi)P_{{\it{l}}m}(\sin\phi^{*})\;\cos
m(\lambda-\lambda^{*})~{}~{}.~{}~{}~{}$ (7)

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

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

$\displaystyle U(\mbox{{\boldmath$\vec{r}$}})\;=\;-\;\sum_{{\it
l}=2}^{\infty}\;k_{\it l}\;\left(\,\frac{R}{a}\,\right)^{\textstyle{{}^{{\it
l}+1}}}\frac{G\,M^{*}_{1}}{a^{*}}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\;\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(8) $\displaystyle~{}~{}~{}~{}\left.-\,\delta_{0m}\right)\sum_{p=0}^{\it
l}F_{{\it l}mp}({\it i}^{*})\sum_{q=-\infty}^{\infty}G_{{\it
l}pq}(e^{*})\sum_{h=0}^{\it l}F_{lmh}({\it
i})\sum_{j=-\infty}^{\infty}G_{lhj}(e)\;\cos\left(\,\left(v_{{\it
l}mpq}^{*}-m\theta^{*}\right)-\left(v_{{\it
l}mhj}-m\theta\right)\,\right)\,~{}_{\textstyle{{}_{\textstyle,}}}$

where

$\displaystyle v_{{\it l}mpq}^{*}\;\equiv\;({\it l}-2p)\,\omega^{*}\,+\,({\it
l}-2p+q){\cal M}^{*}\,+\,m\,\Omega^{*}~{}~{}~{},$ (9) $\displaystyle v_{{\it
l}mhj}\;\equiv\;({\it l}-2h)\,\omega\,+\,({\it l}-2h+j){\cal
M}\,+\,m\,\Omega~{}~{}~{},$ (10)

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

Decomposition (8) was pioneered by Kaula (1961, 1964). However, its partial
sum, with $\,|{\it{l}}|,\,|q|,\,|j|\,\leq\,2\,$, was derived yet by Darwin
(1879). In modern notations, Darwin’s work is discussed by Ferraz-Mello,
Rodríguez & Hussmann (2008). 777 Be mindful that the convention on the meaning
of notations $\vec{r}$ and $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,$ in Ibid. is
opposite to ours.

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

###  4.2 Dynamical tides with no friction

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

The key point is that, to get $\,U(\mbox{{\boldmath$\vec{r}$}})\,$ at the
point $\vec{r}$ at time $\,t\,$, we insert into (7) or (8) the perturber’s
position $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}(t)\,$ taken at that same time
$\,t\,$, and not at an earlier time. Formulae (7) and (8) stay equivalent to
one another, and remain unchanged, except that the distances, the sidereal
angle, and the angular coordinates acquire a simultaneous time dependence: 888
The orbital parameters $\,a\,$, $\,e\,$, $\,i\,$, $\,a^{*}\,$, $\,e^{*}\,$,
$\,i^{*}\,$ acquire no time dependence, insofar as the apsidal and nodal
precession remain the only permitted variations of the orbits. $\,r\,$ becomes
$\,r(t)\,$, $\,r^{\,*}\,$ becomes $\,r^{\,*}(t)\,$; while $\,\theta\,$,
$\,\theta^{*}\,$, $\,\phi\,$, $\,\phi^{*}\,$, $\,\lambda\,$,
$\,\lambda^{*}\,$, $\,\omega\,$, $\,\omega^{*}\,$, $\,\Omega\,$,
$\,\Omega^{*}\,$, $\,{\cal{M}}\,$, $\,{\cal{M}}^{\,*}\,$ become
$\,\theta(t)\,$, $\,\theta^{*}(t)\,$, $\,\phi(t)\,$, $\,\phi^{*}(t)\,$,
$\,\lambda(t)\,$, $\,\lambda^{*}(t)\,$, $\,\omega(t)\,$, $\,\omega^{*}(t)\,$,
$\,\Omega(t)\,$, $\,\Omega^{*}(t)\,$, $\,{\cal{M}}(t)\,$,
$\,{\cal{M}}^{\,*}(t)\,$.

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

###  4.3 Tidal modes and forcing frequencies

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

$\displaystyle
W(\mbox{{\boldmath$\vec{R}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\;=\;-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.~{}~{}~{}-\;\delta_{0m}\,\right)\;P_{{\it{l}}m}(\sin\phi)\;\sum_{p=0}^{\it
l}\;F_{{\it l}mp}({\it i}^{*})\;\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})\left\\{\begin{array}[]{c}\cos\\\ \sin\end{array}\right\\}^{{\it
l}\,-\,m\;\;\mbox{\small even}}_{{\it l}\,-\,m\;\;\mbox{\small
odd}}\;\left(v_{{\it
l}mpq}^{*}-m(\lambda+\theta^{*})\right)~{}~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(13)

Since $\,R\,$ is the radius of the tidally perturbed body, and since the
latitude $\,\phi\,$ and the longitude $\,\lambda\,$ are reckoned from the
equator and a fixed meridian, correspondingly, then (13) is just another
expression for the perturbing potential at the fixed point
$\,(R,\,\phi,\,\lambda)\,$ of the body’s surface. In (13), the expression in
round brackets can be reshaped as

$\displaystyle v_{{\it l}mpq}^{*}-m(\lambda+\theta^{*})\,=\,\omega_{{\it
l}mpq}\,(t\,-\,t_{0})~{}-~{}m~{}\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})~{}~{},$
(14)

where

$\displaystyle\omega_{lmpq}\;\equiv\;({\it l}-2p)\;\dot{\omega}^{*}\,+\,({\it
l}-2p+q)\;n^{*}\,+\,m\;(\dot{\Omega}^{*}\,-\,\dot{\theta}^{*})\,~{}.~{}~{}~{}$
(15)

Here $\,n^{*}\,\equiv\,{\bf{\dot{\cal{M}}}}^{\,*}\,$ is the mean motion of the
perturber, while $\,t_{0}\,$ is the time of perigee passage wherefrom the mean
anomaly $\,{\cal{M}}^{\,*}\,$ of the perturber is reckoned.

We see from (14) that the quantities $\,\omega_{lmpq}\,$ given by (15) are the
Fourier modes over which the tidal perturbation (13) is expanded. While these
modes can be positive or negative, the physical forcing frequencies,

$\displaystyle\chi_{lmpq}~{}\equiv~{}|\,\omega_{lmpq}\,|\,~{},~{}~{}~{}$ (16)

at which the stress oscillates, are positive-definite.

Having developed formulae (8 \- 13), Kaula (1961, 1964) never stipulated 999
The linear combination standing on the right-hand side of our formula (15)
appeared in the denominator of formulae (29) and (50) in Kaula (1961).
However, Kaula did not mention that this combination is a Fourier mode of the
tide. that the Fourier modes of the tide are given by (15). Possibly, he was
not interested in the frequency dependence of the phase or time lags.

In Section 6 of his book, Lambeck (1980) explained some aspects of Kaula’s
theory. While Lambeck’s equation (6.1.13b) indicates that Lambeck could be
aware of how the Fourier modes look, he too never wrote down the formula for
the modes explicitly. Perhaps, like Kaula, Lambeck had no interest in the
frequency dependence of lags – he just introduced a time lag $\,\Delta t\,$,
which in his developments was implicitly regarded frequency independent.

While in the review by Efroimsky & Williams (2009) and in Efroimsky (2012a,
2012b) the expression for $\,\omega_{lmpq}\,$ was written down explicitly, its
derivation was omitted.

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

To conclude, at the point
$\,\mbox{{\boldmath$\vec{R}$}}=(R,\,\phi,\,\lambda)\,$ of the surface of the
perturbed body, the perturbing potential is expressed through the tidal
Fourier modes as:

$\displaystyle
W(\mbox{{\boldmath$\vec{R}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\;=\;-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\;-\;\delta_{0m}\,\right)\;P_{{\it{l}}m}(\sin\phi)\;\sum_{p=0}^{\it
l}\;F_{{\it l}mp}({\it i}^{*})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})\left\\{\begin{array}[]{c}\cos\\\ \sin\end{array}\right\\}^{{\it
l}\,-\,m\;\;\mbox{\small even}}_{{\it l}\,-\,m\;\;\mbox{\small
odd}}\;\left(\,\omega_{lmpq}\,(t\,-\,t_{0})\,-\,m\,\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})\,\right)~{}\,~{},~{}~{}~{}~{}~{}~{}$
(19)

the tidal mode being given by (15). In an idealised situation, when the
extended body is frictionless and its response is instantaneous, we can employ
the static formula (6), as explained in subsection 4.2. Combining that formula
with expression (19), we see that the additional tidal potential generated by
a perfectly elastic body at the point
$\,\mbox{{\boldmath$\vec{r}$}}=(r,\,\phi,\,\lambda)\,$ right above $\vec{R}$
will now read as:

$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\;=\;-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{r}\,\right)^{\textstyle{{}^{l+1}}}\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2-\;\delta_{0m}\,\right)\;P_{{\it{l}}m}(\sin\phi)\;\sum_{p=0}^{\it
l}\;F_{{\it l}mp}({\it i}^{*})~{}~{}~{}~{}$
$\displaystyle\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})~{}k_{l}~{}\left\\{\begin{array}[]{c}\cos\\\
\sin\end{array}\right\\}^{{\it l}\,-\,m\;\;\mbox{\small even}}_{{\it
l}\,-\,m\;\;\mbox{\small
odd}}\;\left(\,\omega_{lmpq}\,(t\,-\,t_{0})\,-\,m\,\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})\,\right)~{}~{}.~{}~{}~{}~{}$
(22)

It is due to the lack of friction that the $\,lmpq\,$ term of (22) is in phase
with the $\,lmpq\,$ term of (19). Below we shall see that inclusion of
friction into the picture renders a phase shift between these terms. It is
also in anticipation of the discussion of friction that we placed the Love
numbers inside the $~{}\sum_{mpq}~{}$ sum in expression (22). Mode-independent
in the perfectly elastic case, the Love numbers may acquire dependence upon
the Fourier modes $\,\omega_{lmpq}\,$, because friction may mitigate
amplitudes of distortion differently at different frequencies.

##  5 Friction and Lagging

In this section, we shall trace, step by step, how internal friction gets
included into the tidal theory. In subsection 4.2, we made an observation
that, for an absolutely elastic (frictionless) body, treatment of dynamical
tides mimics that of static tides, except that all coordinates acquire time-
dependence. Our next step will be to incorporate friction, and therefore
lagging, into the picture.

As a first step, we shall address a simplistic method implying that the
coordinates of the tide-raising body (as seen in a frame corotating with the
perturbed body) get shifted back in time by some fixed time lag $\,\Delta
t\,$. Although implementations of this method into formulae (7) and (8) look
very different, they render results which in fact are equivalent – simply
because (7) and (8) are equivalent, and because the same procedure (shift by
$\,\Delta t\,$) is performed on the quantities with asterisks in both these
formulae.

However, the difference in the mathematical form of these, equivalent, results
also prompts a more consistent way of taking care of friction. This, more
advanced, method will be implementable only in formula (8) and not in (7). The
method is the one used by Kaula (1964). Since the explanation of the method in
Ibid. was extremely concise, we shall elucidate it here in mode detail.

###  5.1 A naive way of bringing in lagging

Naively, dissipation and the ensuing lagging can be included into the picture
by assuming that the exterior body located at point $\vec{r}$ at time $\,t\,$
is subject not to the tidal potential created simultaneously by the perturber
residing at $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}(t)\,$, but to the potential
generated by the perturber lagging in time on its orbit 101010 Here we imply:
on its orbit as seen from the perturbed extended body. The caveat is needed,
since we are considering the physical reaction of the extended body, and thus
are interested in the location of the perturber relative to its surface, and
not to an inertial frame. It is for this reason that in equation (23) we
employ the latitudes and longitudes defined in a frame corotating with the
perturbed extended body. In an inertial frame, a shifting of the perturber
back by $\,\Delta t\,$ should then be accompanied by a shift of the
orientation of the extended body back by the same $\,\Delta t\,$; and this is
why we have $\,\theta^{*}(t-\Delta t)\,$ in equation (24). by some $\,\Delta
t\,$. Speaking loosely, the no-asterisk exterior body located at
$\,\mbox{{\boldmath$\vec{r}$}}(t)\,$ “feels” the tide given by (7) or (8), as
if the tide were generated by the asterisk perturber located on its orbit not
at $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}(t)\,$ but at
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}(t\,-\,\Delta t)\,$.

Mathematically, this implies that the tidal potential
$\,U(\mbox{{\boldmath$\vec{r}$}})\,$ at $\,t\,$ must be calculated via (7) or
(8), by insertion of the no-asterisk coordinates taken at $\,t\,$, and the
coordinates with asterisk taken at $\,t\,-\,\Delta t\,$. The naive strategy
also implies that the Love numbers $\,k_{l}\,$ keep their static values,
though this detail is seldom spelled out.

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

$\displaystyle U(\,\mbox{{\boldmath$\vec{r}$}}(t)\,)\,=$ $\displaystyle-$
$\displaystyle{G\;M^{*}_{1}}\sum_{{\it{l}}=2}^{\infty}k_{\it
l}\;\frac{R^{\textstyle{{}^{2\it{l}+1}}}}{r(t)^{\textstyle{{}^{\it{l}+1}}}{r^{\;*}(t-\Delta
t)}^{\textstyle{{}^{\it{l}+1}}}}\sum_{m=0}^{\it
l}\frac{\left(l-m\right)!}{({\it l}+m)!}\left(2\right.$ (23)
$\displaystyle~{}\left.-\,\delta_{0m}\right)P_{{\it{l}}m}(\,\sin\phi(t)\,)P_{{\it{l}}m}\left(\,\sin\phi^{*}\left(t-\Delta
t\right)\,\right)\;\cos m[\lambda(t)-\lambda^{*}(t-\Delta
t)]~{}\,~{}.~{}\quad~{}\quad~{}$

and

$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})\;=\;-\;\sum_{l=2}^{\infty}\;k_{\it
l}\;\left(\,\frac{R}{a}\,\right)^{\textstyle{{}^{{\it
l}+1}}}\frac{G\,M^{*}_{1}}{a^{*}}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{l}}}\sum_{m=0}^{\it
l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\;\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}\left.-\,\delta_{0m}\right)\sum_{p=0}^{l}F_{lmp}({\it
i}^{*})\sum_{q=-\infty}^{\infty}G_{lpq}(e^{*})\sum_{h=0}^{\it l}F_{lmh}({\it
i})\sum_{j=-\infty}^{\infty}G_{lhj}(e)~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\cos\left(\,\left[v_{{\it l}mpq}^{*}(t-\Delta
t)-\,m\theta^{*}(t-\Delta t)\,\right]-\left[v_{{\it
l}mhj}(t)-m\theta(t)\,\right]\,\right)\,~{}_{\textstyle{{}_{\textstyle.}}}$
(24)

Just as their static precursors (7) and (8), our dynamical formulae (23) and
(24) remain equivalent to one another. They still reflect a mere switch from
the spherical to the Keplerian coordinates, 111111 En route  from (23) to
(24), one changes not only a coordinate system but also a frame of reference.
Our longitude $\,\lambda\,$ being reckoned from a meridian, the switch goes
from the corotating coordinates $\,(r,\,\phi,\,\lambda)\,$ to the Kepler
coordinates $(a,\,e,\,i,\,\Omega,\,\omega,\,{\cal{M}})$ defined in a frame
comoving but not corotating with the perturbed body. Technically, one first
substitutes $\,\lambda\,$ with $\,\tilde{\lambda}-\theta\,$, where
$\,\tilde{\lambda}=\lambda+\theta\,$ is the longitude in the comoving (not
corotating) frame. Then one can resort to the standard formulae connecting the
spherical and Kepler coordinates in the same frame. The formulae apply not to
$\,(r,\,\phi,\,\lambda)\,$ but to $\,(r,\,\phi,\,\tilde{\lambda})\,$, see
Kaula (1961). Thus, the current spin rate $\,\theta(t)\,$ pops up in (24) due
to the transition from a corotating frame to a comoving one. All said relates
equally to both the spherical and Kepler coordinates with asterisks. So the
delayed value $\,\theta^{*}(t-\Delta t)\,$, too, emerges in (24) due to the
frame switch. Recall that $\,\theta^{*}\,$ is the same spin rate as
$\,\theta\,$, except that it gets equipped with an asterisk, when it stands in
expressions corresponding to the perturber. Also recall that, within the
described approach, we model friction by simply shifting the perturber (as
seen in a frame corotating with the perturbed body) back in time by $\Delta
t$. In a frame which is comoving but not corotating, this implies not only
pulling the perturber back by $\,\Delta t\,$ but also rotating the perturbed
body back by $\,\dot{\theta}\,\Delta t\,$. Leaving the coordinates
$\,(r,\,\phi,\,{\lambda})\,$ untouched, and changing only
$\,(r^{\,*},\,\phi^{\,*},\,\lambda^{\,*})\,$ to
$\,(a^{\,*},\,e^{\,*},\,i^{\,*},\,\Omega^{\,*},\,\omega^{\,*},\,{\cal{M}}^{\,*})\,$,
one arrives at (13) and then at (19 \- 22). Applying this machinery also to
the variables with no asterisk, one ends up with (24).  except that now
$\vec{r}$ is taken at time $\,t\,$, while
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,$ is taken at $\,t-\Delta t\,$.

The longitude reckoned from a fixed meridian on the perturbed body is
expressed through the true anomaly $\,\nu\,$, the periapse $\,\omega\,$ and
the node $\,\Omega\,$ as

$\displaystyle\lambda\,=\,-\,\theta\,+\,\Omega\,+\,\omega\,+\,\nu\,+\,O({\it
i}^{2})~{}\,.$

In neglect of the nodal and the apsidal precession, and for a small obliquity,
this results in

$\displaystyle\dot{\lambda}\,\approx\,-\,\dot{\theta}~{}+~{}\dot{\nu}\,+\,O({\it
i}^{2})~{}\,.$

In (23), the argument of cosine may now be written, in a linear approximation
over $\,\Delta t\,$, as

$\displaystyle m\,\left[\,\lambda(t)\,-\,\lambda^{*}(t-\Delta
t)\,\right]\,=\,m\,\left[\,\lambda(t)\,-\,\lambda^{*}(t)\,+\,\dot{\lambda}^{*}\,\Delta
t\,\right]\,=\,m\,\left(\,\lambda\,-\,\lambda^{*}\,\right)\,-\,m\,(\,\dot{\theta}\,-\,\dot{\nu}\,)\,\Delta
t~{}+~{}O({\it i}^{2})~{}\,.~{}\,~{}\,$ (25)

In (24), the argument of cosine may be shaped, in a linear approximation over
$\,\Delta t\,$, as

$\displaystyle\left[v_{{\it l}mpq}^{*}(t-\Delta t)-\,m\,\theta^{*}(t-\Delta
t)\,\right]-\left[v_{{\it l}mhj}(t)-m\,\theta(t)\,\right]\,~{}=$
$\displaystyle~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\left[v_{{\it
l}mpq}^{*}(t)-\,m\,\theta^{*}(t)\,\right]-\left[v_{{\it
l}mhj}(t)-m\,\theta(t)\,\right]\,~{}-~{}\epsilon_{lmpq}$

where

$\displaystyle\epsilon_{lmpq}$ $\displaystyle\equiv$
$\displaystyle\left[\dot{v}_{{\it
l}mpq}^{*}(t)-\,m\,\dot{\theta}^{*}(t)\right]\,\Delta t\,=\,\left[({\it
l}-2p)\;\dot{\omega}^{*}\,+\,({\it
l}-2p+q)\;n^{*}\,+\,m\;(\dot{\Omega}^{*}\,-\,\dot{\theta}^{*})\right]\,\Delta
t$ (27) $\displaystyle=$ $\displaystyle\omega_{lmpq}~{}\Delta t$

is the phase lag corresponding to the mode $\,\omega_{lmpq}\,$.

From here we observe that the above-chosen method of taking the tidal friction
into account fixes the phase lags in a very specific way: through shifting the
perturber back on its orbit by a fixed time $\,\Delta t\,$, we set the phase
lags (27) to be proportional to this $\,\Delta t\,$. It should be emphasised
once again that the shift is performed in a frame corotating with the
perturbed body. In a frame comoving but not corotating with it, the shift will
thus be accompanied by rotation of the perturbed body back by
$\,\dot{\theta}\Delta t\,$, hence the term $\,-\,\dot{\theta}\,\Delta t~{}$ in
the expression (21) for the phase lag.

Our formulae (23) and (24) can be written in another, equivalent form:

$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\;=\;-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{r}\,\right)^{\textstyle{{}^{l+1}}}\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2~{}-\;\delta_{0m}\,\right)\;~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle P_{{\it{l}}m}(\sin\phi)\;\sum_{p=0}^{\it l}\;F_{{\it l}mp}({\it
i}^{*})\;\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})~{}k_{l}~{}\left\\{\begin{array}[]{c}\cos\\\
\sin\end{array}\right\\}^{{\it l}\,-\,m\;\;\mbox{\small even}}_{{\it
l}\,-\,m\;\;\mbox{\small odd}}\;\left(\,\omega_{lmpq}\,(t\,-\,\Delta
t\,-\,t_{0})\,-\,m\,\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})\,\right)~{}~{}.~{}~{}~{}~{}$
(30)

This form is analogous to (22), except for the time lag $\,\Delta t\,$, the
same for each Fourier mode.

Expression (30) is equivalent to expression (23), and is obtained from it by a
switch from the spherical coordinates
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}=(r^{*},\,\phi^{*},\,\lambda^{*})\,$ to
the Kepler elements
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}=(\,a^{*},\,e^{*},\,{\it
i}^{*},\,\Omega^{*},\,\omega^{*},\,{\cal M}^{*}\,)\,$, with the variables
$\,\mbox{{\boldmath$\vec{r}$}}=(r,\,\phi,\,\lambda)\,$ kept unchanged. From
the equivalence of expressions (30) and (23), we observe that, after the same
time lag is added into all terms of (30), all the so-shifted terms add up to
the equilibrium bulge geometrically displaced in such a way as if it were a
static bulge generated by the perturber at a slightly different time. No other
rheology can make this claim because, more generally, the lag in each term in
the expansion would correspond to its own increment in $\,t\,$. This tells us
that by setting $\,\Delta t\,$ frequency-independent we impose a highly
restrictive rheological rule, obedience to which cannot be expected of
realistic mantles.

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

###  5.2 A consistent way of bringing in lagging (Kaula 1964)

The above expression (27) for phase lags contains in itself an obvious hint on
how a general-type rheology should be built into the tidal theory – to that
end, one simply has to endow each mode $\,\omega_{lmpq}\,$ with a time lag
$\,\Delta t_{l}(\omega_{lmpq})\,$ of its own. Another adjustment is the mode
dependence of the Love numbers: $\,k_{l}\,=\,k_{l}(\omega_{lmpq})\,$.
Expression (24) will now become

$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})\;=\;-\;\sum_{l=2}^{\infty}\;\left(\,\frac{R}{a}\,\right)^{\textstyle{{}^{{\it
l}+1}}}\frac{G\,M^{*}_{1}}{a^{*}}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{l}}}\sum_{m=0}^{\it
l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\;\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}\left.-\,\delta_{0m}\right)\sum_{p=0}^{l}F_{lmp}({\it
i}^{*})\sum_{q=-\infty}^{\infty}G_{lpq}(e^{*})\sum_{h=0}^{\it l}F_{lmh}({\it
i})\sum_{j=-\infty}^{\infty}G_{lhj}(e)~{}~{}~{}~{}~{}~{}~{}$ $\displaystyle
k_{\it l}(\omega_{lmpq})\;\cos\left(\,\left[v_{{\it
l}mpq}^{*}(t)-\,m\theta^{*}(t)\,\right]-\left[v_{{\it
l}mhj}(t)-m\theta(t)\,\right]\,-\,\epsilon_{l}(\omega_{lmpq})\,\right)\,~{}_{\textstyle{{}_{\textstyle,}}}~{}~{}~{}~{}$
(31)

where

$\displaystyle\epsilon_{l}(\omega_{lmpq})\,\equiv\,\omega_{lmpq}\,\Delta
t_{l}(\omega_{lmpq})\,=\,\left[\,({\it l}-2p)\;\dot{\omega}^{*}\,+\,({\it
l}-2p+q)\;n^{*}\,+\,m\;(\dot{\Omega}^{*}\,-\,\dot{\theta}^{*})\,\right]\,\Delta
t_{l}(\omega_{lmpq})~{}\,.~{}\,~{}\,~{}\,~{}$ (32)

In (31 \- 32), we prefer to denote the phase and time lags not as
$\,\epsilon_{lmpq}\,$ and $\,\Delta t_{lmpq}\,$, but as
$\,\epsilon_{l}(\omega_{lmpq})\,$ and $\,\Delta t_{l}(\omega_{lmpq})\,$.
Indeed, their dependence on the indices $\,mpq\,$ is solely due to the
argument $\,\omega_{lmpq}~{}$. However, we cannot strip $\,\epsilon\,$ or
$\,\Delta t\,$ of the subscript $\,l\,$, because the functional form of the
frequency dependence of the phase or time lag is different for different
$\,l\,$s. The tidal friction is not the same as the seismic friction, and the
degree $\,l\,$ affects the tidal damping rate.121212 The difference between
the tidal and seismic friction and, accordingly, the difference of dissipation
at different $\,l\,$s is unimportant in small bodies, where only the rheology
matters. However things change in large planets where self-gravitation becomes
a crucial factor in tidal friction (Efroimsky 2012a). Hence in formulae (31 \-
32) we have $\,\epsilon_{l}\,$ and $\,\Delta t_{l}\,$, and not just
$\,\epsilon\,$ or $\,\Delta t\,$.

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

$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*})\;=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{r}\,\right)^{\textstyle{{}^{l+1}}}\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2~{}-\;\delta_{0m}\,\right)\;P_{{\it{l}}m}(\sin\phi)~{}\sum_{p=0}^{\it
l}\;F_{{\it l}mp}({\it i}^{*})\;\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})~{}~{}~{}~{}~{}$ $\displaystyle
k_{l}(\omega_{lmpq})~{}\left\\{\begin{array}[]{c}\cos\\\
\sin\end{array}\right\\}^{{\it l}\,-\,m\;\;\mbox{\small even}}_{{\it
l}\,-\,m\;\;\mbox{\small odd}}\;\left(\,\omega_{lmpq}\,[\,t\,-\,\Delta
t_{l}(\omega_{lmpq})\,-\,t_{0}\,]\,-\,m\,\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})\,\right)~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}$
(35)

This form is analogous to (22) and (30), except for two details. First, each
mode $\,\omega_{lmpq}\,$ now has a time lag $\,\Delta t_{l}=\,\Delta
t_{l}(\omega_{lmpq})\,$ of its own. Likewise, the dynamical Love number at
each Fourier mode is a function of this mode:
$\,k_{l}=\,k_{l}(\omega_{lmpq})\,$.

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

$\displaystyle\delta_{lmpq}\,=~{}\frac{\omega_{lmpq}}{m}~{}\Delta
t_{l}(\omega_{lmpq})\,~{}.$ (36)

For example, the geometric lag angle of the principal, semidiurnal bulge is
$\,\delta_{2200}=\,\frac{\textstyle\omega_{2200}}{\textstyle 2}~{}\Delta
t_{2}\,=\,(n-\dot{\theta})\,\Delta t_{2}\,$, where the time lag is taken at
the appropriate, semidiurnal mode: $\,\Delta t_{2}=\Delta
t_{2}(\omega_{2200})\,$.

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

In practical applications, most important is the special case when the
exterior body located at $\vec{r}$ coincides with the tide-raising perturber
located at $\,\mbox{{\boldmath$\vec{r}$}}^{~{}*}\,$. In this situation, the
perturber is acting upon itself through the tide it creates on the perturbed
body. Keeping the phase lags intact, and identifying
$\,\mbox{{\boldmath$\vec{r}$}}(t)\,$ with
$\,\mbox{{\boldmath$\vec{r}$}}^{~{}*}(t)\,$, we may be tempted to mis-assume
that the expression with no asterisk, standing in the argument of the cosine
in (31), compensates the expression with an asterisk, so the phase lag becomes
all that is left. However, this would only furnish us the secular term of the
tidal potential $\,U(\mbox{{\boldmath$\vec{r}$}})\,$ wherewith the perturber
acts upon itself through the medium of the tidally perturbed body. This term
is proportional to $\,\cos\epsilon_{lmpq}\,$. As the $\,p,\,q\,$ and
$\,h,\,j\,$ are independent pairs of integers, there also will be
contributions with $\,\\{p,\,q\\}\neq\\{h,\,j\\}\,$. These are the oscillating
components of the tidal potential $\,U(\mbox{{\boldmath$\vec{r}$}})\,$.
Although their time average is nil, they do contribute to the heat production,
while the appropriate oscillating components of the tidal torque may influence
free librations. The topic was first addressed in Efroimsky (2012b), and was
revisited by Makarov et al. (2012) who explored whether the oscillating part
of the torque can influence capture into spin-orbit resonances. It has turned
out that, naturally, the oscillating part of the torque alters the outcome of
a particular realisation of the capture scenario, but leaves the statistics
unchanged.

##  6 Calculation of the tidal torque

We would begin with emphasising a key circumstance concerning the two forms of
expansion of the tidal potential. In the absence of friction, these
expansions, (7) and (8), were equivalent. Their amended versions, (23) and
(24), remain equivalent in the presence of friction, provided the latter obeys
a special (often unphysical) restriction that the time lag $\,\Delta t\,$ is
the same at all frequencies. Beyond that threshold, the equivalence of the two
expansions failed to extend. Since expression (23) does not contain the
Fourier modes in it, it is plainly impossible to write (23) in a form that
takes into account different time lags at different frequencies. Therefore,
any calculation based on (23) will unavoidably imply the frequency-
independence of $\,\Delta t\,$ and will thus be incompatible with any other
rheological law.

###  6.1 Tidal torque, as derived from the concise expression (23)

Let us calculate the tidal torque, using (23). By employing this expression,
we automatically set the rheology to be $\,\Delta t=\,$const .

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

$\displaystyle{\cal{T}}_{z}(\mbox{{\boldmath$\vec{r}$}})\;=\;\,M_{1}\;\frac{\partial
U(\mbox{{\boldmath$\vec{r}$}})}{\partial\lambda}\,~{}.$ (37)

Here polar means: orthogonal to the perturbed body’s equator. For small
obliquities (and, therefore, small latitudes), insertion of (23) into (37)
yields

$\displaystyle{\cal{T}}_{z}(\,\mbox{{\boldmath$\vec{r}$}}(t)\,)\,=$
$\displaystyle{G\;M_{1}\;M^{*}_{1}}\sum_{{\it{l}}=2}^{\infty}k_{\it
l}\;\frac{R^{\textstyle{{}^{2\it{l}+1}}}}{r(t)^{\textstyle{{}^{\it{l}+1}}}{r^{\;*}(t-\Delta
t)}^{\textstyle{{}^{\it{l}+1}}}}\sum_{m=0}^{\it
l}~{}m~{}\frac{\left(l-m\right)!}{({\it l}+m)!}\left(2\right.$ (38)
$\displaystyle~{}\left.-\,\delta_{0m}\right)P_{lm}(0)P_{lm}(0)\;\sin\left(\,m\,\left(\lambda-\lambda^{*}\right)\,-\,m\,(\dot{\theta}-\dot{\nu})\,\Delta
t\,\right)~{}+~{}O(i^{2})~{}\,~{},~{}\quad~{}\quad~{}$

where we made use of (25).

Since the integer $\,m\,$ is now entering the above expression as a
multiplier, the term with $\,m=0\,$ becomes nil. As $\,P_{21}(0)=0\,$, the
term with $\,m=1\,$ also vanishes. Hence, the term with $\,l=m=2~{}$ is
leading. Neglecting the smaller terms, we thus obtain:

$\displaystyle{\cal{T}}_{z}(\,\mbox{{\boldmath$\vec{r}$}}(t)\,)\,\approx\,\frac{\textstyle
3}{\textstyle
2}~{}{G\;M_{1}\;M^{*}_{1}}\,k_{2}\;\frac{R^{5}}{r(t)^{\textstyle{{}^{3}}}{r^{\;*}(t-\Delta
t)}^{\textstyle{{}^{3}}}}\;\sin\left(\,2\,\left(\lambda-\lambda^{*}\right)\,-\,\,2\,(\dot{\theta}-\dot{\nu})\,\Delta
t\,\right)~{}\,~{}.~{}\quad~{}\quad~{}$ (39)

Consider the special case when the tide-raising perturber (with the asterisk)
coincides with the other external body (with no asterisk). The perturber
creates tides on the perturbed body, and then interacts with the tides it
itself has created. Hence the perturber becomes subject to a tidal torque
$\,\vec{\bf{T}}\,$ exerted by it upon itself, through the medium of the tidal
bulge it creates on the perturbed body. Evidently, a torque
$\,\vec{\cal{T}}\,=\,-\,\vec{\bf{T}}\,$, of the same magnitude but opposite
direction, will be acting upon the perturbed body. We arrive at this torque by
setting $\,\lambda=\lambda^{*}~{}$ and $~{}M_{1}=M^{*}_{1}~{}$ in the above
expression: 131313 Be mindful that in (40) we chose to make no distinction
between $\,r(t)\,$ and $\,r(t-\Delta t)\,$. Replacement of $\,r(t-\Delta t)\,$
with $\,r(t)\,$ gives birth to an absolute error of order
$\,O(eQ^{-2}n/\chi)\,$. It is however explained in Efroimsky & Williams
(2009), that after averaging of (40) over one orbital period this error
reduces to $\,O(e^{2}Q^{-3}n^{2}/\chi^{2})\,$.

$\displaystyle{\cal{T}}_{z}(\,\mbox{{\boldmath$\vec{r}$}}\,)\,\approx\,\frac{\textstyle
3}{\textstyle
2}~{}{G\;M_{1}^{\,2}}\,k_{2}\;\frac{R^{5}}{r^{\textstyle{{}^{6}}}}\;\sin\left(\,2\,(\dot{\nu}-\dot{\theta})\,\Delta
t\,\right)~{}\,~{}.~{}\quad~{}\quad~{}$ (40)

Naturally, for the model with a frequency-independent $\,\Delta t\,$, the
quantity

$\displaystyle\epsilon_{g}\,\equiv~{}(\dot{\nu}-\dot{\theta})~{}\Delta t$ (41)

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

$\displaystyle\chi~{}=~{}2\,|\,\dot{\nu}-\dot{\theta}\,|$ (42)

acts as an instantaneous tidal frequency.

The quantity

$\displaystyle\epsilon_{ph}\,\equiv~{}2~{}(\dot{\nu}-\dot{\theta})~{}\Delta
t~{}=~{}2~{}\epsilon_{g}$ (43)

is commonly interpreted as an instantaneous phase lag , so the torque in
expression (40) may be written down as

$\displaystyle{\cal{T}}_{z}(\,\mbox{{\boldmath$\vec{r}$}}\,)\,\approx\,\frac{\textstyle
3}{\textstyle
2}~{}{G\;M_{1}^{\,2}}\,k_{2}\;\frac{R^{5}}{r^{\textstyle{{}^{6}}}}\;\sin\epsilon_{ph}~{}=~{}\frac{\textstyle
3}{\textstyle
2}~{}{G\;M_{1}^{\,2}}\,k_{2}\;\frac{R^{5}}{r^{\textstyle{{}^{6}}}}\;\sin
2\epsilon_{g}~{}\,~{},~{}\quad~{}\quad~{}$ (44)

which is exactly the expression (1) of our concern. In Section 2, we mentioned
several popular papers and books, including Murray & Dermott (1999, eqn.
4.159), 141414 Mind a misprint in Eqn. (4.159) of Murray & Dermott (1999): in
the denominator, $\,a^{6}$ must be changed to $\,r^{6}$. The misprint emerged
because in subsection 4.2 the distance was denoted with $\,a\,$. In formulae
(5.2 - 5.3) of Ibid. the misprint gets corrected. where this formula is
employed. Now, that we have derived this formula accurately, we see that its
validity hinges on the time lag being frequency independent.

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

$\displaystyle
Q~{}\equiv~{}1/\,|\,\sin\left(\,2\,(\dot{\nu}-\dot{\theta})\,\Delta
t\,\right)\,|~{}=~{}1/\sin|\,\epsilon_{ph}\,|$ (45)

as an instantaneous quality factor, the validity of this interpretation of
(45) remains questionable. For a nonzero eccentricity, the instantaneous tidal
frequency (42) is varying in time. So it is not readily apparent whether the
instantaneous $\,Q\,$ is connected to the damping rate in the manner the
proper quality factor introduced at a certain frequency links to the damping
rate at that frequency. McDonald (1964) and Goldreich (1966) tried to sidestep
this difficulty by assuming that $\,Q\,$ is a frequency independent constant.
However, at finite eccentricities this assumption does not work, as it is
incompatible with the constant $\,\Delta t\,$ assumption (the latter
assumption being a necessary prerequisite to using formulae 41 \- 45, as we
saw above). For more on this see Williams & Efroimsky (2012).

###  6.2 Tidal torque, as derived from the Fourier expansion (24),
with all Fourier modes delayed by the same time lag $\,\Delta t\,$

When starting out with expression (24), it is convenient to use the formula

$\displaystyle{\cal{T}}_{z}(\mbox{{\boldmath$\vec{r}$}})\;=\;-\;M_{1}\;\frac{\partial
U(\mbox{{\boldmath$\vec{r}$}})}{\partial\theta}\;\;\;,$ (46)

instead of (37). Technically, we should first differentiate $\,U\,$ with
respect to the sidereal angle $\,\theta\,$, and then set
$\,\theta\,=\,\theta^{\,*}\,$. We should also set the orbital elements with
asterisk equal to their counterparts with no asterisk, it being understood
that the tide-raising perturber is the same as the other exterior body which
“feels” the tides on the perturbed body. The development will furnish us the
polar component of the torque with which the perturber acts upon the tidally
deformed body. For exploration of dynamics in a low-obliquity configuration,
this component is sufficient.

Insertion of the Fourier series (24) into equation (46) yields a Fourier
series for the polar component of the torque, which is presented in Efroimsky
(2012b). Here we shall not repeat this long formula, but shall only make an
important comment on it. Insofar as $\,\Delta t\,$ stays frequency independent
(i.e., has the same value for all phase lags (27) entering the expansion for
the torque), the resulting series for the torque stays fully equivalent to
(38), with $\,\lambda\,$ and $\,\lambda^{*}\,$ set equal to one another in the
latter formula.

This equivalence is ensured by the expression (24) for the potential $\,U\,$
being equivalent to the expression (23) whence formula (38) originated, and by
our agreement to keep $\,\Delta t\,$ the same for all phase lags.

###  6.3 Tidal torque, as derived from the Fourier expansion (31),
with each mode $\,\omega_{lmpq}\,$ having a time lag of its own, $\,\Delta
t_{l}(\omega_{lmpq})\,$

As soon as we abandon the assumption that the phase lags (27) contain the same
fixed $\,\Delta t\,$, i.e., as soon as we switch from the lags (27) to those
rendered by (32), we acquire an opportunity to describe a tidal torque acting
on a perturbed body of an arbitrary rheology. Indeed, as the time lags can
have an arbitrary mode-dependence, this also relates to the phase lags. Above
that, we now permit the Love numbers to be mode-dependent.

To derive the tidal torque, we now combine formula (46) not with expansion
(24) but with the expansion (31) where the time lags are, generally, all
different. The rheological emancipation, though, comes at a cost: the Fourier
decomposition for the torque, obtained through (31), with mode-dependent
$\,\Delta t_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$ and
$\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$, will no longer be equivalent to
the concise and elegant formula (38). The customary and widely used leading-
order approximation of (38), given by (40) or by (44), will not work for an
arbitrary rheology.

If, for example, we choose to set the factors $~{}\,\frac{\textstyle
k_{l}(\omega_{\textstyle{{}_{lmpq}}})}{\textstyle
Q_{l}(\omega_{\textstyle{{}_{lmpq}}})}~{}\mbox{Sgn}(\omega_{\textstyle{{}_{lmpq}}})\,=\,k_{l}\,\sin\epsilon_{l}~{}\,$
to be mode independent, then, to calculate the torque, we shall have to plug
the same value of $\,k_{l}\,\sin\epsilon_{l}\,$ into all terms of the Fourier
series for the torque (eqn. 106 from Efroimsky 2012b). However, we shall not
be able to employ the neat formula (44).

##  7 Why the $\,`\mathbf{\emph{constant angular lag}}\,$’  model is wrong

As we explained above, an accurate derivation of the popular formula (1) for
the polar component of the torque hinges on a tacit assumption that the time
lag $\,\Delta t\,$ is the same for all modes in the expansion of the tidal
potential. Through formula (41), this, mode-independent time lag is related to
the geometric lag angle $\,\epsilon_{g}\,$ present in formula (1).

Relation (41) tells us that, since $\,\Delta t\,$ is the same for all the
tidal modes involved, then the geometric lag angle $\,\epsilon_{g}\,$ cannot
be treated as a fixed constant – even if we add a caveat permitting
$\,\epsilon_{g}\,$ to switch signs when the value of $\,\dot{\nu}\,$
transcends $\,\dot{\theta}\,$. With this caveat or not, keeping
$\,|\,\epsilon_{g}\,|\,$ constant is impossible simply for the reason that
(for a nonvanishing eccentricity) the quantity $\,\dot{\nu}\,$ oscillates in
time.151515 While derivation of (1) absolutely requires $\,\Delta t\,$ to be
the same for all tidal modes, it does not require $\,\Delta t\,$ to be fixed
in time. Therefore, in theory, we can save the constant angular lag model by
tuning the time dependence of $\,\Delta t\,$ in such a special way that
$\,\epsilon_{g}\,$ in (41) stays constant in time. This however would require
the dissipative properties of the mantle to be fine-tuned, simultaneously at
all frequencies, to ensure that, first, the time lags at all frequencies
evolve but remain equal to one another and, second, that $\,\Delta
t\,(\dot{\nu}-\dot{\theta})\,$ stays constant in time. As the evolution rate
of $\,\dot{\nu}-\dot{\theta}\,$ is defined by the orbit, such fine tuning of
rheology is unrealistic. So a constant $\,\Delta t\,$ is incompatible with a
constant $\,|\,\epsilon_{g}\,|\,$.

This is the reason why the so-called constant angular lag  model based on (1)
must be discarded wholesale as being inherently contradictive.

While we still retain the right to set the factors $~{}\,\frac{\textstyle
k_{l}(\omega_{\textstyle{{}_{lmpq}}})}{\textstyle
Q_{l}(\omega_{\textstyle{{}_{lmpq}}})}~{}\mbox{Sgn}(\omega_{\textstyle{{}_{lmpq}}})\,=\,k_{l}\,\sin\epsilon_{l}~{}\,$
mode independent, the value of the torque resulting from this assumption has
to be calculated by insertion of these factors into the full Fourier expansion
of the torque and not into (1). We may as well use (1), but only for a
constant time lag, and not for a constant geometric lag angle.

##  8 Conclusions

We have reexamined the common formula (1) for the tidal torque, a formula
which is equivalent to the expressions given in Sections 4 and 5 of Murray &
Dermott (1999) and to the expressions offered in Goldreich (1966) and Kaula
(1968). It has turned out that an accurate derivation of this popular formula
necessarily implies a specific rheology – the assertion that the time lag
$\,\Delta t\,$ is frequency independent. As can be easily seen from (41), this
assertion is incompatible with the assertion of the geometric lag being
frequency independent. Moreover, the quantity $\,\epsilon_{g}\,$ furnished by
formula (41) can be endowed with the meaning of a geometric lag only within
the constant $\,\Delta t\,$ rheological model (and only for a small obliquity
$\,i\,$).

To conclude, whenever the analysis of bodily tides is carried out using (1),
the analysis cannot be combined with a constant geometric lag (or phase lag,
or quality factor) assumption, nor with any other assumption different from
the frequency independence of $\,\Delta t\,$. This circumstance would not, by
itself, prohibit one from considering a material for which the factors 161616
We deliberately equip the quality factors with the subscript $\,l\,$, to
emphasise that they are different from the seismic $\,Q\,$ and have different
frequency dependencies for different $\,l\,$s (Efroimsky 2012a).
$~{}\,\frac{\textstyle k_{l}(\omega_{\textstyle{{}_{lmpq}}})}{\textstyle
Q_{l}(\omega_{\textstyle{{}_{lmpq}}})}~{}\mbox{Sgn}(\omega_{\textstyle{{}_{lmpq}}})\,=\,k_{l}\,\sin\epsilon_{l}~{}\,$
are insensitive to the frequency over some limited frequency band. Consistent
employment of this model will then require insertion of the same value of
$~{}\,k_{l}\,\sin\epsilon_{l}~{}\,$ into all terms of the expansion of the
torque (each term thus changing its sign as the corresponding resonance is
transcended). However, neither the quantity $\,(\dot{\nu}\,-\,\dot{\theta})\,$
nor its sign will come into play in this expression for the torque. So the
outcome will be different from the mathematically incorrect “constant angular
lag” model based on equation (1).

## Acknowledgments

The authors are grateful to Stanton Peale, who refereed the paper and whose
comments and recommendations were of great help. One of the authors (ME) is
indebted to Sylvio Ferraz Mello and James G. Williams for numerous
enlightening discussions on the theory of tides.

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