#
${{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}^{{\rm{Extended~{}version~{}of~{}a~{}paper~{}published~{}in~{}the~{}Astrophysical~{}Journal\,~{}795\;:\;6\,~{}(2014)}}}}$
Tidal dissipation in a homogeneous spherical body.
I. Methods

Michael Efroimsky and Valeri V. Makarov  US Naval Observatory, 3450
Massachusetts Avenue NW, Washington DC 20392<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A formula for the tidal dissipation rate in a spherical body is derived from
first principles, to correct some mathematical inaccuracies found in the
literature. The development is combined with the Darwin-Kaula formalism for
tides. Our intermediate results are compared with those by Zschau (1978) and
Platzman (1984). When restricted to the special case of an incompressible
spherical planet spinning synchronously without libration, our final formula
can be compared with the commonly used expression from Peale & Cassen (1978,
Eqn. 31). The two turn out to differ. In our expression, the contributions
from all Fourier modes are positive-definite, which is not the case with the
formula from Ibid. Examples of the application of our expression for the tidal
damping rate are provided in the work by Makarov & Efroimsky (2014).

##  1 Motivation and plan

The tidal heating of planets and moons has long been a key area of planetary
science. Accurate investigation into this process requires numerical
integration of dissipation over layers of the perturbed body. At the same
time, it is common to infer qualitative conclusions from approximations based
on modeling the body with a homogeneous sphere of a certain rheology. However,
the simplistic nature of the approach limits the precision of the ensuing
conclusions. For example, the presence of a sizable molten core, like in
Mercury, may increase the damping rate, compared to a homogeneous body. Still,
estimates obtained with our simplified, homogeneous-sphere model should be
accurate within a factor of several — thus (1) serving as a useful guidance
for solar system bodies and (2) being completely legitimate for exoplanets, as
our knowledge of their structure is speculative at best.

In our paper, we derive from the first principles a formula for the tidal
heating rate in a tidally perturbed homogeneous sphere. We compare our result
with the formulae used in the literature and point out the differences.

In Sections 3 \- 5, we present an accurate re-examination of the standard
integral expression for the damping rate in a homogeneous incompressible
sphere subject to tides. The check is necessary because in previous studies
the expression was derived in an ad hoc manner, sometimes with demonstrable
mathematical inaccuracies. The conventional derivation begins with the general
formula for the power

$\displaystyle
P\;=\;\int\,\rho_{{}_{E}}\;{\bf{v}}\,\cdot\,\nabla{V_{{}_{E}}}^{\prime}\;d^{3}r$

written in the Eulerian description (i.e., via coordinates associated with a
deformed body). Its time average is then cast into the form of

$\displaystyle\langle P\rangle~{}=~{}\frac{1}{4\pi
GR}~{}\sum_{l=2}^{\infty}\,(2\,l\,+\,1)\,\int~{}\left\langle~{}W_{l}~{}\stackrel{{\scriptstyle\bf\centerdot}}{{U}}_{l}~{}\right\rangle~{}dS$

which is in the Lagrangian language (an integral over an undeformed body). In
the former equation, $\,\rho_{{}_{E}}\,$ is the Eulerian density,
$\,{\bf{v}}\,$ is the Eulerian velocity, $\,{V_{{}_{E}}}^{\prime}\,$ is the
Eulerian perturbation of the potential (perturbation assembled of the tide-
raising potential and the resulting additional tidal potential of the deformed
body), and $\,r\,$ is a perturbed position in the body frame. In the latter
equation, $\,W_{l}\,$ and $\,U_{l}\,$ are the degree-$l\,$ components of the
tide-raising and additional tidal potentials, $\,G\,$ is the Newton gravity
constant, $\,R\,$ is the radius of the planet, and $\,dS\,$ is an element of
the undeformed surface of the sphere.

The transition from the former formula to the latter requires the use of the
boundary conditions on the free surface. At that point, integration is already
carried out within the Lagrangian description (over an undeformed surface),
but the boundary conditions are nonetheless imposed on the Eulerian potential
and its gradient. (The boundary conditions are much simpler in the Eulerian
form.) This mixed treatment requires attention, and its employment by the
early authors (Zschau 1978, Platzman 1984) contained inaccuracies. However
none of those turned out to be critical, and the above expression for the
average power $\,\langle P\rangle\,$ is correct for small deformations.

In Section 6 we explore the standard way of casting the above integral into a
spectral sum over the tidal Fourier modes $\,\omega\,$. It is commonly assumed
that the result should read as in Platzman (1984):

$\displaystyle\langle P\rangle~{}=~{}\frac{1}{4\pi
GR}~{}{\sum_{\omega}}(2l+1)\,\frac{\omega}{2}\,W^{\,2}_{l}(\omega)\,k_{l}(\omega)~{}\sin\epsilon_{l}(\omega)\,~{}.$

Here $\,k_{l}(\omega)\,$ and $\,\epsilon_{l}(\omega)\,$ are the Love number
and phase lag corresponding to the Fourier mode $\,\omega=\omega_{lmpq}\,$,
with $\,lmpq\,$ being the four integers wherewith the Fourier modes are
numbered in the Darwin-Kaula theory of tides (see Efroimsky & Makarov 2013 and
references therein). However, an accurate investigation demonstrates that the
spectral sum differs from the above. The difference originates for two
reasons. One is the degeneracy, i.e., the fact that several different Fourier
modes $\,\omega_{lmpq}\,$ share a numerical value $\,\omega\,$, so the
structure of the above sum is more complex. 111 When calculating $\,W_{l}\,$,
one has first to group together and sum all the terms corresponding to a
particular value of $\,\omega\,$. Each such sum should be squared and
averaged, and only after that should the final summation over the distinct
values of $\,\omega\,$ be carried out. In the original expression for the
average power, $\,~{}(\textstyle 4\pi
GR)^{-1}~{}{\sum_{\omega}}(2l+1)\,\frac{\textstyle\omega}{\textstyle
2}\,W^{\,2}_{l}(\omega)\,k_{l}(\omega)~{}\sin\epsilon_{l}(\omega)\,~{}$, the
$W^{\,2}_{l}(\omega)$ term should be replaced with the squared sum of all the
harmonics of $\,W\,$ that correspond to a particular value of $\,\omega\,$.
The second reason is that the modes can be of either sign, not necessarily
positive. So the resulting power will contain seemingly strange terms with
$\,W_{l}(\omega)\,W_{l}(-\omega)\,k_{l}(\omega)~{}\sin\epsilon_{l}(\omega)\,$.

These difficulties were noticed and analysed by Peale and Cassen (1978), but
their result needs correction too. Some terms in their spectral sum (equation
31 in Ibid.) are not positive definite, whence an underestimate of the heat
production may result. 222 In the expression for $\,\langle P\rangle\,$, an
input from each value of $\,\omega_{lmpq}\,$ must be non-negative. This can be
observed from the fact that the mode $\,\omega=\omega_{lmpq}\,$ and the
corresponding phase lag $\,\epsilon_{l}(\omega)\equiv\omega\,\Delta
t_{l}(\omega)\,$ are always of the same sign (the time lag $\,\Delta
t_{l}(\omega)\,$ being positive definite due to causality). Thus the product
$\,\omega\,\epsilon_{l}(\omega)\,=\,\omega_{lmpq}\,\epsilon_{l}(\omega_{lmpq})\,$
in the spectral sum can always be rewritten as $\,|\omega_{lmpq}|/Q_{lmpq}\,$,
with the tidal quality factor being defined via
$\,1/Q_{lmpq}\,=\,|\,\sin\epsilon_{l}(\omega_{lmpq})\,|\,$. In their spectral
sum, Peale & Cassen (1978, eqn 31) have just $\,\omega_{lmpq}/Q_{lmpq}\,$, and
not $\,|\omega_{lmpq}|/Q_{lmpq}\,$. As a result, some terms come out negative
and the heat production intensity may be underestimated.

The calculation of the power production, developed by Peale & Cassen (1978),
implies averaging not only over the tidal period but also over the apsidal
period. This can be observed from the formulae (20 - 21) in their work. In our
paper, however, we consider two separate cases: those with and without apsidal
precession. In the first case, the period of the apsidal precession is shorter
than the typical time of relaxation in the mantle (which may be identified
with the Maxwell time). The argument of the pericentre of the perturber,
$\,{\omega}^{*}\,$, cannot be treated as constant, wherefore the formula for
the mean power should be averaged not only over the tidal period, but also
over the period of the pericentre motion. (We assume this motion steady.) In
the second case, the evolution of the line of apsides is slow, with its period
being longer than the Maxwell time. The argument of the pericentre should be
regarded as a constant. Accordingly, in the latter case the tidal dissipation
formula is more complicated, because it includes explicit dependence of
Fourier terms on the argument of pericentre.

In a subsequent work, Makarov & Efroimsky (2014), we apply our results in
three case studies: Io, Mercury, and Kepler-10 b. In that paper we, among
other things, hypothesise that the tidal heating rate at spin-orbit resonances
is greatly influenced by libration and, therefore, by the triaxiality of the
tidally perturbed body.

##  2 The Darwin-Kaula formalism in brief

Describing of linear bodily tides consists of two steps. First, it is
necessary to Fourier-expand both the tide-raising potential and the induced
additional potential of the tidally perturbed body. Second, it is necessary to
link each Fourier component of the additional tidal potential to an
appropriate Fourier component of the tide-raising potential. This means:
establishing the phase lag and the ratio of magnitudes called the dynamical
Love number.

Due to interplay of rheology and self-gravitation, the phase lags and Love
numbers have nontrivial frequency dependencies. Things are complicated even
further because different mechanisms of friction become leading over different
frequency bands, wherefore the tidal response cannot be described by one
simple dissipation model (Efroimsky 2012 a,b).

###  2.1 Generalities

The development of the mathematical theory of bodily tides was started by
Darwin (1879) who derived a partial sum of the Fourier expansion of the
additional potential of a tidally perturbed sphere. A decisive contribution
into this theory was offered almost a century later by Kaula (1964) who wrote
down a complete series. In a previous paper (Efroimsky & Makarov 2013), we
provided a detailed presentation of the Darwin-Kaula expansion and explained
how tidal friction and lagging are built into it. We compared the Darwin-Kaula
theory with the one by MacDonald (1964) and demonstrated that the former
theory is superior to the latter, because it can, in principle, be combined
with an arbitrary rheology. Referring the reader to the afore-cited literature
for details, we present several central formulae that will be necessary.

An external body of mass $\,M^{\,*}\,$, located in
$\,{\mbox{{\boldmath$r$}}}^{\;*}=(r^{*},\,\lambda^{*},\,\phi^{*})\,$,
generates the following disturbing potential in a point
$\,\mbox{{\boldmath${R}$}}=(R,\phi,\lambda)\,$ on the surface of a sphere of
radius $\,R\,<\,r^{*}~{}$:

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

Here $\,G\,$ denotes Newton’s gravity constant, $\,\phi\,$ is the latitude
reckoned from the spherical body’s equator, $\,\lambda\,$ is the longitude
measured from a fixed meridian, and $\gamma$ is the angular separation between
the vectors $\,{\mbox{{\boldmath$r$}}}^{\;*}\,$ and ${R}$ pointing from the
perturbed body’s centre. The definitions of the Legendre polynomials
$\,P_{l}(\cos\gamma)\,$ and the associated Legendre polynomials
$\,P_{lm}(\sin\phi)\,$ are given in Appendix A.

While in the above formula the location of the perturber on its trajectory is
expressed through the spherical coordinates
$\,{\mbox{{\boldmath$r$}}}^{\;*}=(r^{*},\,\lambda^{*},\,\phi^{*})\,$, a
trigonometric transformation (developed by Kaula 1961) enables one to switch
to the perturber’s orbital elements
$\,\mbox{{\boldmath$r$}}^{\;*}=(\,a^{*},\,e^{*},\,{\it
i}^{*},\,\Omega^{*},\,\omega^{*},\,{\cal M}^{*}\,)\,$. In terms thereof, the
disturbing potential is expressed as

$\displaystyle
W(\mbox{{\boldmath${R}$}}\,,\;\mbox{{\boldmath$r$}}^{\;*})\;=\;\sum_{lmpq}\,W_{lmpq}\;=\;-\;\frac{G\,M^{*}}{a^{*}}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{a^{*}}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it l}\;\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.~{}~{}~{}-\;\delta_{0m}\,\right)\;P_{{\it{l}}m}(\sin\phi)\;\sum_{p=0}^{\it
l}\;F_{{\it l}mp}({\it i}^{*})\;\sum_{q=\,-\,\infty}^{\infty}\;G_{{\it
l}pq}(e^{*})\left\\{\begin{array}[]{c}\cos\\\ \sin\end{array}\right\\}^{{\it
l}\,-\,m\;\;\mbox{\small even}}_{{\it l}\,-\,m\;\;\mbox{\small
odd}}\;\left(v_{{\it
l}mpq}^{*}-m(\lambda+\theta^{*})\right)~{}~{}~{},~{}~{}~{}~{}~{}~{}$ (4)

where $\,\theta^{*}\,$ is the rotation angle of the tidally perturbed body,
333 When the equinoctial precession may be neglected, $\,\theta^{*}\,$ may be
regarded as the sidereal angle. while $\,F_{lmp}({\it i}^{*})\,$ and
$\,G_{lpq}(e^{*})$ are the inclination functions and the eccentricity
polynomials, respectively. The auxiliary linear combinations
$\,v_{lmpq}^{*}\,$ are defined by

$\displaystyle v_{lmpq}^{*}\;\equiv\;(l-2p)\,\omega^{*}\,+\,(l-2p+q){\cal
M}^{*}\,+\,m\,\Omega^{*}~{}~{}~{}.$ (5)

Conventionally, the letters denoting the elements of the perturber are
accompanied with asterisks: $\,a^{*},\,e^{*},\,{\it
i}^{*},\,\Omega^{*},\,\omega^{*},\,{\cal M}^{*}\,$. Following Kaula (1964),
the sidereal angle also acquires an asterisk, when it appears in a combination
$~{}v_{lmpq}^{*}-\,m\,\theta^{*}~{}$ with the perturber’s elements.

The angle $\,\theta\,$, however, does not acquire an asterisk, when it appears
in a linear combination $~{}v_{lmpq}-\,m\,\theta~{}$ with the orbital elements
of a test body subject to the additional tidal potential of the perturbed
body. This strange nomenclature introduced by Kaula (1964) — two different
notations for one angle — turns out to be helpful and convenient in the
calculation of the back-reaction experienced by the perturber. For
comprehensive explanation of this obscure point, see Section 5 in Efroimsky &
Makarov (2013).

Over timescales shorter than the apsidal-motion period, the expression in
round brackets in the formula (4) can be linearised as

$\displaystyle
v_{lmpq}^{*}-m(\lambda+\theta^{*})\,=\,\omega_{lmpq}\,(t\,-\,t_{0})~{}-~{}m~{}\lambda~{}+~{}v_{lmpq}^{*}(t_{0})~{}-~{}m~{}\theta^{*}(t_{0})~{}~{},$
(6)

where the following quantities act as the Fourier tidal modes:

$\displaystyle\omega_{\textstyle{{}_{lmpq}}}\;\equiv~{}\stackrel{{\scriptstyle\bf\centerdot~{}~{}~{}~{}}}{{v^{*}_{lmpq}}}\,-~{}m\,\stackrel{{\scriptstyle\bf\centerdot\,}}{{\theta^{*}}}~{}=\;(l-2p)\;\dot{\omega}^{*}\,+\,(l-2p+q)\;{\bf{\dot{\cal{M}}}}^{\,*}\,+\,m\;(\dot{\Omega}^{*}\,-\,\dot{\theta}^{*})~{}~{},$
(7)

${\bf{\dot{\cal{M}}}}^{\,*}\,$ being the perturber’s “anomalistic” mean motion
(see Section 2.3 below), and $\,t_{0}\,$ being the time of pericentre passage.
(As ever, we set $\,{\cal{M}}^{\,*}=\,0\,$ in the pericentre.) The modes
$\,\omega_{\textstyle{{}_{lmpq}}}\,$ can assume either sign, but the physical
forcing frequencies are positive definite:

$\displaystyle\chi_{\textstyle{{}_{lmpq}}}\,=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,~{}.$
(8)

###  2.2 Simplifying the notation: less asterisks

In the preceding subsection, we obeyed the convention by Kaula (1964) and
marked with asterisk the orbital elements of the tide-raising body. Kaula
introduced this notation, because within his model he also considered another
exterior body that was disturbed by the tides generated on the planet by the
tide-raising body. This exterior body’s elements were denoted by the same
letters, but without an asterisk.

When the two outer bodies coincide, the asterisks may be dropped, except on
two occasions. The first is writing the masses – while the mass of the planet
is denoted with $\,M\,$, the mass of the perturber (the star) will be written
as $\,M^{\,*}\,$. The other occasion requires writing the additional tidal
potential of the perturbed body – the potential will have a value
$\,U(\mbox{{\boldmath$r$}},\,\mbox{{\boldmath$r$}}^{\,*})\,$ in a point $r$ ,
provided the perturber (the star) is located in an exterior point
$\,\mbox{{\boldmath$r$}}^{\,*}\,$ (both vectors being planetocentric). The
planet’s rotation rate $\,\theta\,$, as well as the orbital elements of the
star as seen from the planet, will hereafter be written without asterisks.

The most important notations employed in this paper are connected in Table 1.

###  2.3 Difficulties

At this point, a word of warning is necessary. Deriving the equation (7), we
differentiated the expression (5), which gave us the terms with
$\,\dot{\omega}\,$ and $\,\dot{\Omega}\,$. Including these terms in the
equation (7) acknowledges the fact that the perturber’s trajectory is
disturbed, not Keplerian. The disturbance may come solely from tides, as in
Kaula (1964, eqn 38), or from both tides and other sources. One way or
another, the perturber’s mean anomaly $\,{\cal{M}}\,$ is no longer equal to
$\,n\,t\,$, but is now given by
$~{}\,{\cal{M}}\,=~{}{\cal{M}}_{0}\,+~{}\int^{\,t}_{t_{0}}n(t)~{}dt~{}\,$,
where $n(t)\,\equiv\,\sqrt{G\,(M\,+\,M^{*})~{}a^{-3}(t)\,}~{}.\,$ Accordingly,
the expression for the modes becomes:

$\displaystyle\omega_{\textstyle{{}_{lmpq}}}\;\equiv~{}\stackrel{{\scriptstyle\bf\centerdot~{}~{}~{}~{}}}{{v_{lmpq}}}\,-~{}m\,\stackrel{{\scriptstyle\bf\centerdot\,}}{{\theta}}~{}=\;(l-2p)\;(\dot{\omega}\,+\,\dot{\cal{M}}_{0})\,+\,q\,\dot{\cal{M}}_{0}\,+\,(l-2p+q)\;n\,+\,m\;(\dot{\Omega}\,-\,\dot{\theta})~{}~{}.\quad$
(9)

It is, of course, tempting to assume that
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{{\cal{M}}_{0}}}\,\ll\,n\,$, thus
accepting the approximation

$\displaystyle\stackrel{{\scriptstyle\bf\centerdot}}{{\cal{M}\,}}\,=~{}\stackrel{{\scriptstyle\bf\centerdot}}{{{\cal{M}}_{0}}}\,+~{}n~{}\approx~{}n~{}~{},$
(10)

as Kaula (1964) did in his equations (46 - 47). Within his theory, however,
this approximation could not be used. 444 In his books, Kaula (1966, 1968)
corrected this oversight. There, he kept the notation $\,n\,$ for the mean
motion defined as in the Kepler law, and never confused it with
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{\cal{M}\,}}\,$. This is explained
in Appendix B where we consider two examples. One is the case where
perturbation of an orbit of a moon is mainly due to the tides the moon creates
in the planet. In that situation, $\,\dot{\omega}\,$ and
$\,\dot{\cal{M}}_{0}\,$ are of the same order but of opposite signs, so they
largely compensate one another. This suggests a simultaneous neglect of both
rates. The second example is when the dominant perturbation of the orbit comes
from the oblateness of the primary. In this case, $\,\dot{\omega}\,$ and
$\,\dot{\cal{M}}_{0}\;$ turn out to be of the same order and of the same sign
– so keeping one of these terms requires keeping the other.

Whether one or both of these rates should be included depends on a particular
setting, and each practical case must be examined separately. In general, both
rates should be kept.

While keeping $\,\stackrel{{\scriptstyle\bf\centerdot\,}}{{\omega}}\,$
complicates the formalism, the emergence of
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{{\cal{M}}_{0}}}\,$ complicates the
treatment even further. To sidestep this issue, we shall define  the mean
motion via

$\displaystyle n\,\equiv\,{\bf{\dot{\cal{M}}}}~{}~{}.$ (11)

This, the so-called anomalistic  mean motion differs from
$\,\sqrt{G(M+M^{*})\,a^{-3}(t)\,}~{}$.

We shall derive the heat-production formulae for two different settings – with
a fixed pericentre $\,\omega\,$ and with $\,\omega\,$ moving uniformly.

###  2.4 Lagging

For a static tide, the incremental tidal potential of the perturbed body
mimics the perturbation (4), except that each term $\,W_{l}\,$ is now equipped
with a mitigating multiplier
$\,k_{\textstyle{{}_{l}}}\left({R}/{r}\right)^{\,l+1}\,$, where
$\,k_{\textstyle{{}_{l}}}\,$ is an $\,l-$degree Love number. With the star
located in $\,\mbox{{\boldmath$r$}}^{\,*}\,$, the additional potential in a
point $r$ will read as

$\displaystyle U(\mbox{{\boldmath$r$}}\,,\;\mbox{{\boldmath$r$}}^{\;*})$
$\displaystyle=$ $\displaystyle\sum_{{\it
l}=2}^{\infty}~{}U_{\it{l}}(\mbox{{\boldmath$r$}})~{}=~{}\sum_{{\it
l}=2}^{\infty}~{}k_{\it l}\;\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\;W_{\it{l}}(\mbox{{\boldmath${R}$}}\,,\;\mbox{{\boldmath$r$}}^{\;*})~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(12)

For time-dependent tides, this expression acquires an extra amendment: the
reaction must lag, compared to the action. Naively, this would imply taking
each $\,W_{l}\,$ at an earlier instant of time. However, in reality lagging
depends on frequency; so each $\,W_{l}\,$ must be first expanded into a
Fourier series over tidal modes, whereafter each term of the series should be
delayed separately. The magnitude of the tidal reaction is frequency dependent
too; so each term of the Fourier series will be multiplied by a dynamical Love
number of its own. Symbolically, this may be written in a manner similar to
the static expression:

$\displaystyle U(\mbox{{\boldmath$r$}}\,,\;\mbox{{\boldmath$r$}}^{\;*})$
$\displaystyle=$ $\displaystyle\sum_{{\it
l}=2}^{\infty}~{}U_{\it{l}}(\mbox{{\boldmath$r$}}\,,\;\mbox{{\boldmath$r$}}^{\;*})~{}=~{}\sum_{l=2}^{\infty}~{}\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\;\hat{k}_{l}\;W_{\it{l}}(\mbox{{\boldmath${R}$}}\,,\;\mbox{{\boldmath$r$}}^{\;*})~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(13)

The hat above $\,\hat{k}_{l}\,$ means that this is not a multiplier but a
linear operator that mitigates and delays each Fourier mode of $\,W_{l}\,$
differently:

$\displaystyle U(\mbox{{\boldmath$r$}}\,,\;\mbox{{\boldmath$r$}}^{\;*})$
$\displaystyle=$ $\displaystyle-\;\frac{G\,M^{*}}{a}\;\sum_{{\it
l}=2}^{\infty}\;\left(\,\frac{R}{r}\,\right)^{\textstyle{{}^{l+1}}}\left(\,\frac{R}{a}\,\right)^{\textstyle{{}^{\it
l}}}\sum_{m=0}^{\it
l}\;\frac{(l-m)!}{(l+m)!}\;\left(\,2~{}-\;\delta_{0m}\,\right)\;P_{lm}(\sin\phi)~{}\sum_{p=0}^{\it
l}\;F_{lmp}(i)~{}~{}~{}~{}$ (16)
$\displaystyle\left.~{}~{}~{}\right.\sum_{q=\,-\,\infty}^{\infty}\;G_{lpq}(e)~{}k_{l}(\omega_{lmpq})~{}\left\\{\begin{array}[]{c}\cos\\\
\sin\end{array}\right\\}^{{\it l}\,-\,m\;\;\mbox{\small
even}}_{l\,-\,m\;\;\mbox{\small
odd}}\;\left(\,v_{lmpq}~{}-\,m\,(\lambda~{}+~{}\theta)~{}-~{}\epsilon_{l}\,\right)~{}~{},~{}~{}\qquad~{}\,$

where the Love numbers $\,k_{l}(\omega_{lmpq})\,$ and the phase lags
$\,\epsilon_{\textstyle{{}_{l}}}(\omega_{lmpq})\,$ are functions of the
Fourier modes. The lags emerge as the products

$\displaystyle\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}=~{}\omega_{\textstyle{{}_{lmpq}}}~{}\Delta
t_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}~{},$ (17a) where $\Delta
t_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$ is the time delay at the mode
$\,\omega_{\textstyle{{}_{lmpq}}}\,$. In reality, the time delays are
functions not of the Fourier modes (which can assume either sign), but of the
actual physical forcing frequencies
$~{}\chi_{\textstyle{{}_{lmpq}}}\,=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|~{}$
which are positive definite. Thus it is more accurate to write the delays not
as $\Delta t_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$ but as $\Delta
t_{l}(\chi_{\textstyle{{}_{lmpq}}})\,$. Accordingly, the phase lags become
$\displaystyle\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}=~{}\omega_{\textstyle{{}_{lmpq}}}~{}\Delta
t_{l}(\chi_{\textstyle{{}_{lmpq}}})~{}~{}.$ (17b) The time delays are positive
definite due to causality, so the sign of the phase lag always coincides with
that of the corresponding Fourier mode. Thus we finally have:
$\displaystyle\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}=~{}|\,\omega_{\textstyle{{}_{lmpq}}}\,|~{}\,\mbox{Sgn}\,(\omega_{\textstyle{{}_{lmpq}}})~{}\,\Delta
t_{l}(\chi_{\textstyle{{}_{lmpq}}})~{}=~{}\chi_{\textstyle{{}_{lmpq}}}~{}\,\mbox{Sgn}\,(\omega_{\textstyle{{}_{lmpq}}})~{}\,\Delta
t_{l}(\chi_{\textstyle{{}_{lmpq}}})~{}~{},$ (17c)

where
$\,\chi_{\textstyle{{}_{lmpq}}}\equiv\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,$
are the positive definite forcing frequencies.

The dynamical Love number $\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$ and the
phase lag $\,\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$ are the absolute
value and the negative phase of the complex Love number
$\,\bar{k}_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$ whose functional dependence
upon the Fourier mode is solely determined by $\,l\,$, provided the body is
spherical. 555 For oblate celestial bodies, the functional form of the complex
$\,\bar{k}_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$ is also
determined by the order $\,m\,$. In that situation, the right notation for the
complex Love number is:
$\,\bar{k}_{\textstyle{{}_{lm}}}(\omega_{\textstyle{{}_{lmpq}}})\,$. Its
absolute value and negative phase will then be denoted with
$\,{k}_{\textstyle{{}_{lm}}}(\omega_{\textstyle{{}_{lmpq}}})\,$ and
$\,\epsilon_{\textstyle{{}_{lm}}}(\omega_{\textstyle{{}_{lmpq}}})\,$.

###  2.5 Physics behind the Love numbers and phase lags

As we saw above, to obtain the decomposition (16) from the Fourier series (4),
each $\,lmpq\,$ term of the latter had to be endowed with its own mitigating
factor
$\,k_{\textstyle{{}_{l}}}=k_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$
and phase lag
$\,\epsilon_{\textstyle{{}_{l}}}=\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$.
In the past, some authors enquired whether this mitigate-and-lag method was
general enough to describe tides. It is, as long as the tides are linear. This
is explained in Appendix C below.

The expression (16) for the additional tidal potential contains both sines and
cosines of the phase lags, and so does the ensuing expression for the surface
elevation. However, the resulting expression for the tidal dissipation rate
turns out to contain only the combination
$\,{k}_{\textstyle{{}_{l}}}(\omega)\;\sin\epsilon_{\textstyle{{}_{l}}}(\omega)\,$
which is the negative imaginary part of the complex Love number:

$\displaystyle{k}_{\textstyle{{}_{l}}}(\omega)\;\sin\epsilon_{\textstyle{{}_{l}}}(\omega)\;=\;|\bar{k}_{\textstyle{{}_{l}}}(\omega)|\;\sin\epsilon_{\textstyle{{}_{l}}}(\omega)\;=\;-\;{\cal{I}}{\it{m}}\left[\bar{k}_{\textstyle{{}_{l}}}(\omega)\right]~{}~{},~{}~{}~{}\mbox{where}\quad\omega=\omega_{\textstyle{{}_{lmpq}}}~{}~{}.$
(18)

This quantity is often denoted as $\,k_{l}/Q\,$, although it would be more
reasonable to employ the notation $\,k_{l}/Q_{l}\,$, with the tidal quality
factors defined through $\,1/Q_{l}\,\equiv\,\sin|\epsilon_{l}|\,$.

A dynamical Love number
$\,{k}_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$ is an even
function of the tidal mode $\,\omega_{\textstyle{{}_{lmpq}}}\,$, while a phase
lag $\,\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$ is
odd, as can be observed from the equation (17b). Thus the expression for the
product $\,{k}_{\textstyle{{}_{l}}}\,\sin\epsilon_{\textstyle{{}_{l}}}\,$ as a
function of the physical frequency
$\,\chi=\chi_{\textstyle{{}_{lmpq}}}\,\equiv\,|\omega_{\textstyle{{}_{lmpq}}}|\,$
is:

$\displaystyle{k}_{\textstyle{{}_{l}}}(\omega)\;\sin\epsilon_{\textstyle{{}_{l}}}(\omega)\;=\;{k}_{\textstyle{{}_{l}}}(\chi)\;\sin\epsilon_{\textstyle{{}_{l}}}(\chi)\;\,\mbox{Sgn}\,(\omega)~{}~{},$
(19)

where $\,\epsilon_{l}(\chi)\,$ is non-negative, because non-negative is the
physical frequency $\,\chi\,$.

The frequency dependence of
$\,{k}_{\textstyle{{}_{l}}}/Q_{l}={k}_{\textstyle{{}_{l}}}(\chi)\,\sin\epsilon_{\textstyle{{}_{l}}}(\chi)\,$
is defined by two major physical circumstances: self-gravitation of the planet
and the rheology of its mantle. A rheological law is expressed by a
constitutive equation, i.e., by an equation interconnecting the strain and the
stress. A particular form of this equation is determined by the friction
mechanisms present in the considered medium. A realistic rheological law
should contain contributions from elasticity, viscosity, and inelastic
processes (mainly, dislocation unjamming). Self-gravitation suppresses the
tidal bulge. At low frequencies this effectively adds to the mantle’s
rigidity, whereas at higher frequencies the interplay of rheology and gravity
is more complex (Efroimsky 2012 b, Figure 2).

The calculation of the frequency dependence
$\,{k}_{\textstyle{{}_{l}}}(\chi)\,\sin\epsilon_{\textstyle{{}_{l}}}(\chi)\,$
for a homogeneous body of a known size, mass and rheology is presented in
detail in Efroimsky (2012a,b). See also the Appendix to Makarov & Efroimsky
(2014).

While quadrupole ($\,l=2\,$) terms are sufficient in most problems, exceptions
are known. For the orbital evolution of Phobos, the $\,l=3\,$ and, possibly,
even $\,l=4\,$ terms of the Martian tidal potential may be of relevance (Bills
et al. 2005). Studying close binary asteroids, Taylor & Margot (2010) took
into account the Love numbers up to $\,l=6\,$.

The question of how rapidly $\,l>2\,$ terms fall off with the increase of the
degree $\,l\,$ is also interesting. Most authors only rely on the geometric
factor $\,(R/a)^{2l+1}\,$ to answer this question. As was explained in
Efroimsky (2012 b), the $\,l$-dependence of
$\,k_{l}(\omega_{lmpq})\,\sin\epsilon(\omega_{lmpq})\,$, too, comes into play
and changes the result considerably.

##  3 The Eulerian and Lagrangian descriptions

${\left.~{}~{}~{}~{}~{}~{}\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\right.}^{\mbox{\small\it
What~{}we~{}hope~{}ever~{}to~{}do~{}with~{}ease,}}$
${\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\right.}^{\mbox{\small\it
we~{}must~{}learn~{}first~{}to~{}do~{}with~{}diligence.}}$
${\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\right.}^{\mbox{\small\it~{}~{}~{}Samuel~{}Johnson
}}$

###  3.1 Notations and definitions

To compare the varying shape of a deformable body against some benchmark
configuration, we use $X$ to denote the initial position occupied by a
particle at $\,t=0\,$. At another time $\,t\,$, the particle finds itself in a
new place

$\displaystyle\mbox{{\boldmath$x$}}~{}=~{}\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,~{},$

where the function
$\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,$ is a
trajectory, i.e., a solution to the equation of motion, with the initial
condition $\,\mbox{{\boldmath$x$}}=\mbox{{\boldmath$X$}}~{}$ set at $\,t=0\,$.

The current values of all physical and kinematic properties of the medium can
be expressed as functions of the instantaneous coordinates $x$ of a point
where these properties are being measured at the present moment $\,t\,$. When
referred to the present time and position, such properties are named Eulerian
and are equipped with a subscript E ; for example:
$\,q_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)\,$. The Eulerian description is fit
to answer the question “where” and therefore is convenient in fluid dynamics
where the displacement $\,\mbox{{\boldmath$x$}}-\mbox{{\boldmath$X$}}\,$ can
become arbitrarily large and the initial position $X$ is soon forgotten.

While $x$ denotes a place in space, the initial condition $X$ acts as the
“number” of a particle presently residing at the place $x$ . Although located
now at $x$ , the particle originally came from $X$ and will carry the label
$X$ forever.

Knowing the trajectories of all particles, we can express the properties as
functions of the time $\,t\,$ and the initial conditions $X$ . To that end, we
employ the change of variables
$\,\mbox{{\boldmath$x$}}~{}=~{}\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,$.
Expressed through the initial conditions, a property $\,q\,$ will be termed as
Lagrangian and equipped with the subscript $\,L~{}$:

$\displaystyle
q_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}\equiv~{}q_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)\,~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(20a) or, in more detail: $\displaystyle
q_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}\equiv~{}q_{{}_{E}}(\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}}\,,\,t)\,,~{}t\,)\,~{}.$
(20b)

So $\,q_{{}_{L}}\,$ has the same value as $\,q_{{}_{E}}\,$, but has a
different functional form, as it is now understood as a function of the
initial conditions (the particles’ ‘numbers’) $X$ , and not of the present-
time coordinates $x$ . Relating the quantities to the initial positions $X$ ,
the Lagrangian description tells us “which particle” and is thus practical in
description of deformable solids.

In anticipation of perturbative treatment, we regard the trajectory
$\,\mbox{{\boldmath$x$}}~{}=~{}\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,$
as fiducial and equip the appropriate functional dependencies with a
superscript $\,0\,$:

$\displaystyle
q^{0}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}\equiv~{}q^{0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(21a) which is: $\displaystyle
q^{0}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}\equiv~{}q^{0}_{{}_{E}}(\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,,~{}t\,)\,~{}.$
(21b)

###  3.2 Perturbative approach

Under disturbance, two changes will take place in a point $r$ at a time
$\,t~{}$:

  * 1\.

Properties will now assume different values in this point at this time. So we
substitute the unperturbed Eulerian dependencies
$\,q^{0}_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)\,$ with

$\displaystyle
q_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}=~{}q^{0}_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}+~{}q\,^{\prime}_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)\,~{}.$
(22)

This equality, in fact, serves as a definition of the variation
$\,q\,^{\prime}_{{}_{E}}(\mbox{{\boldmath$r$}},~{}t)~{}$: the variation is a
change in the functional dependence of a physical property upon the present
position $r$

  * 2\.

A different particle will now appear in the point $r$ at the time $\,t\,$. It
will not be the same particle as the one expected there at the time $\,t\,$ in
the absence of perturbation.

Accordingly, a particle, which starts in $X$ at $~{}t=0\,$, will show up, at
the time $\,t\,$, not in the point
$\,\mbox{{\boldmath$x$}}=\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,$
but in some other location displaced by $\,{\bf{u}}~{}$:

$\displaystyle\mbox{{\boldmath$r$}}~{}=~{}\mbox{{\boldmath$x$}}~{}+~{}{\bf{u}}~{}=~{}\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)~{}+~{}{\bf{u}}(\mbox{{\boldmath$X$}},\,t)\,~{}.$
(23)

Both OF these changes, 1 and 2, will affect the Lagrangian dependencies of the
properties upon the initial conditions, so the dependency of each property
will acquire a variation
$~{}q\,^{\prime}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}$:

$\displaystyle
q_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}=~{}q^{0}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}+~{}q\,^{\prime}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)\,~{}.$
(24)

In Appendix D, we provide a self-sufficient introduction into the perturbative
treatment of a deformable body, both in the Eulerian and Lagrangian languages.
There we derive a relation between the perturbations of the Lagrangian and
Eulerian quantities:

$\displaystyle
q\,^{\prime}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}=~{}q\,^{\prime}_{{}_{E}}(\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,,~{}t\,)~{}+~{}{\bf{u}}(\mbox{{\boldmath$X$}},\,t)~{}\nabla_{\textstyle{{}_{x}\,}}q_{{}_{E}}~{}+~{}O({\bf{u}}^{2})~{}~{},$
(25)

with the gradient in the second term acting on the unperturbed history: 666 To
derive (25), we expanded
$\,q_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)=q_{{}_{E}}(\mbox{{\boldmath$x$}}+{\bf{u}},\,t)$
into the Taylor series near the unperturbed
$q_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)$.

$\displaystyle\nabla_{\textstyle{{}_{x\,}}}q_{{}_{E}}\,\equiv\,\nabla_{\textstyle{{}_{x}\,}}q_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)~{}\,~{},\,\quad\mbox{where}\qquad\mbox{{\boldmath$x$}}\,=\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)~{}~{}.$
(26)

In the formula (25), the first term on the right-hand side,
$~{}q\,^{\prime}_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)~{}$, accounts for the
change of the final spatial distribution of properties. The other two terms
show up because perturbation alters the mapping from $X$ to the present
position.

###  3.3 Summary of linearised formulae for the density
of a periodically deformed solid

We need several formulae for density perturbations, which are obtained in
Appendix D.

${\bf{\divideontimes}}~{}~{}~{}$ In the Eulerian description:

$\displaystyle\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,=~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$r$}})~{}+~{}{\rho_{{}_{E}}}^{\prime}(\mbox{{\boldmath$r$}},\,t)\,~{},~{}\qquad~{}\quad~{}$
(27)
$\displaystyle\rho_{{}_{E}}\,^{\prime}\,+\,\nabla_{\textstyle{{}_{r}\,}}\cdot(\rho^{\,0}_{{}_{E}}\,{\bf{u}})\,=\,0\,~{},~{}\qquad~{}\quad~{}$
(28)

Formula (27) renders the interrelation between the functions of the same
variable. The unperturbed density $\,\rho^{\,0}_{{}_{E}}\,$ appears here as a
function of the perturbed present positions $r$ , not of the unperturbed
reference positions $x$ . This can be traced through the derivation (D21 \-
D24). There, the unperturbed density initially shows up as a function of
$\,\mbox{{\boldmath$x$}}=\mbox{{\boldmath$r$}}-{\bf{u}}\,$. It then ends up as
a function of $r$ , after the Taylor expansion around $r$ over powers of
$\,{\bf{u}}\,$ is performed.

Accordingly, the symbol $\,\nabla_{\textstyle{{}_{r}\,}}\,$ denotes
differentiation with respect to the perturbed position $r$ upon which
$\,\rho^{\,0}_{{}_{E}}\,$ is set to depend in the above equations. Also
remember that in
$\,{\bf{u}}(\mbox{{\boldmath$x$}},\,t)={\bf{u}}(\mbox{{\boldmath$r$}},\,t)\,+\,O({\bf{u}}^{2})\,$
we can neglect $\,O({\bf{u}}^{2})\,$, in the linear approximation. Thus the
Lagrangian and Eulerian values of the displacement coincide in the first
order. Specifically, in the equation (28), our $\,{\bf{u}}\,$ can be treated
as a function of $r$ . So all entities in that equation are functions of the
same variable, the perturbed location.

${\bf{\divideontimes}}~{}~{}~{}$ In the Lagrangian description:

$\displaystyle\rho_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)\,=~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$X$}})~{}+~{}{\rho_{{}_{L}}}^{\prime}(\mbox{{\boldmath$X$}},\,t)\,~{},~{}\quad~{}\quad$
(29)
$\displaystyle\rho_{{}_{L}}\,^{\prime}\,+\,\rho^{\,0}_{{}_{E}}\,\nabla_{\textstyle{{}_{X}\,}}\cdot{\bf{u}}\,=\,0\,~{}.~{}\qquad~{}~{}\quad~{}$
(30)

Recall that this is an interrelation between functions of the same variable.
This time, it is the initial position $X$ . Had we altered the notation from
$X$ to $r$ , nothing would have changed (except that we would write
$\,\nabla_{\textstyle{{}_{r}\,}}\,$ instead of
$\,\nabla_{\textstyle{{}_{X}\,}}\,$) — it would still be the same relation
between three functions of the same argument.

${\bf{\divideontimes}}~{}~{}~{}$ Relation between the increments
$\,\rho_{{}_{L}}\,^{\prime}\,$ and $\,\rho_{{}_{E}}\,^{\prime}\,\;$:
This relation originates from the general formula (25). In our case the
reference trajectory
$\,\mbox{{\boldmath$x$}}=\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,$
stays identical to the initial position $X$ , so we obtain:

$\displaystyle\rho_{{}_{L}}\,^{\prime}(\mbox{{\boldmath$x$}},\,t)~{}=~{}\rho_{{}_{E}}\,^{\prime}(\mbox{{\boldmath$x$}},\,t)~{}+~{}{\bf{u}}(\mbox{{\boldmath$x$}},\,t)\cdot\nabla_{\textstyle{{}_{x}\,}}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)\,~{}.$
(31)

Once again, we are dealing with a relation between several functions taken all
at one and the same point. Here the point is denoted with $x$ . Had we denoted
it with $r$ , the only change would be a switch from
$\,\nabla_{\textstyle{{}_{x}}}\,$ to $\,\nabla_{\textstyle{{}_{r}}}\,$, no
matter what meaning we instill into these $x$ and $r$ .

For an initially homogeneous body, $\,\nabla\rho^{\,0}_{{}_{E}}\,=\,0\,$; so
the forms (28) and (30) of the linearised conservation law coincide and can
both be conveniently written as

$\displaystyle\rho^{\,0}~{}\nabla_{\textstyle{{}_{r}}}\cdot{{\bf{v}}}\,+\,\frac{\textstyle\partial\rho}{\textstyle\partial
t\,}\,=\,0\,~{},$ (32)

where $~{}\rho^{\,0}\equiv\rho^{\,0}_{{}_{E}}~{}$ and the velocity is

$\displaystyle{\bf{v}}~{}=~{}\frac{\partial{\bf{u}}}{\partial t}~{}~{}.$ (33)

###  3.4 Potentials and their increments

In each point, the density $\,\rho\,$ and potential $\,V\,$ comprise a mean
value and a perturbation:

$\displaystyle\mbox{density:}\qquad\quad\rho$ $\displaystyle=$
$\displaystyle\rho^{\,0}~{}+~{}\rho\,^{\prime}\,~{},\,\qquad\qquad$ (34a)
$\displaystyle\mbox{potential:}\qquad\quad V$ $\displaystyle=$ $\displaystyle
V^{\,0}\,+~{}V\,^{\prime}\,=~{}V^{\,0}\,+~{}(\,W~{}+~{}U\,)\,~{},$ (34b)

where $\,V^{\,0}\,$ is the constant-in-time spherically symmetrical potential
of an undeformed body, while $\,V\,^{\prime}\,$ denotes the potential’s
perturbation. The perturbation consists of the external tide-raising potential
$\,W\,$ and the resulting additional potential $\,U\,$ of the perturbed body:

$\displaystyle V\,^{\prime}~{}=~{}W~{}+~{}U\,~{}.$ (35)

The potentials and densities will be endowed with a subscript “$\,L\,$”  or
“$\,E\,$”  pointing at the Lagrangian or Eulerian descriptions, accordingly.
Owing to the general expression (25), we have:

$\displaystyle{V_{{}_{E}}}^{\prime}(\mbox{{\boldmath$r$}},\,t)~{}=~{}{V_{{}_{L}}}^{\prime}(\mbox{{\boldmath$x$}},\,t)~{}-~{}{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{\,0}_{{}_{E}}\,~{},$
(36)

the same being valid for $\,\rho\,$, see the equation (31). For unperturbed
properties, however, subscripts may be dropped without causing any confusion:

$\displaystyle
V^{\,0}~{}\equiv~{}V^{\,0}_{{}_{E}}\,\quad,\qquad\rho^{\,0}~{}\equiv~{}\rho^{\,0}_{{}_{E}}\,~{}.$
(37)

###  3.5 The Poisson equation in the Eulerian description

In both the perturbed and unperturbed settings, the density and potential are
always linked through the Poisson equation:

$\displaystyle\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,V_{{}_{E}}$
$\displaystyle=$ $\displaystyle-~{}4\,\pi\,G\,\rho_{{}_{E}}~{}~{},$ (38a)
$\displaystyle\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,V^{\,0}_{{}_{E}}$
$\displaystyle=$ $\displaystyle-~{}4\,\pi\,G\,\rho^{\,0}_{{}_{E}}~{}~{},$
(38b) while the perturbing potential $\,W\,$ obeys the Laplace equation
outside the perturber:
$\displaystyle\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,W_{{}_{E}}$
$\displaystyle=$ $\displaystyle 0~{}~{}.~{}\qquad\qquad$ (38c)

Subtraction of (38b) from (38a) results in a Poisson equation for the density
perturbation:

$\displaystyle\left.~{}\quad~{}\right.\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,{V_{{}_{E}}}^{\prime}~{}=~{}-~{}4~{}\pi~{}G~{}{\rho_{{}_{E}}}^{\prime}~{}~{}~{}$
(39)

The Poisson equation in the Lagrangian description is presented in Appendix D.

##  4 The power produced by the tidal force

###  4.1 In the Eulerian description

The power $\,P\,$ exerted on the perturbed body is an integral, over its
volume, of the rate of working by tidal forces on displacements. In the
Eulerian language, the power reads as

$\displaystyle
P\;=\;\int\,\rho_{{}_{E}}\;{{\bf{v}}}\,\cdot\,\nabla_{\textstyle{{}_{r}\,}}{V_{{}_{E}}}^{\prime}\;d^{3}r\,~{},$
(40)

the integration being performed over an instantaneous, deformed volume.
Together with

$\displaystyle\rho_{{}_{E}}~{}{{\bf{v}}}\cdot\nabla_{\textstyle{{}_{r}\,}}{V_{{}_{E}}}^{\prime}~{}=~{}\nabla_{\textstyle{{}_{r}\,}}\cdot(\,\rho_{{}_{E}}\,{{\bf{v}}}~{}{V_{{}_{E}}}^{\prime}\,)~{}-~{}{V_{{}_{E}}}^{\prime}~{}\nabla_{\textstyle{{}_{r}\,}}\cdot(\,\rho_{{}_{E}}\,{{\bf{v}}}\,)\,~{},~{}~{}$
(41)

the mass-conservation law

$\displaystyle\nabla_{\textstyle{{}_{r}\,}}\cdot(\,\rho_{{}_{E}}\,{\bf{v}}\,)\,+\,\frac{\textstyle\partial\rho_{{}_{E}}}{\textstyle\partial
t\,}\,=\,0\,~{}$ (42)

simplifies the expression under the integral to the following form:

$\displaystyle\rho_{{}_{E}}~{}{{\bf{v}}}\cdot\nabla_{\textstyle{{}_{r}\,}}{V_{{}_{E}}}^{\prime}~{}=~{}\nabla_{\textstyle{{}_{r}\,}}\cdot(\,\rho_{{}_{E}}\,{{\bf{v}}}\,{V_{{}_{E}}}^{\prime}\,)\,+\,{V_{{}_{E}}}^{\prime}~{}\frac{\,\partial{\rho_{{}_{E}}}^{\prime}}{\partial
t\,}\;\;~{}.$ (43a) Further employment of the Poisson equation in the Eulerian
form, (39), gives us
$\displaystyle\rho_{{}_{E}}~{}{{\bf{v}}}\cdot\nabla_{\textstyle{{}_{r}\,}}{V_{{}_{E}}}^{\prime}~{}=~{}\nabla_{\textstyle{{}_{r}\,}}\cdot(\,\rho_{{}_{E}}\,{{\bf{v}}}\,{V_{{}_{E}}}^{\prime}\,)~{}-~{}\frac{1}{4\,\pi\,G}~{}{V_{{}_{E}}}^{\prime}~{}\frac{\partial}{\partial
t}\,\nabla_{\textstyle{{}_{r}\,}}^{2}{V_{{}_{E}}}^{\prime}\;\;~{}.$ (43b)

So the power becomes

$\displaystyle P$ $\displaystyle=$
$\displaystyle\int\,\nabla_{\textstyle{{}_{r}\,}}\cdot(\,\rho_{{}_{E}}\,{{\bf{v}}}\,{V_{{}_{E}}}^{\prime}\,)\;d^{3}r~{}+~{}\int\,{V_{{}_{E}}}^{\prime}~{}\frac{\,\partial{\rho_{{}_{E}}}^{\prime}}{\partial
t\,}~{}d^{3}r$ (44a) $\displaystyle=$
$\displaystyle\int_{\Sigma^{t}}\,\rho_{{}_{E}}~{}{V_{{}_{E}}}^{\prime}\;{{\bf{v}}}\cdot
d{\bf{S}}^{t}~{}-~{}\frac{1}{4\,\pi\,G}~{}\int\,{V_{{}_{E}}}^{\prime}~{}\frac{\partial}{\partial
t\,}\,\nabla_{\textstyle{{}_{r}\,}}^{2}{V_{{}_{E}}}^{\prime}~{}d^{3}r\,~{},$
(44b)

where $\,d{\bf{S}}^{t}\,\equiv~{}{\bf{\hat{n}}}^{t}\;d\Sigma^{t}\,$, with
$\,{\bf{\hat{n}}}^{t}\,$ and $\,d\Sigma^{t}\,$ being a unit normal to the
deformed surface and an element of area on that surface, both taken at the
time $\,t\,$. Correct to the first order in the displacement $\,{\bf{u}}\,$,
these are related to their unperturbed analogues via

$\displaystyle{\bf{\hat{n}}}^{t}\,=\;(\,1~{}-~{}\nabla^{\Sigma}\otimes{\bf{u}}\,)\,{\bf{\hat{n}}}\qquad~{}\mbox{and}~{}\qquad
d\Sigma^{t}\,=\;\left(\,1~{}+~{}\nabla^{\Sigma}\cdot{\bf{u}}\,\right)\,d\Sigma~{}~{},$
(45)

where the surface gradient is defined as

$\displaystyle\nabla^{\Sigma}\,\equiv\,\nabla_{\textstyle{{}_{x}\,}}\,-~{}{\bf{\hat{n}}}~{}\partial_{{\bf{\hat{n}}}}\,~{},$
(46)

so $\,\nabla^{\Sigma}\otimes{\bf{u}}\,$ is a three-dimensional second-rank
tensor (Dahlen & Tromp 1998). Altogether,

$\displaystyle
d{\bf{S}}^{t}\,\equiv~{}{\bf{\hat{n}}}^{t}\;d\Sigma^{t}\,=\,\left(\,1\,+\,\nabla^{\Sigma}\cdot{\bf{u}}\,\right)\,{\bf{\hat{n}}}~{}d\Sigma~{}-~{}(\nabla^{\Sigma}\otimes{\bf{u}})\,{\bf{\hat{n}}}~{}d\Sigma~{}=~{}\left(\,1\,+\,\nabla^{\Sigma}\cdot{\bf{u}}\,\right)\,d{\bf
S}~{}-~{}(\nabla^{\Sigma}\otimes{\bf{u}})\,d{\bf S}\,~{},~{}~{}\,~{}$ (47a)
with $d{\bf S}\equiv{\bf{\hat{n}}}d\Sigma$ pertaining to the unperturbed
surface. In a shorter form, the above reads as $\displaystyle
d{\bf{S}}^{t}\,=~{}{\mathbb{J}}~{}d{\bf S}~{}~{},$ (47b)

where the three-dimensional second-rank tensor

$\displaystyle{\mathbb{J}}~{}\equiv~{}(\,1\,+\,\nabla^{\Sigma}\cdot{\bf{u}}\,)\,{\mathbb{I}}\,-\,\nabla^{\Sigma}\otimes{\bf{u}}$
(48)

is, loosely speaking, playing the role of a Jacobian for elements of area.
This is fully analogous to the formula

$\displaystyle
d^{3}r\,=\,J\,d^{3}x\,=\,(1\,+\,\nabla_{\textstyle{{}_{x}}}\cdot{\bf{u}})\,d^{3}x\,=\,[1\,+\,\nabla_{\textstyle{{}_{r}}}\cdot{\bf{u}}\,+\,O(u^{2})\,]\,d^{3}x$
(49)

linking the deformed volume $\,d^{3}r\,$ to the undeformed volume
$\,d^{3}x\,$. (See Appendix D.4.1.)

###  4.2 In the Lagrangian description

Applied to the density, the general formula (20) renders:

$\displaystyle\rho_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)\,\equiv\,\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,~{}~{}.$
(50)

This, together with the formula (36) for the potential perturbation, enables
us to express the power in the Lagrangian description:

$\displaystyle
P\;=\;\int\,\rho_{{}_{L}}\;{{\bf{v}}}\,\cdot\,\nabla_{\textstyle{{}_{x}\,}}{V_{{}_{L}}}^{\prime}\;d^{3}x~{}-~{}\int\,\rho_{{}_{L}}\;{{\bf{v}}}\,\cdot\,\nabla_{\textstyle{{}_{x}\,}}(\,{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{0}\,)\;d^{3}x\,~{},$
(51)

the integral now being taken over the undeformed body. Be mindful that
$\,d^{3}r\,\nabla_{\textstyle{{}_{r}\,}}\,=\,d^{3}x\,\nabla_{\textstyle{{}_{x}\,}}\,$,
so no Jacobian shows up on the right-hand side.

The velocity and displacement being in quadrature, the second term should be
dropped after time averaging (denoted with angular brackets):

$\displaystyle\langle
P\rangle\;=\;\int\,\rho_{{}_{L}}\;{{\bf{v}}}\,\cdot\,\nabla_{\textstyle{{}_{x}\,}}{V_{{}_{L}}}^{\prime}\;d^{3}x\,~{},$
(52a) For a periodically deformed solid, we set the equilibrium state to play
the role of the unperturbed configuration, for which reason 777 The mass is
conserved along both trajectories, perturbed and unperturbed. So both
$\,\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,d^{3}\mbox{{\boldmath$r$}}\,$ and
$\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)\,d^{3}\mbox{{\boldmath$x$}}\,$
must be equal to the initial mass
$\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$X$}})\,d^{3}\mbox{{\boldmath$X$}}\,$,
and therefore to one another:
$\,\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,d^{3}\mbox{{\boldmath$r$}}\,=\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)\,d^{3}\mbox{{\boldmath$x$}}\,$.
Thence,
$\,\rho_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}J~{}=~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$X$}})\,$,
where $\,J\equiv d^{3}\mbox{{\boldmath$r$}}/d^{3}\mbox{{\boldmath$x$}}\,$. In
combination with (50), this yields:
$\displaystyle\rho_{{}_{L}}(\mbox{{\boldmath$X$}}\,,~{}t)~{}J~{}=~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)~{}~{}.$
When the unperturbed configuration is the equilibrium state,
$\,\mbox{{\boldmath$x$}}=\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,$
coincides with $X$ at all times. So
$\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)\,$ bears no dependence on
time, and the above equality becomes simply
$\,\rho_{{}_{L}}(\mbox{{\boldmath$X$}}\,,~{}t)~{}J~{}=~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}})\,$.
See Appendix D.4.1 for a detailed discussion.
$\,~{}\rho_{{}_{L}}(\mbox{{\boldmath$X$}}\,,~{}t)~{}J~{}=~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}})\,$.
Insertion of this equality into the expression (52a) gives us:
$\displaystyle\langle
P\rangle\;=\;\int\,\rho^{\,0}\;\,{{\bf{v}}}\,\cdot\,\nabla_{\textstyle{{}_{x}\,}}{V_{{}_{L}}}^{\prime}\;J^{-1}\,d^{3}x\,~{}.$
(52b)

The dot-product can be easily rearranged via the formulae analogous to (41 \-
43). Due to

$\displaystyle\rho^{\,0}~{}{{\bf{v}}}\cdot\nabla_{\textstyle{{}_{x}\,}}{V_{{}_{L}}}^{\prime}~{}=~{}\nabla_{\textstyle{{}_{x}\,}}\cdot(\,\rho^{\,0}\,{{\bf{v}}}\,{V_{{}_{L}}}^{\prime}\,)\,-\,{V_{{}_{L}}}^{\prime}\,\nabla_{\textstyle{{}_{x}\,}}\cdot(\,{{\bf{v}}}\,\rho^{\,0}\,)$
(53)

and

$\displaystyle\rho^{\,0}~{}\nabla_{\textstyle{{}_{x}\,}}\cdot{{\bf{v}}}\,+\,\frac{\textstyle\partial\rho}{\textstyle\partial
t\,}\,=\,0\,~{},$ (54)

the expression under the integral becomes

$\displaystyle\rho^{\,0}~{}{{\bf{v}}}\cdot\nabla_{\textstyle{{}_{x}\,}}{V_{{}_{L}}}^{\prime}~{}=~{}\nabla_{\textstyle{{}_{x}\,}}\cdot(\,\rho^{\,0}\,{{\bf{v}}}\,{V_{{}_{L}}}^{\prime}\,)\,+\,{V_{{}_{L}}}^{\prime}~{}\frac{\,\partial\rho_{{}_{L}}\,^{\prime}}{\partial
t\,}\;\;~{},$ (55)

provided we set $\,\nabla_{\textstyle{{}_{x}\,}}\rho^{\,0}\,=0\,$, i.e.,
provided we assume that the unperturbed body is homogeneous. 888 No such
assumption was required to obtain the Eulerian analogue (42) of the Lagrangian
formula (54). Then the time-averaged power, for an initially homogeneous body,
acquires the form of

$\displaystyle\langle
P\rangle\;=\;\int\,\nabla_{\textstyle{{}_{x}\,}}\cdot(\,\rho^{\,0}\,{{\bf{v}}}\,{V_{{}_{L}}}^{\prime}\,)\;d^{3}x~{}+~{}\int\,{V_{{}_{L}}}^{\prime}~{}\frac{\,\partial\rho_{{}_{L}}\,^{\prime}}{\partial
t\,}\;d^{3}x\,~{},$ (56)

where we approximated the Jacobian with unity, thus neglecting higher-order
terms.

##  5 Tidal dissipation rate in a homogeneous sphere

Although the Eulerian and Lagrangian descriptions are equivalent, the boundary
conditions look simpler in the Eulerian picture. On the other hand, for
periodic deformations, practical calculations are easier carried out in the
Lagrangian description, as it implies integrations over the unperturbed volume
and surface corresponding to the equilibrium shape. It is, unfortunately, not
unusual for the authors to refrain from pointing out which description is
employed, leaving this to the discernment of the readers. The easiest way to
trace an author’s choice is to look at the way they write the expression for
the power and the Poisson equation.

The often-cited authors Zschau (1978) and Platzman (1984) started in the
Eulerian language and then switched to the Lagrangian description. This can be
seen from the fact that the time-average power was eventually written by both
of them as an integral over the undeformed  body. Both works contained some
mathematical omissions which, fortunately, did not influence the final form of
the integral.

Below we present these authors’ method in a more mathematically complete
manner. While our expression for the power, written as an integral over the
unperturbed surface, will coincide with the integrals derived by the said
authors, our final result (the power written as a spectral sum over the
Fourier modes) will differ. In one important detail, our result also differs
from that by Peale & Cassen (1978).

###  5.1 A mixed, Eulerian-Lagrangian treatment

Similar to Zschau (1978, eqn. 2), we begin with the formula (40) for the power
in the Eulerian variables. The next natural step is (44), whereafter
integration by parts renders:

$\displaystyle P~{}=\int\rho_{{}_{E}}\,{V_{{}_{E}}}^{\prime}\;{{\bf{v}}}\cdot
d{\bf{S}}^{t}\,-~{}\frac{1}{4\pi G}\,\int
d^{3}r~{}\nabla_{\textstyle{{}_{r}}}\cdot\left({V_{{}_{E}}}^{\prime}~{}\,\frac{\partial\nabla_{\textstyle{{}_{r}\,}}{V_{{}_{E}}}^{\prime}}{\partial
t}\,\right)~{}+~{}\frac{1}{4\pi G}\,\int
d^{3}r~{}\frac{\partial\nabla_{\textstyle{{}_{r}\,}}{V_{{}_{E}}}^{\prime}}{\partial
t}\,\cdot\,\nabla_{\textstyle{{}_{r}\,}}{V_{{}_{E}}}^{\prime}~{}~{}~{}~{}~{}~{}~{}$
(57a)
$\displaystyle=\int\rho_{{}_{L}}\,{V_{{}_{E}}}^{\prime}\,{{\bf{v}}}\cdot\left(\,{\mathbb{J}}~{}d{\bf{S}}\,\right)\,-~{}\frac{1}{4\pi
G}\,\int{V_{{}_{E}}}^{\prime}~{}\frac{\partial\nabla_{\textstyle{{}_{r}\,}}{V_{{}_{E}}}^{\prime}}{\partial
t}\,\cdot\left(\,{\mathbb{J}}\,d{\bf{S}}\,\right)\,+~{}\frac{1}{8\pi
G}~{}\frac{\partial}{\partial
t}\int(J\,d^{3}x)~{}\nabla_{\textstyle{{}_{r}\,}}{V_{{}_{E}}}^{\prime}\cdot\nabla_{\textstyle{{}_{r}\,}}{V_{{}_{E}}}^{\prime}\,~{}.~{}~{}~{}~{}~{}$
(57b)

En route from the former expression to the latter, we switch from
$\,d{\bf{S}}^{t}\,$ and $\,d^{3}r\,$ to $\,{\mathbb{J}}~{}d{\bf{S}}\,$ and
$\,J\,d^{3}x\,$, respectively. Thereby we switch from integration over a
deformed body to that over the undeformed one. So $\,\rho_{{}_{E}}\,$ becomes
$\,\rho_{{}_{L}}\,$, see the equation (50). A similar switch from
$\,{V_{{}_{E}}}^{\prime}\,$ to $\,{V_{{}_{L}}}^{\prime}\,$ can be performed
using the equation (36), but we prefer to stick to $\,{V_{{}_{E}}}^{\prime}\,$
for some time, for it will be easier to impose the boundary conditions on the
Eulerian potential.

In a leading-order calculation, both the Jacobian and its tensorial analogue
may be set unity: $\;{\mathbb{J}}\approx{\mathbb{I}}\;$ and $\;J\approx 1~{}$,
as evident from the formulae (48) and (49). In the same order, we can
substitute $\,\nabla_{\textstyle{{}_{r}\,}}\,$ with
$\,\nabla_{\textstyle{{}_{x}\,}}\,$. In addition, as was explained in Footnote
7, we can substitute $\,\rho_{{}_{L}}\,=\,\rho^{\,0}/J\,$ with
$\,\rho^{\,0}\,$, and can treat the latter as time-independent. Thus the time
average of the power becomes:

$\displaystyle\langle P\rangle$ $\displaystyle=$
$\displaystyle\int\,\left\langle\,\rho^{\,0}~{}{V_{{}_{E}}}^{\prime}\;{{\bf{v}}}\,\right\rangle\,\cdot\,d{\bf{S}}~{}-~{}\frac{1}{4\pi
G}~{}\int~{}\left\langle\,{V_{{}_{E}}}^{\prime}~{}\,\frac{\partial\,\nabla_{\textstyle{{}_{x}}}{V_{{}_{E}}}^{\prime}}{\partial
t}\,\right\rangle\,\cdot\,d{\bf{S}}$ (58a) $\displaystyle=$
$\displaystyle-~{}\frac{1}{4\pi
G}~{}\int~{}\left\langle~{}{V_{{}_{E}}}^{\prime}~{}\,\frac{\partial}{\partial
t}\,\left(\,\nabla_{\textstyle{{}_{x}\,}}{V_{{}_{E}}}^{\prime}\,-\,4\,\pi\,G\,\rho^{\,0}\,{\bf{u}}\,\right)\;\right\rangle\,\cdot\,d{\bf{S}}\,~{}.$
(58b) with the volume integral dropped. 999 As previously agreed, in our
approximation the Jacobian is set unity. The potential variation
$\,{V_{{}_{E}}}^{\prime}\,$ is a sum of sinusoidal harmonics, and so is its
gradient $\,\nabla_{\textstyle{{}_{x}\,}}{V_{{}_{E}}}^{\prime}\,$. After time
averaging of (57b), the cross terms in the product
$\,\nabla_{\textstyle{{}_{x}\,}}{V_{{}_{E}}}^{\prime}\,\cdot\,\nabla_{\textstyle{{}_{x}\,}}{V_{{}_{E}}}^{\prime}\,$
will vanish, while the products of harmonics of the same frequency will render
constants. The potential $\,{V_{{}_{E}}}^{\prime}\,$ in the above developments
was the interior  potential, so the above formula should, rigorously speaking,
have been written as $\displaystyle\langle P\rangle~{}=~{}-~{}\frac{1}{4\pi
G}~{}\int~{}\left\langle~{}{{V_{{}_{E}}}^{\prime}}^{\textstyle{{}^{\,(interior)}}}~{}\frac{\partial}{\partial
t}\,\left(\,\nabla_{\textstyle{{}_{x}\,}}{{V_{{}_{E}}}^{\prime}}^{\textstyle{{}^{\,(interior)}}}\,-\,4\,\pi\,G\,\rho^{\,0}\,{\bf{u}}\,\right)\;\right\rangle\,\cdot\,d{\bf{S}}\,~{}.$
(58c)

The expression (58c) is somewhat formal. On the one hand, it contains
integration over an undeformed surface, an operation appropriate to the
Lagrangian description. On the other hand, the quantity under the integral is
Eulerian, i.e., is a function of the perturbed positions. Thus, to employ the
expression (58c) in practical calculations, one would first have to express
the integrated average product
$~{}\left\langle~{}{{V_{{}_{E}}}^{\prime}}^{\textstyle{{}^{\,(interior)}}}~{}\frac{\textstyle\partial}{\textstyle\partial
t}\,\left(\,\nabla_{\textstyle{{}_{x}\,}}{{V_{{}_{E}}}^{\prime}}^{\textstyle{{}^{\,(interior)}}}\,-\,4\,\pi\,G\,\rho^{\,0}\,{\bf{u}}\,\right)\;\right\rangle~{}$
as a function of the unperturbed positions, i.e., of the coordinates on the
undeformed surface. Simply speaking, one would have to switch from a Eulerian
function under the integral to a Lagrangian function, using the formula (36).
The reason for our procrastination with this step is the convenience of the
Eulerian description for imposing boundary conditions.

###  5.2 Comparing the intermediate result (58c) with analogous
formulae from Zschau (1978) and Platzman (1984)

Our expression (58c) is equivalent to the formula (12) in Zschau (1978). The
sole difference is how we justify the substitution of the Lagrangian density
$\,\rho_{{}_{L}}\,$ with the unperturbed $\,\rho^{\,0}\,$. Whereas we
approximated the Jacobian with $\,1+O(|{\bf{u}}|)\,$, Zschau (1978, eqn. 10)
employed a clever trick that did not rely on the smallness of disturbance. In
our notation, the trick looks like this: if in the first term of our
expression (58a) we also keep the first-order perturbation
$\,{\rho_{{}_{L}}}^{\prime}\,$ of the density, the time average of the product
$\,{\rho_{{}_{L}}}^{\prime}\,{\bf{v}}\,{V_{{}_{E}}}^{\prime}\,\,$ will always
be zero, provided all three oscillate at the same frequency. While elegant,
Zschau’s argument works only for a perturbation at one frequency, not for a
spectrum of frequencies.

The treatment by Platzman (1984) contains more inaccuracies. The author’s
formula (2) looks like our equation (52b), with the actual density substituted
from the beginning by its unperturbed value $\,\rho^{\,0}\,$. Such a start
indicates the use of the Lagrangian description. This however comes into
contradiction with the way the author writes down the conservation law.
Platzman’s form of that law is equivalent to our equation (28), i.e., is
written in the Eulerian language. The following Poisson equation, too, is
Eulerian. That the author eventually arrives at the right integral expression
(equation 5 in Ibid.) is more due to luck than to accuracy. In the subsequent
derivation, the author’s formulae (7) and (10) are incorrect, because the fact
that the Fourier modes in the Darwin-Kaula theory can be of either sign is
neglected. We shall address this point at the end of Section 5.4.

###  5.3 Employment of the boundary conditions

The Eulerian boundary conditions mimic those from electrostatics (see Appendix
E):

$\displaystyle{{V_{{}_{E}}}^{\prime}}^{\,\textstyle{{}^{(interior)}}}~{}=~{}{{V_{{}_{E}}}^{\prime}}^{\,\textstyle{{}^{(exterior)}}}~{}~{}$
(59)

and

$\displaystyle\left[~{}\frac{\partial~{}}{\partial\hat{\bf{n}}}\,{V_{{}_{E}}}^{\prime}~{}-~{}4~{}\pi~{}G~{}\rho^{\,0}~{}{\bf
u}~{}\right]^{\,\textstyle{{}^{(exterior)}}}\,=~{}\left[~{}\frac{\partial~{}}{\partial\hat{\bf{n}}}\,{V_{{}_{E}}}^{\prime}~{}-~{}4~{}\pi~{}G~{}\rho^{\,0}~{}{\bf
u}~{}\right]^{\,\textstyle{{}^{(interior)}}}\,~{}.$ (60)

Insertion thereof into the equation (58c) makes the power look

$\displaystyle\langle P\rangle~{}=~{}-~{}\frac{1}{4\pi
G}~{}\int~{}\left\langle~{}{{V_{{}_{E}}}^{\prime}}^{\textstyle{{}^{\,(exterior)}}}~{}\frac{\partial}{\partial
t}~{}\nabla_{\textstyle{{}_{x}\,}}{{V_{{}_{E}}}^{\prime}}^{\textstyle{{}^{\,(exterior)}}}\;\right\rangle\,\cdot\,d{\bf{S}}\,~{}.$
(61)

It is now high time to write the expression under the integral (61) as a
function of the coordinates on the unperturbed surface, the one over which we
integrate. The formula (36) prescribes us to substitute
$\,{{V_{{}_{E}}}^{\prime}}\,$ with
$\,{V_{{}_{L}}}^{\prime}-\,{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V_{0}\,$.
As $\,{\bf{u}}\,$ is zero outside the body, we get 101010 For the first
multiplier under the integral (61), we simply substitute
$\,{{V_{{}_{E}}}^{\prime}}^{\textstyle{{}^{\,(exterior)}}}\,$ with
$\,{{V_{{}_{L}}}^{\prime}}^{\textstyle{{}^{\,(exterior)}}}\,$, omitting the
term
$\,\left[\,-\,{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V_{0}\,\right]^{\textstyle{{}^{\,(exterior)}}}\,$
because $\,{\bf{u}}\,$ is zero outside the body.
The case of the second multiplier,
$\,\frac{\textstyle\partial}{\textstyle\partial
t}~{}\nabla_{\textstyle{{}_{x}\,}}{{V_{{}_{E}}}^{\prime}}^{\textstyle{{}^{\,(exterior)}}}\,$,
is less obvious. Employment of the formula (36) furnishes us
$\,\frac{\textstyle\partial}{\textstyle\partial
t}\,\left[\,\nabla_{\textstyle{{}_{x}\,}}{V_{{}_{L}}}^{\prime}\,-\,\nabla_{\textstyle{{}_{x}}}\cdot({\bf{u}}\,V^{\,0})\,\right]^{\textstyle{{}^{\,(exterior)}}}$.
The vanishing of $\,{\bf{u}}\,$ on the exterior side of the boundary does not
imply the vanishing of its gradient there. On the contrary,
$\,\nabla_{\textstyle{{}_{x}}}\cdot({\bf{u}}\,V^{\,0})\,$ performs a finite
step – but so also does the gradient of $\,{V_{{}_{L}}}^{\prime}\,$, so that
altogether the gradient $\,{V_{{}_{E}}}^{\prime}\,$ remains continuous. To
sidestep these intricacies, we can expand the volume of integration slightly
outward from the actual volume of the planet (Platzman 1984, p. 74).

$\displaystyle\langle P\rangle~{}=~{}-~{}\frac{1}{4\pi
G}~{}\int~{}\left\langle~{}{{V_{{}_{L}}}^{\prime}}^{\textstyle{{}^{\,(exterior)}}}~{}\frac{\partial}{\partial
t}~{}\nabla_{\textstyle{{}_{x}\,}}{{V_{{}_{L}}}^{\prime}}^{\textstyle{{}^{\,(exterior)}}}\;\right\rangle\,\cdot\,d{\bf{S}}\,~{},$
(62)

To analyse the behaviour of $\,V\,^{\prime}\,$ outside the perturbed body,
recall that its two components, $\,U\,$ and $\,W\,$, scale differently with
the planetocentric radius. As can be seen from (1), the degree-$l\,$ Legendre
component of the perturbing potential changes as 111111 Do not be misled by
the planetocentric distance in (1) being denoted with $\,R\,$. There we needed
the value of $\,W\,$ on the surface, whereas here we need to know $\,W\,$ at
an arbitrary planetocentric distance. $~{}W_{l}\,\propto\,\,r^{\,l}\,$.
According to (13), the degree-$l\,$ component of the tidal potential obeys
$~{}U_{l}\,\propto\,r^{-(l+1)}\,$. All in all, the $\,l-$degree part of the
exterior $\,V\,^{\prime}\,$ assumes the form of

$\displaystyle{{V_{{}_{L}}}^{\prime}}^{\,\textstyle{{}^{(exterior)}}}=~{}\sum_{l=2}^{\infty}\,\left[\,\left(\,\frac{r}{R}\,\right)^{\textstyle{{}^{l}}}\,W_{l}(R)\,+\,\left(\,\frac{r}{R}\,\right)^{\textstyle{{}^{-(l+1)}}}\,U_{l}(R)\,\right]\,~{},$
(63)

while the normal part of its gradient on the free surface is

$\displaystyle\frac{\partial\,}{\partial
r}~{}{{V_{{}_{L}}}^{\prime}}_{\textstyle{{}_{l}}}^{\,\textstyle{{}^{(exterior)}}}~{}=~{}R^{-1}\,\sum_{l=2}^{\infty}\,\left[\,l~{}W_{l}\,-\,(l+1)~{}U_{l}\,\right]\,~{}.$
(64)

Plugging it into (62), and benefitting from the orthogonality of surface
harmonics, we obtain: 121212 On the boundary, we have:
$\,{{V_{{}_{L}}}^{\prime}}(R)\,=\,\sum_{l=2}^{\infty}\,\left[\,W_{l}(R)\,+\,U_{l}(R)\,\right]~{}$,
as evident from (63). Together with (64), this expression was inserted in
(62). By doing so, we omitted the diagonal products $\,W_{l}\,\dot{W}_{l}\,$
and $\,U_{l}\,\dot{U}_{l}\,$ that vanish after time averaging. (Indeed,
$\,W_{l}\,$ is in quadrature with $\,\dot{W}_{l}\,$, while $\,U_{l}\,$ is in
quadrature with $\,\dot{U}_{l}\,$.) En route from (65a) to (65b), we took into
account that the time averages of $\,\partial(U_{l}\,W_{l})/\partial t\,$ also
vanish.

$\displaystyle\langle P\rangle$ $\displaystyle=$
$\displaystyle-~{}\frac{1}{4\pi
GR}~{}\sum_{l=2}^{\infty}\,\int~{}\left\langle~{}l~{}U_{l}\,\stackrel{{\scriptstyle\bf\centerdot}}{{W}}_{l}\,-\,(l+1)~{}W_{l}\,\stackrel{{\scriptstyle\bf\centerdot}}{{U}}_{l}~{}\right\rangle~{}dS$
(65a) $\displaystyle=$ $\displaystyle\frac{1}{4\pi
GR}~{}\sum_{l=2}^{\infty}\,(2\,l\,+\,1)\,\int~{}\left\langle~{}W_{l}~{}\stackrel{{\scriptstyle\bf\centerdot}}{{U}}_{l}~{}\right\rangle~{}dS\,~{},$
(65b)

which is equivalent to the formulae (18) in Zschau (1978) and (5) in Platzman
(1984). This, however, is the last point on which we are still in agreement
with our predecessors.

###  5.4 Writing the integral as a spectral sum

Bringing in the dynamical Love numbers $\,k_{l}\,$ and the phase lags defined
in (16), one can express the products $\,W_{l}(t)\,\dot{U}_{l}(t)\,$ via the
spectral components of the disturbance $\,W(t)~{}$. 131313 In this subsection,
$\,\omega\,$ is a shortened notation for the mode $\,\omega_{lmpq}\,$, not the
argument of the pericentre.

Although the formula

(66)

is often used in the literature (Zchau 1978, Platzman 1984, Segatz et al.
1988), 141414 Our expression (66) is identical to the upper line of the
equation (10) in Platzman (1984). (Note a misprint on that line of Platzman’s
equation: a missing factor of $\,\omega\,$.)
Our formula (66), when truncated to $\,l=2\,$, also becomes equivalent to the
equation (22) in Zschau (1978) and to the equation (12) in Segatz et al.
(1988). (Both authors kept only the degree-2 terms.) accurate examination
demonstrates that it is incorrect. To appreciate this, one simply has to
insert the expansions (4) and (16) into the formula (65b) and see what
happens.

That the answer differs from (66) was noticed by Peale & Cassen (1978).
However, their development also needs correction. Below, we dwell upon this
matter in great detail and provide a full inventory of the terms emerging in
the spectral expansion for damping rate. At this point, we only mention the
two key circumstances:

  * (a)

The conventional expression (66) ignores the degeneracy of modes, i.e., a
situation where several modes $\,\omega_{\textstyle{{}_{lmpq}}}\,$ with
different sets $\,lmpq\,$ take the same numerical value $\,\omega\,$. As will
be demonstrated in the Section 6, the sum over modes $\,\omega\,$ in (66)
should be substituted with a sum over distinct values  of the modes:

$\displaystyle\mbox{instead~{}of}~{}~{}\sum_{\omega}W_{l}^{2}(\omega)~{}~{}\mbox{in~{}(\ref{26})$\,,\,$~{}use~{}this}~{}:~{}~{}\sum_{\omega}\left[\,\sum_{{\omega_{\textstyle{{}_{{}_{lmpq}}}}=~{}\omega}}W_{l}(\omega_{lmpq})\,\right]^{2}$
$\displaystyle\mbox{where}~{}\sum_{{\omega_{\textstyle{{}_{{}_{lmpq}}}}=~{}\omega}}W_{l}(\omega_{lmpq})~{}~{}\mbox{denotes~{}a~{}sum~{}of~{}all~{}terms~{}for~{}which~{}}\,\omega_{lmpq}\,~{}\mbox{takes~{}a~{}value~{}}\,\omega~{}.\qquad\qquad\qquad\qquad\qquad$

In short: first sum all the terms corresponding to one value of $\,\omega~{}$,
then square the sum, and only thereafter sum over all the values of
$\,\omega\,$.

  * (b)

Much less intuitive is the fact that the spectral sum will contain extra terms
missing completely in the expression (66). As we shall see in Appendix G,
these terms look (up to some caveat) as $\,W_{l}(\omega)\,W_{l}(-\omega)\,$.
They show up because two modes of opposite values, $\,\omega\,$ and
$~{}-\omega\,$, correspond to the same physical frequency $\,|\,\omega\,|\,$.

For the time being, we use the notation $\,\sum^{\,\textstyle\sharp}~{}$:

$\displaystyle\sum_{l=2}^{\infty}(2l+1)\,\left\langle
W_{l}(t)\,\dot{U}_{l}(t)\right\rangle~{}=~{}{\sum_{\omega}}^{\textstyle{\sharp}~{}}(2l+1)\,\frac{\omega}{2}\,W^{\,2}_{l}(\omega)\,k_{l}(\omega)~{}\sin\epsilon_{l}(\omega)\,~{},$
(67)

where the superscript $\,{}^{\textstyle{\sharp}~{}}\,$ reminds the reader that
the spectral sum needs to be amended down the road.

Insertion of (67) into (65b) results in: 151515 Were we using complex
potentials, we would have $\,W_{l}\,\dot{U}_{l}^{*}\,$ instead of
$\,W_{l}\,\dot{U}_{l}\,$ in (65b), and would have $\,W_{l}\,W_{l}^{*}\,$
instead of $\,W_{l}\,W_{l}\,$ in (67).

$\displaystyle\langle\,P\,\rangle~{}=~{}\frac{1}{8\pi
GR}~{}{\sum_{\omega}}^{\textstyle{\sharp}~{}}\,(2\,l\,+\,1)~{}\omega\,k_{l}(\omega)~{}\sin\epsilon_{l}(\omega)\int~{}W^{{\,2}}_{l}(\omega)~{}dS\,~{}~{}.~{}~{}$
(68)

If not for the superscript $\,{}^{\textstyle{\sharp}~{}}\,$, this expression
would coincide with the results by Zschau (1978) and Platzman (1984). 161616
Our expression (68) should be compared to the equation (22) from Zschau
(1978), in understanding that our expression furnishes the mean damping rate
summed over the entire spectrum, whereas Zschau’s formula renders the energy
loss over a period, at a certain frequency. With these details taken into
account, the formulae are equivalent. They are also equivalent to the formulae
(10) and (12) in Segatz et al. (1988) and (10) in Platzman (1984). Note,
however, that in the first line of Platzman’s formula a factor of $\,\omega\,$
is missing.  The superscript reminds us of the important caveat in the
evaluation of the sum: the factors $\,W^{\,2}_{l}(\omega)\,$ should be
substituted with more complicated expressions, whereas the sum should be
carried not over all modes $\,\omega=\omega_{\textstyle{{}_{lmpq}}}\,$, but
over all distinct values  of $\,\omega\,$, see Section 6.

##  6 Heat production over tidal modes

We must insert the expansions (4) and (16) into the formula (65b) for the
heating rate, in order to obtain a comprehensive version of the somewhat
symbolic sum (67) and to see what the modified sum
$\,\sum^{\textstyle{{}^{\,\sharp}}}\,$ actually means. A sketchy version of
this calculation (which takes into account that the modes may have either
sign, but neglects the degeneracy of modes) is given in Appendix G.
Extraordinarily laborious, the full calculation is presented in Appendix H.
Here we provide the final results.

In the case of a uniformly  moving pericentre, the average dissipation rate
is:

$\displaystyle\langle\,P\,\rangle~{}=~{}$
$\displaystyle\frac{G\,{M^{*}}^{\,2}}{a}\sum_{l=2}^{\infty}\left(\frac{R\,}{a}\right)^{\textstyle{{}^{2l+1}}}\sum_{m=0}^{l}\frac{(l-m)!}{({\it
l}+m)!}\left(2-\delta_{0m}\right)\sum_{p=0}^{l}F^{\,2}_{lmp}(i)\sum_{q\,=-\infty}^{\infty}G^{\,2}_{lpq}(e)\,\chi_{\textstyle{{}_{lmpq}}}\,k_{l}(\chi_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\chi_{\textstyle{{}_{lmpq}}})\,~{},~{}~{}~{}$
(69)

where the physical frequencies are the absolute values of the Fourier modes:

$\displaystyle\chi_{\textstyle{{}_{lmpq}}}\,=~{}|\,\omega_{\textstyle{{}_{lmpq}}}\,|~{}=~{}|\,(l-2p)\;\dot{\omega}\,+\,(l-2p+q)\;{\bf{\dot{\cal{M}}}}\,+\,m\;(\dot{\Omega}\,-\,\dot{\theta})\,|~{}\approx~{}|\,(l-2p+q)\;n~{}-~{}m~{}\dot{\theta}\,|\,~{},~{}~{}~{}$
(70)

and $\,\sin\epsilon_{l}(\chi_{\textstyle{{}_{lmpq}}})\,$ is what they often
call $\,1/Q_{\textstyle{{}_{l}}}\,$ in the literature. 171717 It would not
hurt to reiterate that the Fourier modes $\,\omega_{\textstyle{{}_{lmpq}}}\,$
can be of either sign, while the physical forcing frequencies (70) are
positive definite. Obviously,
$~{}\chi_{\textstyle{{}_{lmpq}}}\,k_{l}(\chi_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\chi_{\textstyle{{}_{lmpq}}})\,=\,\omega_{\textstyle{{}_{lmpq}}}\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$,
because the dynamical Love numbers are even functions, whereas the phase lags
are odd and of the same sign as their argument. This is why the tidal quality
factors may be expressed as
$\,1/Q_{\textstyle{{}_{l}}}\,=\,\sin\epsilon_{l}(\chi_{\textstyle{{}_{lmpq}}})\,$
and also as
$\,1/Q_{\textstyle{{}_{l}}}\,=\,|\,\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,|\,$,
with the absolute value symbols being redundant in the former formula and
needed in the latter.

In the Appendix H, we also derive a formula for an idle pericentre; but the
applicability realm of that formula is limited. 181818 For an idle pericentre,
the time-averaged tidal-heating power reads as:
$\displaystyle\langle\,P\,\rangle~{}=~{}\frac{G\,{M^{*}}^{\,2}}{a}\,\sum_{l=2}^{\infty}\,\left(\frac{R}{a}\right)^{\textstyle{{}^{2\,l\,+\,1}}}\sum_{m=0}^{l}~{}\frac{(l-m)!}{({\it
l}+m)!}\;\left(\,2-\delta_{0m}\,\right)\,\sum_{p=0}^{l}F_{lmp}(i)\;\sum_{p\,^{\prime}=0}^{l}F_{lmp\,^{\prime}}(i)\qquad\,\qquad\qquad\qquad\qquad\qquad\qquad~{}$
$\displaystyle\left.\qquad\qquad\qquad\right.\sum_{q\,=-\infty}^{\infty}G_{lpq}(e)~{}\left[\,G_{lp\,^{\prime}q\,^{\prime}}(e)\,\right]_{\textstyle{{}_{q\,^{\prime}\,=\,q\,-\,2\,(p-p\,^{\prime})}}}~{}\cos\left(\,2\,(p\,^{\prime}\,-\,p)\,\omega_{0}\,\right)\,~{}\omega_{\textstyle{{}_{lmpq}}}\,\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\,~{},~{}~{}\qquad$
$\omega_{0}\,$ being the value of the pericentre. This formula is of a limited
practical value, since $\,\omega_{0}\,$ seldom stays idle. For example, if we
are computing tidal damping in a planet perturbed by the star,
$\,\omega_{0}\,$ of the star as seen from the planet will be evolving due to
the equinoctial precession of the planet equator.

Our formula (69) differs from the appropriate expression in Kaula (1964, Eqn
28) that contains a redundant factor $\,(1+k_{\,l})/2\,$.

In the special situation where

  * (a)

$~{}l=2\,$,

  * (b)

the body is incompressible, so $\,k_{2}\,=\,3\,h_{2}/5\,$, 191919 Static Love
numbers of an incompressible spherical planet satisfy the relation
$\;(2l+1)\,k_{l}\,=\,3\,h_{l}\;$. As explained in Appendix F, an analogue of
this equality for dynamical Love numbers is
$\;(2l+1)\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})\,=3\,h_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$.

  * (c)

the spin is synchronous, with no libration,

the expression (69) should be compared to the formula (31) from Peale & Cassen
(1978). This comparison is carried out in Appendix H. In our expression, all
terms are positive-definite, because the factors
$\,\omega_{\textstyle{{}_{2mpq}}}\,k_{2}(\omega_{\textstyle{{}_{2mpq}}})\,\sin\epsilon_{2}(\omega_{\textstyle{{}_{2mpq}}})\,$
are even functions of the tidal mode $\,\omega_{\textstyle{{}_{2mpq}}}\,$.
Peale & Cassen (1978), however, have their terms proportional to the products
202020 In the notation of Peale & Cassen (1978), these products are written as
$~{}\,\frac{\textstyle 3}{\textstyle
5}\,h_{2}\,\frac{\textstyle{2-2p+q-m}}{\textstyle{Q_{2mpq}}}\,$.
$\,\omega_{\textstyle{{}_{2mpq}}}\,k_{2}(\omega_{\textstyle{{}_{2mpq}}})\,\sin|\epsilon_{2}(\omega_{\textstyle{{}_{2mpq}}})|\,~{}$
which are negative for negative $\,\omega_{\textstyle{{}_{2mpq}}}\,$. In the
considered setting, the largest of such terms were of the order of
$\,e^{4}\,$. Such inputs lead to an underestimation of the heat production
rate in the situations where the eccentricity is sufficiently high (like in
the case of the Moon whose eccentricity might attain high values in the past
due to a three-body resonance with the Sun).

##  7 Conclusions

We have derived from the first principles a formula for the tidal dissipation
rate in a homogeneous spherical body. En route  to that formula, we compared
our intermediate results with those by Zschau (1978) and Platzman (1984). When
restricted to the special case of an incompressible spherical planet spinning
synchronously without libration, our final formula can be compared with the
commonly used expression from Peale & Cassen (1978, Eqn. 31). The two turn out
to differ. In our expression, the contributions from all Fourier modes are
positive-definite, which is not the case of the formula from Ibid. As a
result, employment of the formula from Peale & Cassen (1978) may cause
underestimation of tidal heat production. For example, our calculations for
the rate of energy dissipation in the Moon with its current parameters yield a
value approximately a factor of 2 greater than the result derived from the
classic equation, with the difference coming from the non-resonant terms
having the correct sign.

We therefore propose to use our equation (69) for rocky planets and moons,
instead of the classic formula from Peale & Cassen (1978, eqn 31), because (a)
our expression is more accurate for the basic case of objects captured into
the 1:1 resonance, and (b) it correctly captures the frequency dependence of
tidal dissipation for objects outside the 1:1 resonance.

Several applications are provided in the work by Makarov and Efroimsky (2014).

## Acknowledgments

M.E. is indebted to Jeroen Tromp and Mikael Beuthe for pointing out the
advantage of the Lagrange description, and to Gabriel Tobie for a very useful
exchange on tidal heating.

The authors are grateful to James G. Williams for meticulous reading of the
manuscript and very judicious comments that were of great help.

The authors’ special thanks go to the referee, Patrick A. Taylor, whose
thoughtful and comprehensive report enabled the authors to improve the quality
of the paper significantly.

This research has made use of NASA’s Astrophysics Data System.

Appendix

##  Appendix A The associated Legendre functions and their normalisation

The Legendre polynomials are usually defined by the Rodriguez formula:

$\displaystyle P_{l}(x)\,=\,\frac{\textstyle 1}{\textstyle
2^{\,l}\,l!}~{}\frac{\textstyle d^{\,l}}{\textstyle
dx^{\,l}}\left(\,x^{2}\,-\,1\,\right)^{l}~{}.~{}$ (A1)

The associated Legendre functions $\,P_{lm}(x)\,$ (termed associated Legendre
polynomials  when their argument is sine or cosine of some angle) were
introduced by Ferrers (1877) as 212121 Sometimes in the literature they also
use the functions  $\displaystyle
P_{l}^{m}(x)=(-1)^{m}\,\left(1-x^{2}\right)^{m/2}\,\frac{d^{m}\,}{dx^{m}}\,P_{l}(x)=(-1)^{m}\,P_{lm}(x)\,~{},~{}~{}~{}$
as defined, e.g., in Abramowitz & Stegun (1972, p. 332). There also exists a
different convention wherein $\,P_{l}^{m}(x)\,$ lacks the $\,(-1)^{m}\,$
multiplier and thus coincides with $\,P_{lm}(x)\,$, as in Arfken & Weber
(1995, p. 623).

$\displaystyle
P_{lm}(x)=\left(1-x^{2}\right)^{m/2}\,\frac{d^{m}\,}{dx^{m}}\,P_{l}(x)~{}\,~{},\quad\mbox{where}\quad
l\geq m\geq 0\,~{}.$ (A2)

The so-defined associated Legendre functions are sometimes called
unnormalised, although a more accurate term would be: in Ferrers’
normalisation. This normalisation reads as:

$\displaystyle\int_{-1}^{1}P_{lm}(x)~{}P_{\,l\,^{\prime}\,m}(x)~{}dx~{}=~{}\frac{2}{2\,l\,+\,1}~{}\,\frac{(l+m)!}{(l-m)!}\,~{}\delta_{\,l\,l\,^{\prime}}~{}$
(A3a) or, equivalently:
$\displaystyle\int_{-\pi/2}^{\pi/2}P_{lm}(\sin\phi)~{}P_{\,l\,^{\prime}\,m}(\sin\phi)~{}\cos\phi~{}d\phi~{}=~{}\frac{2}{2\,l\,+\,1}\,~{}\frac{(l+m)!}{(l-m)!}~{}\,\delta_{\,l\,l\,^{\prime}}~{}~{},$
(A3b) another equivalent form being
$\displaystyle\int_{0}^{\pi}P_{lm}(\cos\varphi)~{}P_{\,l\,^{\prime}\,m}(\cos\varphi)~{}\sin\varphi~{}d\varphi~{}=~{}\frac{2}{2\,l\,+\,1}\,~{}\frac{(l+m)!}{(l-m)!}\,~{}\delta_{\,l\,l\,^{\prime}}~{}~{}.$
(A3c)

The associated Legendre functions in Ferrers’ normalisation should not be
confused with the associated Legendre functions $\,\tilde{P}_{lm}(x)\,$ which
are written in the Schmidt partial normalisation:

$\displaystyle\int_{-1}^{1}\tilde{P}_{lm}(x)~{}\tilde{P}_{\,l\,^{\prime}\,m}(x)~{}dx~{}=~{}\frac{2}{2\,l\,+\,1}\,~{}(2\,-\,\delta_{\,0\,m})~{}\,\delta_{\,l\,l\,^{\prime}}\,~{}.$
(A4)

For more on these normalisations, see Winch et al. (2005).

##  Appendix B Keeping $\,\dot{\omega}\,$ implies either keeping
$\,\dot{\cal{M}}_{0}\,~{}$ or defining $\,{\cal{M}}\,$ as $\,dn/dt\,$

Under disturbance, the mean anomaly is written as

$\displaystyle{\cal{M}}\,=~{}{\cal{M}}_{0}\,+~{}\int^{\,t}_{t_{0}}n(t)~{}dt~{}~{},~{}~{}~{}\mbox{where}\qquad
n(t)\,\equiv\,\sqrt{G\,(M\,+\,M^{*})~{}a^{-3}(t)\,}~{},$ (B1)

so the expression (7) for the Fourier tidal modes acquires the form of

$\displaystyle\omega_{\textstyle{{}_{lmpq}}}\;\equiv~{}\stackrel{{\scriptstyle\bf\centerdot~{}~{}~{}~{}}}{{v_{lmpq}}}\,-~{}m\,\stackrel{{\scriptstyle\bf\centerdot\,}}{{\theta}}~{}=\;(l-2p)\;(\dot{\omega}\,+\,\dot{\cal{M}}_{0})\,+\,q\,\dot{\cal{M}}_{0}\,+\,(l-2p+q)\;n\,+\,m\;(\dot{\Omega}\,-\,\dot{\theta})~{}~{}.\quad$
(B2)

Kaula (1964, equation 40) makes an oversight by accepting the approximation

$\displaystyle\stackrel{{\scriptstyle\bf\centerdot}}{{\cal{M}\,}}\,=~{}\stackrel{{\scriptstyle\bf\centerdot}}{{{\cal{M}}_{0}}}\,+~{}n~{}\approx~{}n~{}~{}.$
(B3)

Indeed, as $\,\dot{\omega}\,$ and $\,\dot{\cal{M}}_{0}\;$ are often of the
same order, it is incorrect to keep the former while neglecting the latter. We
present two examples. In the first, $\,\dot{\omega}\,$ and
$\,\dot{\cal{M}}_{0}\;$ are of the same order but of opposite signs, so they
largely compensate one another. This suggests a simultaneous neglect of both
terms. In the second example, $\,\dot{\omega}\,$ and $\,\dot{\cal{M}}_{0}\;$
turn out to be of the same order and the same sign, so keeping one of these
terms requires keeping the other.

###  B.1 Example 1. Tidal perturbation of a low-inclination, low-eccentricity
orbit

Consider a low-inclined perturber. From the tides it creates, the perturber
gets predominantly transversal orbital disturbance. We need two planetary
equations in the Gauss form (Brouwer & Clemence 1961, page 301, eqn 33):
222222 The system (33) in Brouwer & Clemence (1961, page 301) contains an
equation for the rate $\,d\epsilon/dt\,$, where (as explained on the preceding
page in Ibid.) $\,\epsilon\,$ is understood as
$\,\epsilon^{{{\,I}}}\,\equiv\,{\cal{M}}_{0}\,+\,\tilde{\omega}\,=\,{\cal{M}}_{0}\,+\,\omega\,+\,\Omega\,$.

$\displaystyle\stackrel{{\scriptstyle\bf\centerdot\,}}{{\omega}}~{}=~{}\frac{\sqrt{1\,-\,e^{2}}}{n~{}a~{}e}~{}\left[\,-~{}R~{}\cos
f~{}+~{}\left(\,1\,+\,\frac{r}{p}\,\right)~{}T~{}\sin
f\,\right]~{}-~{}\frac{\,\sin(\omega\,+\,f)~{}\,\cot
i\,}{n~{}a~{}\sqrt{1\,-\,e^{2}}}~{}\,\frac{\,r\,}{a}~{}W~{}~{},$
$\displaystyle\stackrel{{\scriptstyle\bf\centerdot}}{{{\cal{M}}_{0}}}~{}=~{}\frac{1\,-\,e^{2}}{n~{}a~{}e}~{}\left[\,\left(\,\cos
f~{}-~{}2~{}\,\frac{\,r\,}{p}\,~{}e\,\right)\,R~{}-~{}\left(\,1\,+\,\frac{r}{p}\,\right)~{}T~{}\sin
f\,\right]~{}~{},$

where $\,f\,$ is the true anomaly, $\,p\,\equiv\,a\,(1\,-\,e^{2})\,$ is the
semilatus rectum , while $\,R\,$, $\,T\,$, and $\,W\,$ are the radial,
transversal, and normal-to-orbit forces, respectively. In a situation where
the perturbation is predominantly transversal and the terms with $\,R\,$ and
$\,W\,$ may be neglected, we obtain:

$\displaystyle\stackrel{{\scriptstyle\bf\centerdot\,}}{{\omega}}\,+~{}\stackrel{{\scriptstyle\bf\centerdot}}{{{\cal{M}}_{0}}}~{}\approx~{}\frac{e}{1\,+\,\sqrt{1\,-\,e^{2}}}~{}\frac{\sqrt{1\,-\,e^{2}}}{n~{}a}~{}\left(\,1\,+\,\frac{r}{p}\,\right)~{}T~{}\sin
f~{}~{}.$ (B4)

A low-inclined moon gets predominantly transversal orbital disturbance from
the tides it creates in the planet. Inserting the latter expression in the
formula (B2) for the Fourier modes, we see that for the modes with a zero
$\,q\,$ (like the semidiurnal tide parameterised with $\,lmpq\,=\,2200\,$) the
input from the pericentre rate may be omitted, if the eccentricity is not too
large. Indeed, for $\,q=0\,$, the term $\,q\dot{\cal{M}}_{0}\,$ vanishes,
while the term $\,(l-2p)(\dot{\omega}+\dot{\cal{M}}_{0})\,$ is now
approximated with $\,(l-2p)\,$ multiplied by the expression (B4). Although
$\,\dot{\omega}\,$ and $\,\dot{\cal{M}}_{0}\,$ can, separately, be
substantial, their sum (B4) is smaller by the order of $\,e\,$. Being (in this
particular case) of the same order but of opposite sign, $\,\dot{\omega}\,$
and $\,\dot{\cal{M}}_{0}\,$ largely compensate one another. Therefore, if we
choose to drop $\,\dot{\cal{M}}_{0}\,$, we should also drop
$\,\dot{\omega}\,$. In this special case, dropping of both will be legitimate.

As a useful aside, we would remind that the mean longitude is defined through
$\,L\,\equiv\,{\cal{M}}\,+\,\omega\,+\,\Omega\,$, its rate being
$~{}\dot{L}\,\equiv\,\sqrt{G\,(M\,+\,M^{*})~{}a^{-3}(t)\,}\,+\,\dot{\cal{M}}_{0}\,+\,\dot{\omega}\,+\,\dot{\Omega}~{}$.
As we have just seen, the rates $\,\dot{\cal{M}}_{0}\,$ and $\,\dot{\omega}\,$
largely compensate one another and may both be neglected in the considered
case. If, above that, the rate of the node happens to be negligible, then the
mean motion from the Kepler law will be close to the mean longitude rate.

###  B.2 Example 2. Orbital perturbation due to oblateness

The situation is different where the principal perturbation is due to the
oblateness of the tidally perturbed primary. The mean rates (Vallado 2007, pp.
647 - 648)

$\displaystyle\stackrel{{\scriptstyle\bf\centerdot}}{{{\cal{M}}_{0}}}~{}=~{}\frac{\,3\,}{\,4\,}~{}n~{}J_{2}~{}\frac{R^{2}}{a^{2}}~{}\frac{\,2~{}-~{}3~{}\sin^{2}i\,}{(1~{}-~{}e^{2})^{3/2}}~{}~{}\qquad\mbox{and}\qquad~{}\stackrel{{\scriptstyle\bf\centerdot\,}}{{\omega}}~{}=~{}\frac{\,3\,}{\,4\,}~{}n~{}J_{2}~{}\frac{R^{2}}{a^{2}}~{}\frac{\,4~{}-~{}5~{}\sin^{2}i\,}{(1~{}-~{}e^{2})^{2}}~{}~{}.$

are of the same order and sign. Therefore, when keeping
$\,\stackrel{{\scriptstyle\bf\centerdot\,}}{{\omega}}\,$, we must also include
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{{\cal{M}}_{0}}}\,$.

The easiest way to get rid of
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{{\cal{M}}_{0}}}\,$ is to define the
mean motion as $\,n\,\equiv\,{\bf{\dot{\cal{M}}}}\,$. This, the so-called
anomalistic  mean motion will, however, differ from
$~{}\,\sqrt{G\,(M\,+\,M^{*})~{}a^{-3}(t)\,}~{}$.

##  Appendix C Universality of the Darwin-Kaula description

As we saw earlier, to obtain the decomposition (16) from the Fourier series
(4), each term of the latter series must be endowed with a mitigating factor
$\,k_{\textstyle{{}_{l}}}=k_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$
of its own and, likewise, must acquire its own phase lag
$\,\epsilon_{\textstyle{{}_{l}}}=\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$.
In the literature, some authors enquired whether this mitigate-and-lag method
is general enough to describe tides. The answer to this question is
affirmative, insofar as the tides are linear. Without going into details (to
be found in Efroimsky 2012 a, b), we would mention that an $\,l-$degree part
of the operator (13) is a convolution called the Love operator:

$\displaystyle
U_{l}(\mbox{{\boldmath$r$}},\,t)\;=\;\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{-\infty}^{t}{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})~{}W_{\it{l}}(\mbox{{\boldmath${R}$}}\,,\;\mbox{{\boldmath$r$}}^{\;*},\;t\,^{\prime})\,dt\,^{\prime}~{},$
(C1)

Indeed, linearity of tides means that, at each time $\,t\,$, the overall
magnitude of reaction depends linearly on the magnitudes of the disturbance at
all preceding instants of time, $\,t\,^{\prime}\leq t\,$. The emergence of
inputs from earlier times stems from the inertia (“memory”) of the material. A
disturbance that took place at an instant $\,t\,^{\prime}\,$ appears in the
integral for $\,U_{l}(\mbox{{\boldmath$r$}},\,t)\,$ with a weight
$\,{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$ that depends on
the elapsed time. Following Churkin (1998), who gave this formalism its
present shape, we call these weights Love functions.

In the frequency domain, the above convolution becomes:

$\displaystyle\bar{U}_{\textstyle{{}_{l}}}(\omega)\;=\;\left(\,\frac{R}{r}\,\right)^{l+1}\bar{k}_{\textstyle{{}_{l}}}(\omega)\;\,\bar{W}_{\textstyle{{}_{l}}}(\omega)\;\;,$
(C2)

where $\,\omega=\omega_{\textstyle{{}_{lmpq}}}\,$ is the tidal mode (not the
periapse); $\,\bar{U}_{\textstyle{{}_{l}}}(\omega)\,$ and
$\,\bar{W}_{\textstyle{{}_{l}}}(\omega)\,$ are the Fourier or Laplace images
of the potentials $\,{U}_{\textstyle{{}_{l}}}(\mbox{{\boldmath$r$}},\,t)\,$
and
$\,{W}_{\textstyle{{}_{l}}}(\mbox{{\boldmath${R}$}},\,\mbox{{\boldmath$r$}}^{*},\,t)\,$;
while the complex Love numbers

$\displaystyle\bar{k}_{\textstyle{{}_{l}}}(\omega)\;=\;|\bar{k}_{\textstyle{{}_{l}}}(\omega)|\;e^{\textstyle{{}^{-i\,\epsilon_{{{}_{l}}}(\omega)}}}\;=\;{k}_{\textstyle{{}_{l}}}(\omega)\;e^{\textstyle{{}^{-i\,\epsilon_{{{}_{l}}}(\omega)}}}~{}~{}$
(C3)

are the Fourier or Laplace components of the Love functions
$\,{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$. The actual
dynamical Love numbers are the real parts of the complex Love numbers,
$\,{k}_{\textstyle{{}_{l}}}(\omega)\,=\,|\bar{k}_{\textstyle{{}_{l}}}(\omega)|~{}$;
while the tidal lags are the complex Love numbers’ negative phases.

The frequency dependencies $\,\bar{k}_{\textstyle{{}_{l}}}(\omega)\,$ and,
consequently, $\,{k}_{\textstyle{{}_{l}}}(\omega)\,$ and
$\,{\epsilon}_{\textstyle{{}_{l}}}(\omega)~{}$ can be derived from the
expression for the complex compliance $\,\bar{J}(\chi)\,$ or the complex
rigidity $\,\bar{\mu}(\chi)=1/\bar{J}(\chi)\,$ of the mantle (with
$\,\chi=\chi_{\textstyle{{}_{lmpq}}}\,\equiv\,|\omega_{\textstyle{{}_{lmpq}}}|\,$
being the physical forcing frequency). The dependency $\,\bar{J}(\chi)\,$
follows from the rheological model.

Evidently, the formula (C2) is but a concise version of (16). Thus we see that
the mitigate-and-lag method ensues directly from the linearity assumption.

##  Appendix D The Eulerian and Lagrangian descriptions.
Perturbative approach to a periodically deformed body

###  D.1 Perturbative treatment

Under perturbation, two changes will happen in a point $r$ at a time $\,t~{}$:

  * 1\.

Physical fields will now acquire different values in this point at this time.

For example, a moon fixed in a certain position relative to the planetary
surface will render some distribution of its potential over the volume of the
host planet. The same moon fixed in a different position will generate a
different distribution of its potential. This, in its turn, will yield a
different deformation of the planet and therefore a different spatial
distribution of its tidal-response potential and of all other quantities.

Thus, instead of the unperturbed Eulerian dependencies
$\,q^{0}_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)\,$, we now have

$\displaystyle
q_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}=~{}q^{0}_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}+~{}q\,^{\prime}_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)\,~{}.$
(D1)

  * 2\.

A different particle will now arrive in the point $r$ at the time $\,t\,$. It
will not be the same particle as the one expected there at the time $\,t\,$ in
the absence of perturbation.

On the other hand, a particle that starts in $X$ at $~{}t=0\,$, will appear,
at the time $\,t\,$, not in the point
$\,\mbox{{\boldmath$x$}}=\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,$
but in some other place

$\displaystyle\mbox{{\boldmath$r$}}~{}=~{}\mbox{{\boldmath$x$}}~{}+~{}{\bf{u}}~{}=~{}\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)~{}+~{}{\bf{u}}(\mbox{{\boldmath$X$}},\,t)\,~{}.$
(D2)

These two changes will influence the Lagrangian dependencies on the initial
conditions. The dependency of each field will obtain a variation
$~{}q\,^{\prime}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}$:

$\displaystyle
q_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}=~{}q^{0}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}+~{}q\,^{\prime}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)\,~{}.$
(D3)

In the absence of perturbation, the particle $X$ was destined to arrive in $x$
, wherefore $\,q^{0}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)\,$ was defined
through (21). Under perturbation, the same particle $X$ is expected to end up
in $r$ , so the Lagrangian dependency becomes

$\displaystyle q_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)$ $\displaystyle\equiv$
$\displaystyle q_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)$ (D4a)
$\displaystyle=$ $\displaystyle
q_{{}_{E}}(\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,+\,{\bf{u}}(\mbox{{\boldmath$X$}},\,t)\,,~{}t\,)$
(D4b) $\displaystyle=$ $\displaystyle
q_{{}_{E}}(\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,,~{}t\,)~{}+~{}{\bf{u}}(\mbox{{\boldmath$X$}},\,t)~{}\nabla_{\textstyle{{}_{x}\,}}q_{{}_{E}}~{}+~{}O({\bf{u}}^{2})$
(D4c) $\displaystyle=$ $\displaystyle
q^{0}_{{}_{E}}(\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,,~{}t\,)~{}+~{}q\,^{\prime}_{{}_{E}}(\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,,~{}t\,)~{}+~{}{\bf{u}}(\mbox{{\boldmath$X$}},\,t)~{}\nabla_{\textstyle{{}_{x}\,}}q_{{}_{E}}~{}+~{}O({\bf{u}}^{2})~{}~{}.\qquad~{}\qquad$
(D4d)

Insertion of (D3) into the left-hand side of (D4) will give us:

$\displaystyle
q^{0}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}+~{}q\,^{\prime}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}=~{}q^{0}_{{}_{E}}(\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,,~{}t\,)~{}+~{}q\,^{\prime}_{{}_{E}}(\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,,~{}t\,)~{}+~{}{\bf{u}}(\mbox{{\boldmath$X$}},\,t)~{}\nabla_{\textstyle{{}_{x}\,}}q_{{}_{E}}~{}+~{}O({\bf{u}}^{2})~{}~{}.$

Subtracting (21b) from this formula, we arrive at a relation between the
perturbations of the Lagrangian and Eulerian quantities:

$\displaystyle
q\,^{\prime}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}=~{}q\,^{\prime}_{{}_{E}}(\,\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,,~{}t\,)~{}+~{}{\bf{u}}(\mbox{{\boldmath$X$}},\,t)~{}\nabla_{\textstyle{{}_{x}\,}}q_{{}_{E}}~{}+~{}O({\bf{u}}^{2})~{}~{},$
(D5)

where the first term on the right-hand side,
$~{}q\,^{\prime}_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)~{}$, expresses the change
in the final spatial distribution of the field $\,q\,$. The other two terms
show up because perturbation changes the mapping from $X$ to the current
location.

###  D.2 An equivalent description

A slightly different, although equally valid viewpoint is possible. In a
reference setting, at time $\,t\,$, an observer located in $x$ will see the
arrival of a particle that started from $X$ :

$\displaystyle
q^{0}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}\equiv~{}q^{0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(D6)

In a perturbed situation, the same observer in $x$ will register, at the time
$\,t\,$, the arrival of a different particle, one that started from
$\,\mbox{{\boldmath$X$}}-{\bf{U}}~{}$:

$\displaystyle
q_{{}_{L}}(\mbox{{\boldmath$X$}}-{\bf{U}},\,t)~{}\equiv~{}q_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}$
(D7a) which is: $\displaystyle
q_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}-~{}{\bf{U}}~{}\nabla_{{}_{X}}q_{{}_{L}}~{}+~{}O(U^{2})~{}=~{}q_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}$
(D7b)

Subtraction of (D6) from (D7b) gives us the variations:

$\displaystyle
q_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}-~{}q^{0}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}-~{}{\bf{U}}~{}\nabla_{{}_{X}}q_{{}_{L}}~{}+~{}O(U^{2})~{}=~{}q_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)~{}-~{}q^{0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)$
(D8a) or, simply, $\displaystyle
q\,^{\prime}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)~{}=~{}q\,^{\prime}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)~{}+~{}{\bf{U}}~{}\nabla_{{}_{X}}q_{{}_{L}}~{}+~{}O(U^{2})~{}~{}.$
(D8b)

Introducing the Jacobian $\;J\equiv\,\frac{\textstyle dV^{t}}{\textstyle
dV^{0}}\,=\,\mbox{det}\,\frac{\textstyle\partial x_{i}}{\textstyle\partial
X_{j}}\;\,$, we write:

$\displaystyle{\bf{U}}~{}\nabla_{{}_{X}}q_{{}_{L}}~{}=~{}{\bf{U}}~{}J~{}\nabla_{\textstyle{{}_{x}\,}}q_{{}_{E}}~{}=~{}{\bf{u}}~{}\nabla_{\textstyle{{}_{x}\,}}q_{{}_{E}}~{}~{},$
(D9)

with $\,{\bf{u}}\equiv{\bf{U}}J\,$. Thus (D8b) and (D5) are equivalent insofar
as $\,O({\bf{U}}^{2})=O({\bf{u}}^{2})\,$.

While the language of (D5) is more conventional than that of (D8), the latter
description is easier for physical interpretation. Suppose we are observing
gradual cooling of a flow. In an unperturbed setting, a particle, that started
in $X$ at the time $\,t=0\,$, will show up in $x$ at the time $\,t\,$.
Accordingly, a measurement of the temperature in $x$ at the time $\,t\,$ will
render, in the absence of perturbation, a value to which the particle $X$ has
cooled down by this time — see the equality (D6).

Under perturbation, the rate of cooling of each particle will change. In
addition, owing to the change of trajectories, a different particle will show
up in $x$ at the time $\,t\,$. Now this will be a particle that started its
movement at $\,t=0\,$ from some point $\,\mbox{{\boldmath$X$}}-{\bf{U}}\,$. So
a measurement of the temperature in the point $x$ at the time $\,t\,$ will now
give us a temperature value to which the particle
$\,\mbox{{\boldmath$X$}}-{\bf{U}}\,$ has cooled down — see the equation (D7a).

The difference between the measurements performed in $x$ in the perturbed and
unperturbed cases will, according to (D8b), read as
$\,q\,^{\prime}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)\,=\,q\,^{\prime}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)\,-\,{\bf{U}}~{}\nabla_{{}_{X}}q_{{}_{L}}~{}+~{}O(U^{2})\,$.
The first term on the right renders the cooling down of the particle arriving
in $x$ at the time $\,t\,$, while the second and third terms reflect the fact
that, under disturbance, we register a particle arriving from a point
displaced by $\,{\bf{U}}\,$, compared to the particle that would be brought to
$x$ by an unperturbed flow.

With aid of (D9), the expression (D8b) can be equivalently rewritten as

$\displaystyle
D_{t}~{}=~{}\partial_{t}~{}+~{}{\bf{v}}\,\nabla_{\textstyle{{}_{x}}}~{}~{},$
(D10)

where $\,{\bf{v}}\equiv\partial{\bf{u}}/\partial t\,$, while $\,D\equiv
d/dt\,$ is the comoving  derivative. The physical interpretation of (D10) is
obvious: the rate of cooling of a moving particle, $\,dq/dt\,$, can be
measured by a quiescent observer. The observer, however, must amend his
result, $\,\partial q/\partial t\,$, with a correction taking into account the
fact that, being quiescent, he is measuring the difference between the
temperature of different particles passing by, not of the same particle.

###  D.3 Periodically deformed solids. Linearisation

Hereafter, we shall restrict our consideration to the case of a periodically
deformed solid. It is natural to associate the reference trajectory
$\,\mbox{{\boldmath$x$}}~{}=~{}\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,$
with the equilibrium configuration. In this configuration, the particles stay
idle, so $x$ coincides with the initial value $X$ :

$\displaystyle\mbox{{\boldmath$x$}}~{}=~{}\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)~{}~{}~{},\qquad\mbox{where}\qquad\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)~{}=~{}\mbox{{\boldmath$X$}}\qquad\mbox{for~{}~{}all~{}}~{}~{}t~{}~{},$
(D11)

while all properties keep in time their fiducial values:

$\displaystyle
q^{0}_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)\,=\,q^{0}_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)\,=\,q^{0}\,~{}.$
(D12)

Be mindful that the superscript $\,0\,$ did not originally mark a value fixed
in time but a trajectory chosen to be reference. It is only now, that the role
of a reference configuration is played by an equilibrium body, the superscript
$\,0\,$ begins to denote an unchanging value.

For a particle originally located in $X$ , its perturbed trajectory $r$
differs from its reference trajectory $x$ by some $\,{\bf{u}}\,$:

$\displaystyle\mbox{{\boldmath$r$}}~{}=~{}\mbox{{\boldmath$x$}}~{}+~{}{\bf{u}}~{}=~{}\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)~{}+~{}{\bf{u}}(\mbox{{\boldmath$X$}},\,t)\,~{}.$
(D13)

When the reference trajectory is the equilibrium, insertion of (D11) into
(D13) results in

$\displaystyle\mbox{{\boldmath$r$}}~{}=~{}\mbox{{\boldmath$X$}}~{}+~{}{\bf{u}}(\mbox{{\boldmath$X$}},\,t)\,~{},$
(D14a) which can also be written as
$\displaystyle\mbox{{\boldmath$r$}}~{}=~{}\mbox{{\boldmath$x$}}~{}+~{}{\bf{u}}(\mbox{{\boldmath$x$}},\,t)\,~{},$
(D14b)

because, in this case, the unperturbed trajectory $x$ always coincides with
the initial value $X$ .

We work in a linearised approximation, neglecting the term
$\,O({\bf{u}}^{2})\,$ in (D5) and writing all expansions up to terms linear in
the displacement $\,{\bf{u}}\,$ or velocity $\,{\bf{v}}\equiv d{\bf{u}}/dt\,$.

For a short and simple explanation of the linearised Lagrangian and Eulerian
descriptions of tides, see Wang (1997). A more comprehensive treatment is
offered in the book by Dahlen & Tromp (1998, Section 3.1.1).

###  D.4 Conservation of mass in the Lagrangian and Eulerian descriptions

Denote the Eulerian value of the mass density with
$\,\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,$. As mass cannot be destroyed or
created, its amount in a comoving volume $\,V^{t}\,$ of a flow stays constant:

$\displaystyle\frac{d}{dt}\,\int_{V^{t}}\rho_{{}_{E}}\,dV^{t}~{}=~{}0~{}~{}.$
(D15)

For the reference history
$\,\mbox{{\boldmath$x$}}=\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)\,$,
this would imply:

$\displaystyle\int_{V^{t}_{ref}}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)\,d^{4}\mbox{{\boldmath$x$}}~{}=~{}\int_{V^{0}}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$X$}}\,,~{}0)\,d^{4}\mbox{{\boldmath$X$}}~{}~{}.$
(D16a) For a perturbed history
$\,\mbox{{\boldmath$r$}}=\mbox{\boldmath${\boldmath{f}}$}(\mbox{{\boldmath$X$}},\,t)+{\bf{u}}(\mbox{{\boldmath$X$}},\,t)\,$,
we have:
$\displaystyle\int_{V^{t}_{pert}}\rho_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)\,d^{4}\mbox{{\boldmath$r$}}~{}=~{}\int_{V^{0}}\rho_{{}_{E}}(\mbox{{\boldmath$X$}}\,,~{}0)\,d^{4}\mbox{{\boldmath$X$}}~{}~{}.$
(D16b)

For each individual particle, its perturbed trajectory $r$ stems from the same
initial position $X$ as the appropriate reference trajectory $x$ , so the
initial densities are the same:

$\displaystyle\rho_{{}_{E}}(\mbox{{\boldmath$X$}}\,,~{}0)\;=\;\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$X$}}\,,~{}0)\,~{}.$
(D17)

At later times, however, $\,\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,$ and
$\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)\,$ have different functional
forms.

####  D.4.1 The continuity law in the Eulerian description

The right-hand sides of the formulae (D16a) and (D16b) coincide, as they
render the mass of the same initial distribution
$\,\rho_{{}_{E}}(\mbox{{\boldmath$X$}},\,0)=\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$X$}},\,0)\,$.
Thus the left-hand sides of the two formulae also coincide:

$\displaystyle
0~{}=~{}\int_{V^{t}_{pert}}\rho_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)\,d^{4}\mbox{{\boldmath$r$}}~{}-~{}\int_{V^{t}_{ref}}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)\,d^{4}\mbox{{\boldmath$x$}}~{}~{},$
(D18)

as the mass stays unchanged, no matter whether the system follows the
reference history or a perturbed one. Now switch from the perturbed
coordinates, $r$ , to the reference ones, $x$ :

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int\,d^{4}\mbox{{\boldmath$x$}}~{}\left[\,J~{}\rho_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}-~{}\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)\right]$
(D19a) $\displaystyle=$
$\displaystyle\int\,d^{4}\mbox{{\boldmath$x$}}~{}\left[\,\left(\,1\,+\,\nabla_{\textstyle{{}_{{x}}}}\cdot{\bf{u}}\,\right)~{}\rho_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}-~{}\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)\right]~{}~{},$
(D19b)

where the Jacobian is:

$\displaystyle
J~{}\equiv~{}\frac{dV^{t}_{pert}}{dV^{t}_{ref}}~{}=~{}\mbox{det}~{}\frac{\partial
r_{i}}{\partial
x_{j}}~{}=~{}1~{}+~{}\nabla_{\textstyle{{}_{{x}}}}\cdot{\bf{u}}~{}+~{}O({\bf{u}}^{2})~{}=~{}1~{}+~{}J~{}\nabla_{\textstyle{{}_{r}}}\cdot{\bf{u}}~{}+~{}O({\bf{u}}^{2})~{}~{}.~{}~{}~{}$
(D20a) From this we see that the Jacobian can also be written as 232323 The
expression (D20b) indicates that, within a linear approximation, we do not
need to distinguish between $\,\nabla_{\textstyle{{}_{r}\,}}\,$ and
$\,\nabla_{\textstyle{{}_{{x}}}}\,=\,J\,\nabla_{\textstyle{{}_{{r}}}}\,=\,\nabla_{\textstyle{{}_{{r}}}}\,+\,(\nabla_{\textstyle{{}_{{r}}}}\cdot{\bf{u}})\,\nabla_{\textstyle{{}_{{r}}}}\,+\,O({\bf{u}}^{2})\,$,
when the operators are applied to a quantity that, by itself, is of the first
order of smallness. $\displaystyle J~{}=~{}\frac{\textstyle
1~{}+~{}O({\bf{u}}^{2})}{\textstyle
1~{}-~{}\nabla_{\textstyle{{}_{r}}}\cdot{\bf{u}}}~{}=~{}1~{}+~{}\nabla_{\textstyle{{}_{r}}}\cdot{\bf{u}}~{}+~{}O({\bf{u}}^{2})~{}~{}.$
(D20b)

From (D19a), we obtain the exact equality

$\displaystyle\rho_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}J~{}=~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)~{}~{},$
(D21)

a linearised version thereof being

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\rho_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}(1\,+\,\nabla_{\textstyle{{}_{r}}}\cdot{\bf{u}})~{}-~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$r$}}-{\bf{u}}\,,~{}t)~{}+~{}O({\bf{u}}^{2})$
(D22a) $\displaystyle=$
$\displaystyle~{}\rho_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}-~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}+~{}\rho_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}\nabla_{\textstyle{{}_{r}}}\cdot{\bf{u}}~{}+~{}{\bf{u}}~{}\nabla_{\textstyle{{}_{r}\,}}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$r$}}\,,~{}t)~{}+~{}O({\bf{u}}^{2})~{}~{}.$
(D22b)

For a small $\,t\,$, the deviation $\,{\bf{u}}\,$ between the two trajectories
is linear in time, and so is the difference between the perturbed and
reference density functions. Thus, we may change
$\,\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,\nabla_{\textstyle{{}_{r}}}\cdot{\bf{u}}~{}\,$
to
$\,~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,\nabla_{\textstyle{{}_{r}}}\cdot{\bf{u}}~{}$,
to obtain an expression correct to first order in $\,{\bf{u}}~{}$:

$\displaystyle{\rho_{{}_{E}}}^{\prime}~{}+~{}\nabla_{\textstyle{{}_{r}}}\cdot\left(\rho^{\,0}_{{}_{E}}\,{\bf{u}}\right)~{}=~{}0~{}~{},$
(D23)

where the finite variation is

$\displaystyle{\rho_{{}_{E}}}^{\prime}(\mbox{{\boldmath$r$}},\,t)~{}\equiv~{}\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)~{}-~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)~{}~{}.$
(D24)

We would reiterate that the perturbative approach to Eulerian quantities
implies comparison between their present spatial distributions. So the two
histories are compared in the same point $r$ and at the same time $\,t\,$.

In (D22b), we could also have changed
$\,{\bf{u}}\nabla_{\textstyle{{}_{r}\,}}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)~{}\,$
to
$\,~{}{\bf{u}}\nabla_{\textstyle{{}_{r}\,}}\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,$.
Then, instead of (D23), we would have obtained
$\,{\rho_{{}_{E}}}^{\prime}~{}+~{}\nabla_{\textstyle{{}_{r}}}\cdot\left(\rho_{{}_{E}}\,{\bf{u}}\right)~{}=~{}0\,$,
without the superscript $\,0\,$ in the second term. In the linear
approximation, however, this would be no better than (D23). Traditionally, the
form (D23) is preferred in the literature.

However, when switching to a differential form of the conservation law, we no
longer need to keep the superscript $\,0\,$, because the difference between
$\,\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,$ and
$\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)\,$ becomes infinitesimally
small. So the differential law reads as:

$\displaystyle\frac{\partial\rho_{{}_{E}}}{\partial
t}~{}+~{}\nabla_{\textstyle{{}_{r}}}\cdot\left(\rho_{{}_{E}}\,{\bf{v}}\right)~{}=~{}0~{}~{},$
(D25)

where $\,{\bf{v}}\equiv\partial{\bf{u}}/\partial t\,$, the partial derivative
giving the rate of change with coordinates fixed.

Employment of the perturbative formula (D23) near a deformable free boundary
requires some care. On the one hand, the reference density
$\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$r$}})\,$ makes an abrupt step there.
On the other hand, due to deformation of the boundary, we may get a finite
present density in a point where the reference density used to be zero, and
vice versa.

####  D.4.2 The continuity law in the Lagrangian description

The Lagrangian density is introduced in the standard way (D4a):

$\displaystyle\rho_{{}_{L}}(\mbox{{\boldmath$X$}},\,t)\,\equiv\,\rho_{{}_{E}}(\mbox{{\boldmath$r$}},\,t)~{}~{},$
(D26)

so the formula (D21) becomes:

$\displaystyle\rho_{{}_{L}}(\mbox{{\boldmath$X$}}\,,~{}t)~{}J~{}=~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)~{}~{}.$
(D27a) For a periodically deformed solid, the reference density
$\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)\,$ is the density of the
undeformed, stable configuration. So
$\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}},\,t)\,=\,\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$X$}},\,0)\,=\,\rho^{\,0}(\mbox{{\boldmath$X$}})\,$
is time-independent, and the equality (D27a) becomes simply
$\displaystyle\rho_{{}_{L}}(\mbox{{\boldmath$X$}}\,,~{}t)~{}J~{}=~{}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$X$}})~{}~{}.~{}~{}$
(D27b)

In accordance with the general formula (D5), we interrelate the density
variations as

$\displaystyle{\rho_{{}_{L}}}^{\prime}(\mbox{{\boldmath$X$}}\,,~{}t)~{}=~{}{\rho_{{}_{E}}}^{\prime}(\mbox{{\boldmath$x$}}\,,~{}t)~{}+~{}{\bf{u}}~{}\nabla_{\textstyle{{}_{x}}}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$x$}}\,,~{}t)~{}~{}.$
(D28a) In the considered case of small periodic variations, the reference
trajectory is simply $\,\mbox{{\boldmath$x$}}=\mbox{{\boldmath$X$}}\,$ at all
times; so on the right-hand side of the above formula we have a gradient of a
constant-in-time stationary distribution:
$~{}\,\nabla_{\textstyle{{}_{x}\,}}\rho^{\,0}(\mbox{{\boldmath$x$}},\,t)\,=\,\nabla_{\textstyle{{}_{X}\,}}\rho^{\,0}(\mbox{{\boldmath$X$}})~{}$.
Thence we obtain:
$\displaystyle{\rho_{{}_{L}}}^{\prime}(\mbox{{\boldmath$X$}}\,,~{}t)~{}=~{}{\rho_{{}_{E}}}^{\prime}(\mbox{{\boldmath$X$}}\,,~{}t)~{}+~{}{\bf{u}}~{}\nabla_{\textstyle{{}_{X}\,}}\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$X$}})~{}~{}.$
(D28b)

Combining this formula with (D23), we arrive at 242424 When combining (D28b)
with (D23), we should not be confused by the fact that in (D28b) all
quantities are functions of $\,X\,$, while in (D23 \- D24) these quantities
show up as functions of $r$ . Nor should we be confused by $\,\nabla\,$
denoting $\,\nabla_{r}\,$ in (D23) and $\,\nabla_{X}\,$ in (D28b). As our
intention is simply to compare the functions, we are free to change the
notations in (D23 \- D24) from $r$ to $X$ , whereafter (D29) will come out
trivially.

$\displaystyle{\rho_{{}_{L}}}^{\prime}~{}+~{}\rho^{\,0}_{{}_{E}}~{}\nabla_{\textstyle{{}_{X}}}\cdot{\bf{u}}~{}=~{}0~{}~{},$
(D29)

where
$\,{\rho_{{}_{L}}}^{\prime}={\rho_{{}_{L}}}^{\prime}(\mbox{{\boldmath$X$}},\,t)\,$,
while $\,\rho^{\,0}_{{}_{E}}=\rho^{\,0}_{{}_{E}}(\mbox{{\boldmath$X$}})\,$.

###  D.5 The Poisson equation

####  D.5.1 In the Eulerian description

Perturbed or not, the density always obeys the Poisson equation; while the
perturbing potential $\,W\,$ obeys the Laplace equation outside the perturber:

$\displaystyle\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,V_{{}_{E}}$
$\displaystyle=$ $\displaystyle-~{}4\,\pi\,G\,\rho_{{}_{E}}~{}~{},$ (D30a)
$\displaystyle\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,V^{\,0}_{{}_{E}}$
$\displaystyle=$ $\displaystyle-~{}4\,\pi\,G\,\rho^{\,0}_{{}_{E}}~{}~{},$
(D30b) $\displaystyle\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,W_{{}_{E}}$
$\displaystyle=$ $\displaystyle 0~{}~{}.~{}\qquad\qquad$ (D30c)

Subtraction of (D30b) from (D30a) results in a Poisson equation for the
density perturbation:

$\displaystyle\left.~{}\quad~{}\right.\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,{V_{{}_{E}}}^{\prime}~{}=~{}-~{}4~{}\pi~{}G~{}{\rho_{{}_{E}}}^{\prime}~{}~{}~{}$
(D31)

or, equivalently:

$\displaystyle\left.~{}\quad~{}\right.\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,{U_{{}_{E}}}~{}=~{}-~{}4~{}\pi~{}G~{}{\rho_{{}_{E}}}^{\prime}~{}~{},$
(D32)

where we took into account the relations (35) and (D30c).

####  D.5.2 In the Lagrangian description

Insertion of the formulae (31) and (36) into the Eulerian version of the
Poisson equation, (D31), results in the Lagrangian version of this equation:

$\displaystyle\left.~{}\quad~{}\right.\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,\left(\,{V_{{}_{L}}}^{\prime}\,-~{}{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{\,0}\,\right)~{}=~{}-~{}4~{}\pi~{}G~{}\left(\,{\rho_{{}_{L}}}^{\prime}\,-~{}{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}\rho^{\,0}\,\right)~{}~{}.$
(D33a) A switch to differentiation over the initial position,
$\,\nabla_{{}_{x}}\,$, would entail corrections of the order of
$\,O({\bf{u}}^{2})\,$. In neglect of those, the equation may be written as
$\displaystyle\left.~{}\quad~{}\right.\nabla_{\textstyle{{}_{x\,}}}^{\,2}\,\left(\,{V_{{}_{L}}}^{\prime}\,-~{}{\bf{u}}\cdot\nabla_{\textstyle{{}_{x\,}}}V^{\,0}\,\right)~{}=~{}-~{}4~{}\pi~{}G~{}\left(\,{\rho_{{}_{L}}}^{\prime}\,-~{}{\bf{u}}\cdot\nabla_{\textstyle{{}_{x\,}}}\rho^{\,0}\,\right)~{}~{}.$
(D33b)

For an initially homogeneous body, the above formulae simplify to:

$\displaystyle\left.~{}\quad~{}\right.\nabla_{\textstyle{{}_{r}\,}}^{\,2}\,\left(\,{V_{{}_{L}}}^{\prime}\,-~{}{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{\,0}\,\right)~{}=~{}-~{}4~{}\pi~{}G~{}{\rho_{{}_{L}}}^{\prime}~{}~{}$
(D34a) and
$\displaystyle\left.~{}\quad~{}\right.\nabla_{\textstyle{{}_{x}\,}}^{\,2}\,\left(\,{V_{{}_{L}}}^{\prime}\,-~{}{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{\,0}\,\right)~{}=~{}-~{}4~{}\pi~{}G~{}{\rho_{{}_{L}}}^{\prime}~{}~{}.$
(D34b)

##  Appendix E Boundary conditions

The boundary condition on the total Eulerian potential
$\,V_{\textstyle{{}_{E}}}\,$ is trivial. To avoid infinite forces, the
potential must be continuous:

$\displaystyle{V}^{\,\textstyle{{}^{(exterior)}}}_{\textstyle{{}_{E}}}\,=~{}{V}^{\,\textstyle{{}^{(interior)}}}_{\textstyle{{}_{E}}}\,~{}.$
(E1)

The boundary condition on the potential’s gradient emerges as a corollary of
the Gauss theorem and therefore mimics a similar condition from
electrostatics. 252525 Melchior (1972) attributes the derivation of the
boundary condition to Michel Chasles. Let a small area $\,\vec{\bf
s}\,=\,s\,\hat{\bf{n}}\,$ of the free surface be sandwiched between the top
and bottom of a cylinder of an infinitesimal height $\,u\,=\,\vec{\bf
u}\cdot\vec{\bf s}/s\,$, with the vector $\,\vec{\bf u}\,$ being the tidal
displacement. The top and bottom should each have the principal curvature
radii coinciding with those of the free surface, but in the leading order this
can be ignored, with the enclosed volume thus being $\,u\,s\,=\,\vec{\bf
u}\cdot\vec{\bf s}\,$. In neglect of the contributions from the
infinitesimally small side areas of the cylinder, employment of the Gauss
theorem for the Eulerian potential gives:

$\displaystyle-~{}4~{}\pi~{}G~{}\rho^{0}~{}\vec{\bf u}\,\cdot\,\vec{\bf
s}~{}=~{}\int_{s}\,\nabla_{\textstyle{{}_{x}\,}}V_{\textstyle{{}_{E}}}\,\cdot\,d\vec{\bf
s}~{}=~{}\nabla_{\textstyle{{}_{x}\,}}{V}^{\,\textstyle{{}^{(exterior)}}}_{\textstyle{{}_{E}}}\cdot\,\vec{\bf
s}~{}-~{}\nabla_{\textstyle{{}_{x}\,}}{V}^{\,\textstyle{{}^{(interior)}}}_{\textstyle{{}_{E}}}\cdot\,\vec{\bf
s}~{}~{}.~{}~{}~{}~{}$ (E2)

Over a surface between layers, the condition will read as

$\displaystyle\left(\,4~{}\pi~{}G~{}\rho^{\,0}~{}\vec{\bf u}\,\cdot\,\vec{\bf
s}\,\right)^{\,\textstyle{{}^{(exterior)}}}~{}-~{}\left(4~{}\pi~{}G~{}\rho^{\,0}~{}\vec{\bf
u}\,\cdot\,\vec{\bf
s}\,\right)^{\,\textstyle{{}^{(interior)}}}~{}=~{}\nabla_{\textstyle{{}_{x}\,}}{V}^{\,\textstyle{{}^{(exterior)}}}_{\textstyle{{}_{E}}}\cdot\,\vec{\bf
s}~{}-~{}\nabla_{\textstyle{{}_{x}\,}}{V}^{\,\textstyle{{}^{(interior)}}}_{\textstyle{{}_{E}}}\cdot\,\vec{\bf
s}~{}~{}~{}~{}$ (E3)

or, equivalently,

$\displaystyle\left[~{}-~{}4~{}\pi~{}G~{}\rho^{\,0}~{}{\bf
u}~{}+~{}\frac{\partial~{}}{\partial\hat{\bf{n}}}\,{V}_{\textstyle{{}_{E}}}\,\right]^{\,\textstyle{{}^{(exterior)}}}\,=~{}\left[~{}-~{}4~{}\pi~{}G~{}\rho^{\,0}~{}{\bf
u}~{}+~{}\frac{\partial~{}}{\partial\hat{\bf{n}}}\,{V}_{\textstyle{{}_{E}}}\,\right]^{\,\textstyle{{}^{(interior)}}}\,.~{}~{}~{}$
(E4)

In application to tides, it can be interpreted like this: the discontinuity in
attraction is equal to the attraction of the deformation bulge (Legros et al.
2006). Since $\,V_{0}\,$, $\,W\,$ and their normal gradients are continuous on
the boundary, the conditions on $\,U\,$ and $\,V\,^{\prime}\,$ look exactly
like (E1 \- E4). Specifically, in Section 5.1 we need the conditions on the
total variation $\,V\,^{\prime}~{}$:

$\displaystyle{{V_{{{}_{E}}}}^{\prime}}^{\,\textstyle{{}^{(exterior)}}}\,=~{}{{V_{{{}_{E}}}}^{\prime}}^{\,\textstyle{{}^{(interior)}}}$
(E5) $\displaystyle\left[~{}-~{}4~{}\pi~{}G~{}\rho^{0}~{}\vec{\bf
u}~{}+~{}\frac{\partial~{}}{\partial\hat{\bf{n}}}\,{V_{{}_{E}}}^{\prime}\,\right]^{\,\textstyle{{}^{(exterior)}}}\,=~{}\left[~{}-~{}4~{}\pi~{}G~{}\rho^{0}~{}\vec{\bf
u}~{}+~{}\frac{\partial~{}}{\partial\hat{\bf{n}}}\,{V_{{}_{E}}}^{\prime}\,\right]^{\,\textstyle{{}^{(interior)}}}\,~{}.$
(E6)

The Eulerian and Lagrangian potentials are interrelated through

$\displaystyle{V_{{{}_{E}}}}^{\prime}\,=~{}{V_{{}_{L}}}^{\prime}\,-\,{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{0}\,~{}.$
(E7)

Thence, in the Lagrangian description, the conditions will acquire the form of

$\displaystyle\left[\,{V_{{}_{L}}}^{\prime}\,-~{}{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{\,0}\,\right]^{\,\textstyle{{}^{(exterior)}}}\,=~{}\left[\,{V_{{}_{L}}}^{\prime}\,-~{}{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{\,0}\,\right]^{\,\textstyle{{}^{(interior)}}}$
(E8)

and

$\displaystyle\left[\,\frac{\partial~{}}{\partial\hat{\bf{n}}}\,\left({V_{{}_{L}}}^{\prime}-\,{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{\,0}\,\right)\,-\,4\,\pi\,G\,\rho^{0}\,{\bf
u}\,\right]^{\,\textstyle{{}^{(exterior)}}}=\,\left[\,\frac{\partial~{}}{\partial\hat{\bf{n}}}\,\left({V_{{}_{L}}}^{\prime}-\,{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{\,0}\,\right)\,-\,4\,\pi\,G\,\rho^{0}\,{\bf
u}\,\right]^{\,\textstyle{{}^{(interior)}}}.~{}\,~{}$ (E9)

When the boundary is welded or its normal is parallel to
$\,\nabla_{\textstyle{{}_{x}\,}}V^{\,0}\,$, the term
$~{}-{\bf{u}}\cdot\nabla_{\textstyle{{}_{x}\,}}V^{\,0}~{}$ becomes continuous
(Wang 1997). It, thus, can be removed from (E8), rendering the incremental
Lagrangian potential continuous. This term, however, cannot be omitted in
(E9).

##  Appendix F Interrelation between dynamical Love numbers,
for an incompressible homogeneous sphere

For an incompressible homogeneous spherical body, the static  Love numbers
read as

$\displaystyle
k_{l}=\frac{3}{2\,(l\,-\,1)}\;\,\frac{1}{1\,+\,{\cal{A}}_{l}}~{}\qquad~{}\mbox{and}~{}\qquad~{}h_{l}=\frac{2\,l\,+\,1}{2\,(l\,-\,1)}\;\,\frac{1}{1\,+\,{\cal{A}}_{l}}~{}\,~{},$
(F1)

where

$\displaystyle{\cal{A}}_{l}\,\equiv\,\frac{\textstyle{(2\,l^{\,2}+\,4\,l\,+\,3)\,\mu}}{\textstyle{l}\,\mbox{g}\,\rho\,R}\,=\,\frac{\textstyle{3\,(2\,{\it{l}}^{\,2}+\,4\,{\it{l}}\,+\,3)\,\mu}}{\textstyle{4\,l\,\pi\,G\,\rho^{2}\,R^{2}}}\,=\,\frac{\textstyle{3\,(2\,{\it{l}}^{\,2}+\,4\,{\it{l}}\,+\,3)}}{\textstyle{4\,l\,\pi\,G\,\rho^{2}\,R^{2}\,J}}~{}~{}.~{}\quad$
(F2)

$\mu\,$ and $\,J\,=\,1/\mu\,$ being the relaxed rigidity and compliance, and
$\,G\,$ being Newton’s gravity constant. The formulae (F1) yield a well-known
relation connecting the static Love numbers:

$\displaystyle(2\,l\,+\,1)~{}k_{l}~{}=~{}3~{}h_{l}\,~{}.$ (F3)

Expressions (F1) are obtained by solving a system comprising the static
version of the Second Law of Newton and the constitutive equation
interconnecting the stress and strain through the rigidity $\,\mu\,$. A
wonderful theorem, called the correspondence principle  or the elastic-
viscoelastic analogy , tells us that in many situations the dynamical versions
of the Second Law of Newton and constitutive equation, when written in the
frequency domain as algebraic equations for operational moduli, mimic the
static versions of these equations. In order for this correspondence to take
place, the accelerations and inertial forces should be negligibly small (see,
e.g., Appendix B to Efroimsky 2012 a). In that case, the complex Love numbers
$\,\bar{k}_{l}(\omega)\,$ and $\,\bar{h}_{l}(\omega)\,$ will be expressed
through the complex operational moduli $\,\bar{\mu}\,$ or $\,\bar{J}\,$ in the
same algebraic manner as the static $\,{k}_{l}\,$ and $\,{h}_{l}\,$ are
expressed via the static $\,{\mu}\,$ or $\,{J}\,$. Also recall that the static
expressions (F1) were derived under an extra assumption of incompressibility.
If this assumption is also valid in the dynamical case, then the complex
$\,\bar{k}_{l}(\omega)\,$ and $\,\bar{h}_{l}(\omega)\,$ are expressed through
the complex $\,\bar{\mu}\,$ or $\,\bar{J}\,$ by formulae mimicking (F1),
whence an expression like (F3) ensues for $\,\bar{k}_{l}(\omega)\,$ and
$\,\bar{h}_{l}(\omega)\,$. Its imaginary part will read as:

$\displaystyle(2\,l\,+\,1)~{}k_{l}(\omega)~{}\sin\epsilon(\omega)~{}=~{}3~{}h_{l}(\omega)~{}\sin\epsilon(\omega)\,~{},$
(F4)

where $\,k_{l}(\omega)\,\equiv\,|\bar{k}_{l}(\omega)|\,$,
$~{}h_{l}(\omega)\,\equiv\,|\bar{h}_{l}(\omega)|\,$ and
$\,\omega\,=\,\omega_{\textstyle{{}_{lmpq}}}\,$.

To draw to a close, we would emphasise that in the static expression (F2) the
letters $\,\mu\,$ and $\,J\equiv 1/\mu\,$ stand for the static (relaxed)
values of the rigidity and compliance. In a dynamical analogue of this
expression, the same letters will denote the unrelaxed values.

##  Appendix G Heat production over tidal modes. A sketchy derivation of the
formula (69), in neglect of the “degeneracy”

To compute the dissipation rate at separate tidal modes, it is necessary to
insert the expansions (4) and (16) into the formula (65b) for the heating
rate. This will render a comprehensive version of the somewhat symbolic sum
(67) and will enable us to understand what the modified sum
$\,\sum^{\textstyle{{}^{\,\sharp}}}\,$ actually means. A full calculation is
presented in Appendix H below. Here we present a simplified sketch of that
derivation.

Recall that several different Fourier modes
$\,\omega_{\textstyle{{}_{lmpq}}}\,$ can share the same value $\,\omega\,$.
Borrowing a term from quantum mechanics, we call this degeneracy of modes. As
a prelusory exercise, we calculate dissipation at different modes, neglecting
the degeneracy. In other words, suppose that all Fourier modes
$\,\omega\equiv\omega_{\textstyle{{}_{lmpq}}}\,$ have different values. Under
this simplifying assumption, the expression under the integral in (65b)
becomes:

$\displaystyle\sum_{l=2}^{\infty}(2l+1)\,\left\langle
W_{l}(t)\,\dot{U}_{l}(t)\right\rangle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad~{}~{}~{}$
$\displaystyle=$
$\displaystyle\sum_{\omega,\,\omega\,^{\prime}}(2l+1)~{}\left\langle
W_{l}(\omega\,^{\prime})\,\cos\left[\omega\,^{\prime}t+\varphi_{\textstyle{{}_{W_{l}}}}(\omega\,^{\prime})\right]\,(\,-\,\omega)\,U_{l}(\omega)\,\sin\left[\omega
t+\varphi_{\textstyle{{}_{U_{l}}}}(\omega)\right]\,\right\rangle~{}$ (G1a)
$\displaystyle=$
$\displaystyle-\sum_{\omega,\,\omega\,^{\prime}}(2l+1)~{}\frac{\omega}{2}\,W_{l}(\omega\,^{\prime})\,U_{l}(\omega)~{}\left\langle\,\sin\left[\,(\omega\,-\,\omega\,^{\prime})~{}t~{}+~{}\varphi_{\textstyle{{}_{U_{l}}}}(\omega)-\varphi_{\textstyle{{}_{W_{l}}}}(\omega\,^{\prime})\,\right]~{}~{}~{}~{}\,\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\left.+\,\sin\left[\,(\omega\,+\,\omega\,^{\prime})~{}t~{}+~{}\varphi_{\textstyle{{}_{U_{l}}}}(\omega)+\varphi_{\textstyle{{}_{W_{l}}}}(\omega\,^{\prime})\,\right]\,\right\rangle~{}~{},\qquad$
where $~{}\langle\,.\,.\,.\,\rangle~{}$ denotes time averaging. Of the two
sine functions on the right-hand side, we would have kept only the first one,
had the Fourier tidal modes been positive-definite. In the tidal theory,
however, the Fourier modes $\,\omega\,=\,\omega_{\textstyle{{}_{lmpq}}}\,$ can
assume either sign, so both sine functions must be taken into account:
$\displaystyle\sum_{l=2}^{\infty}(2l+1)\,\left\langle
W_{l}(t)\,\dot{U}_{l}(t)\right\rangle~{}=~{}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad~{}~{}~{}$
$\displaystyle-\sum_{\omega}\,(2l+1)\,\frac{\omega}{2}\,\left\\{\,W_{l}(\omega)\,U_{l}(\omega)~{}\sin\left[\varphi_{\textstyle{{}_{U_{l}}}}(\omega)-\varphi_{\textstyle{{}_{W_{l}}}}(\omega)\right]+\,W_{l}(-\omega)\,U_{l}(\omega)~{}\sin\left[\varphi_{\textstyle{{}_{U_{l}}}}(\omega)+\varphi_{\textstyle{{}_{W_{l}}}}(-\omega)\right]\,\right\\}~{}~{}\,~{}~{}$
(G1b)
$\displaystyle=\sum_{\omega}\,(2l+1)\,\frac{\omega}{2}\,W^{\,2}_{l}(\omega)\,k_{l}(\omega)~{}\sin\epsilon_{l}(\omega)~{}+~{}\sum_{\omega}\,(2l+1)\,\frac{\omega}{2}\,W_{l}(\omega)\,W_{l}(-\omega)\,k_{l}(\omega)~{}\sin\epsilon\,^{\prime}_{l}(\omega)\,~{},\qquad$
(G1c)

where we recalled that the dynamical Love number is an even function of the
Fourier mode.

On the right-hand side of (G1c), the first sum is a much expected input
coinciding with the expression obtained by other authors – see, e.g., the
first line of formula (10) in Platzman (1984). 262626 The second line in
Platzman’s formula renders oceanic and atmospheric inputs. This input is
proportional to $\,k_{l}(\omega)~{}\sin\epsilon_{l}(\omega)\,$, where

$\displaystyle\epsilon_{l}(\omega)\,\equiv\,\varphi_{\textstyle{{}_{W_{l}}}}(\omega)~{}-~{}\varphi_{\textstyle{{}_{U_{l}}}}(\omega)$
(G2)

is the tidal phase lag at the frequency
$\,\omega\,=\,\omega_{\textstyle{{}_{lmpq}}}\,$.

The second sum in (G1c) comes into being due to the fact that the Fourier
modes are not positive-definite. This input contains a factor of
$\,k_{l}(\omega)~{}\sin\epsilon\,^{\prime}_{l}(\omega)\,$, where the angle
$\,\epsilon\,^{\prime}_{l}(\omega)\,$ is, generally, different from the phase
lag (G2) appropriate to the mode
$\,\omega\,=\,\omega_{\textstyle{{}_{lmpq}}}\,$. Indeed,