As we just saw, both the secular and short-period parts of the torque may be
approximated with the following degree-2 and degree-3 components:

$\displaystyle\overline{\cal T}$ $\displaystyle=$ $\displaystyle\overline{\cal
T}_{\textstyle{{}_{{}_{\textstyle{{}_{l=2}}}}}}\,+~{}\overline{\cal
T}_{\textstyle{{}_{{}_{\textstyle{{}_{l=3}}}}}}\,+~{}O\left(\,\epsilon\,(R/a)^{9}\,\right)~{}$
(118) $\displaystyle=$ $\displaystyle\overline{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(220)}}}}}\,+\left[\,\overline{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(311)}}}}}\,+~{}\overline{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(330)}}}}}\right]~{}+~{}O(\epsilon\,i^{2})~{}+~{}O\left(\,\epsilon\,(R/a)^{9}\,\right)~{}~{}~{},\quad\quad~{}~{}~{}~{}$

and

$\displaystyle\widetilde{\cal T}$ $\displaystyle=$
$\displaystyle\widetilde{\cal
T}_{\textstyle{{}_{\textstyle{{}_{l=2}}}}}\,+~{}\widetilde{\cal
T}_{\textstyle{{}_{\textstyle{{}_{l=3}}}}}\,+~{}O\left(\,\epsilon\,(R/a)^{9}\,\right)~{}$
(119) $\displaystyle=$ $\displaystyle\widetilde{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(220)}}}}}\,+\left[\,\widetilde{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(311)}}}}}\,+~{}\widetilde{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(330)}}}}}\right]~{}+~{}O(\epsilon\,i^{2})~{}+~{}O\left(\,\epsilon\,(R/a)^{9}\,\right)~{}~{}~{},\quad\quad$

were the $\,l=2\,$ and $\,l=3\,$ inputs are of the order $\,(R/a)^{5}\,$ and
$\,(R/a)^{7}\,$, accordingly; while the $\,l=4,\,5,\,.\,.\,.\,$ inputs
constitute $\,O\left(\,\epsilon\,(R/a)^{9}\,\right)\,$.

Expressions for $~{}\overline{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(220)}}}}}~{}$, $~{}\overline{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(311)}}}}}~{}$, and $~{}\overline{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(310)}}}}}~{}$ are furnished by
formulae (297), (299), and (301) in Appendix H. As an example, here we provide
one of these components:

$\displaystyle\overline{\cal
T}_{\textstyle{{}_{{}_{\textstyle{{}_{(lmp)=(220)}}}}}}$ $\displaystyle=$
$\displaystyle\frac{3}{2}~{}G\;M_{sec}^{2}\,R^{5}\,a^{-6}\,\left[~{}\frac{1}{2304}~{}e^{6}~{}k_{2}~{}\sin|\epsilon_{\textstyle{{}_{\textstyle{{}_{220~{}-3}}}}}|~{}~{}\mbox{sgn}\,\left(\,-\,n\,-\,2\,\dot{\theta}\,\right)\right.$
(120) $\displaystyle+$
$\displaystyle\left(~{}\frac{1}{4}~{}e^{2}-\,\frac{1}{16}~{}e^{4}~{}+\,\frac{13}{768}~{}e^{6}~{}\right)~{}k_{2}~{}\sin|\epsilon_{\textstyle{{}_{\textstyle{{}_{220~{}-1}}}}}|~{}~{}\mbox{sgn}\,\left(\,n\,-\,2\,\dot{\theta}\,\right)~{}$
$\displaystyle+$
$\displaystyle\left(1\,-\,5\,e^{2}\,+~{}\frac{63}{8}\;e^{4}\,-~{}\frac{155}{36}~{}e^{6}\right)~{}k_{2}~{}\sin|\epsilon_{\textstyle{{}_{\textstyle{{}_{2200}}}}}|~{}~{}~{}~{}\mbox{sgn}\,\left(\,n\,-\,\dot{\theta}\,\right)$
$\displaystyle+$
$\displaystyle\left(\frac{49}{4}~{}e^{2}-\;\frac{861}{16}\;e^{4}+~{}\frac{21975}{256}~{}e^{6}\right)~{}k_{2}~{}\sin|\epsilon_{\textstyle{{}_{\textstyle{{}_{2201}}}}}|~{}~{}\mbox{sgn}\,\left(\,3\,n\,-\,2\,\dot{\theta}\,\right)$
$\displaystyle+$
$\displaystyle\left(~{}\frac{289}{4}\,e^{4}\,-\,\frac{1955}{6}~{}e^{6}~{}\right)~{}k_{2}~{}\sin|\epsilon_{\textstyle{{}_{\textstyle{{}_{2202}}}}}|~{}\mbox{sgn}\,\left(\,2\,n\,-\,\dot{\theta}\,\right)$
$\displaystyle+$
$\displaystyle\left.\frac{714025}{2304}~{}e^{6}~{}k_{2}~{}\sin|\epsilon_{\textstyle{{}_{\textstyle{{}_{2203}}}}}|~{}~{}\mbox{sgn}\,\left(\,5\,n\,-\,3\,\dot{\theta}\,\right)~{}\,\right]+\,O(e^{8}\,\epsilon)\,+\,O({\it
i}^{2}\,\epsilon)~{}~{}~{}.\quad\quad\quad\quad$

Here each term changes its sign on crossing the appropriate resonance. The
change of the sign takes place smoothly, as the value of the term goes through
zero – this can be seen from formula (95) and from the fact that the tidal
mode $\,\omega_{\textstyle{{}_{lmpq}}}\,$ vanishes in the $\,lmpq\,$
resonance.

Expressions for $~{}\widetilde{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(220)}}}}}~{}$, $~{}\widetilde{\cal
T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(311)}}}}}~{}$, and
$~{}\widetilde{\cal T}_{\textstyle{{}_{\textstyle{{}_{(lmp)=(330)}}}}}~{}$ are
given by formulae (303 \- 305) in Appendix I. Although the average of the
short-period part of the torque vanishes, this part does contribute to
dissipation. Oscillating torques contribute also to variations of the surface
of the tidally-distorted primary, the latter fact being of importance in
laser-ranging experiments.

Whether the short-period torque may or may not influence also the process of
entrapment is worth exploring numerically. The reason why this issue is raised
is that the frequencies $\,n(q-j)\,$ of the components of the short-period
torque are integers of $\,n\,$ and thus are commensurate with the spin rate
$\,\dot{\theta}\,$ near an $A/B$ resonance, $\,A\,$ and $\,B\,$ being integer.
It may be especially interesting to check the role of this torque when
$\,q-j=1\,$ and $\,A/B\,=\,N\,$ is integer.

The hypothetical role of the short-period torque in the entrapment and
libration dynamics has never been discussed so far, as the previous studies
employed expressions for the tidal torque, which were obtained through
averaging over the period of the secondary’s orbital motion.

###  8.3 Librations

Consider a tidally-perturbed body caught in a $\,A:B\,$ resonance with its
tide-raising companion, $A$ and $B$ being integer. Then the spin rate of the
body is

$\displaystyle\dot{\theta}\;=\;\frac{A}{B}\;n\;+\;\dot{\psi}~{}~{}~{},$ (121)

where the physical-libration angle is

$\displaystyle\psi\,=~{}-~{}\psi_{0}~{}\sin\left(\omega_{{{}_{PL}}}\,t\right)~{}~{}~{},$
(122)

$\,\omega_{{{}_{PL}}}\,$ being the physical-libration frequency. The
oscillating tidal torque exerted on the body is comprised of the modes

$\displaystyle\omega_{\textstyle{{}_{lmpq}}}\;=\;\left(l~{}-~{}2~{}p~{}+~{}q\right)~{}n~{}-~{}m~{}\dot{\theta}~{}=~{}\left(l~{}-~{}2~{}p~{}+~{}q~{}-~{}\frac{A}{B}~{}m\right)\,n~{}-~{}m~{}\dot{\psi}~{}~{}.~{}~{}~{}$
(123)

In those $\,lmpq\,$ terms for which the combination $\,l-2p+q-\frac{\textstyle
A}{\textstyle B}\,m\,$ is not zero, the small quantity $\,-m\dot{\psi}\,$ may
be neglected.171717 The physical-libration input $~{}-\,m\,\dot{\alpha}\,$ may
be neglected in the expression for $\,\omega_{\textstyle{{}_{lmpq}}}\,$ even
when the magnitude of the physical libration is comparable to that of the
geometric libration (as in the case of Phobos). The remaining terms will
pertain to the geometric libration. The phase lags will be given by the
standard formula
$\,\epsilon_{\textstyle{{}_{lmpq}}}\,=\,\omega_{\textstyle{{}_{lmpq}}}\,\Delta
t_{\textstyle{{}_{lmpq}}}\,$.

In those $\,lmpq\,$ terms for which the combination $\,l-2p+q-\frac{\textstyle
A}{\textstyle B}\,m\,$ vanishes, the physical-libration input
$~{}-m\,\dot{\psi}\,$ is the only one left. Accordingly, the multiplier
$~{}\sin\left[\,\left(v_{\textstyle{{}_{lmpq}}}^{*}\,-\,m\,\theta^{*}\,\right)\,-\right.$
$\left.\,\left(\,v_{\textstyle{{}_{lmpq}}}\,-\,m\,\theta\,\right)\,\right]~{}$
in the $\,lmpq\,$ term of the tidal torque will reduce to
$~{}\sin\left[\,-\,m\,\left(\,\psi^{*}\,-\,\psi\,\right)\,\right]\,\approx~{}$
$-m\,\dot{\psi}\,\Delta t\,=\,m\,\psi_{0}\,\omega_{{{}_{PL}}}\,\Delta
t\,\cos\omega_{{{}_{PL}}}t~{}$. Here the time lag $\,\Delta t\,$ is the one
corresponding to the physical-libration frequency $\,\omega_{{{}_{PL}}}~{}$
which may be very different  from the usual tidal frequencies for
nonsynchronous rotation – see Williams & Efroimsky (2012) for a comprehensive
discussion.

##  9 Marking the minefield

The afore-presented expressions for the secular and purely short-period parts
of the tidal torque look cumbersome when compared to the compact and elegant
formulae employed in the literature hitherto. It will therefore be important
to explain why those simplifications are impractical.

###  9.1 Perils of the conventional simplification

Insofar as the quality factor is large and the lag is small (i.e., insofar as
$\,\sin\epsilon\,$ can be approximated with $\,\epsilon\,$), our expression
(296) assumes a simpler form:

${}^{\textstyle{{}^{\tiny{(Q\,>\,10)\,}}}}\overline{\cal
T}_{\textstyle{{}_{{}_{\textstyle{{}_{l=2}}}}}}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\;G\;M_{sec}^{2}\,R^{5}\,a^{-6}\,k_{2}\;\sum_{q=-3}^{3}\,G^{\textstyle{{}^{\,2}}}_{\textstyle{{}_{\textstyle{{}_{20\mbox{\it{q}}}}}}}(e)\,\epsilon_{\textstyle{{}_{\textstyle{{}_{220\mbox{\it{q}}}}}}}~{}+\,O(e^{6}\,\epsilon)\,+\,O({\it
i}^{2}\,\epsilon)\,+\,O(\epsilon^{3})~{}~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(124)

where the error $\,O(\epsilon^{3})\,$ originates from
$\;\sin\epsilon=\epsilon+O(\epsilon^{3})\;$.

The simplification conventionally used in the literature ignores the
frequency-dependence of the Love number and attributes the overall frequency-
dependence to the lag. It also ignores the difference between the tidal lag
$\,\epsilon\,$ and the lag in the material, $\,\delta\,$. This way, the
conventional simplification makes $\,\epsilon\,$ obey the scaling law (79b).
At this point, most authors also set $\,\alpha=-1\,$. Here we shall explore
this approach, though shall keep $\,\alpha\,$ arbitrary. From the formula
181818 Let $~{}\frac{\textstyle 1}{\textstyle\sin\epsilon}=\left(\,{\cal
E}\,\chi\right)^{\alpha}~{}$, where $\,{\cal{E}}\,$ is an empirical parameter
of the dimensions of time, while $\,\epsilon\,$ is small enough, so
$\,\sin\epsilon\approx\epsilon\,$. In combination with
$\,\epsilon_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}\equiv\,\omega_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}\,\Delta
t_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}\,$ and
$\,\chi_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}\,=\,|\omega_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}|\,$,
these formulae entail (125).

$\displaystyle\Delta
t_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}\;=\;{\cal{E}}^{\textstyle{{}^{-\,\alpha}}}\;\chi_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}^{\textstyle{{}^{-\,(\alpha+1)}}}$
(125)

derived by Efroimsky & Lainey (2007) in the said approach, we see that the
time lags are related to the principal-frequency lag $\;\Delta
t_{\textstyle{{}_{\textstyle{{}_{2200}}}}}\;$ via:

$\displaystyle\Delta t_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}\,=\,\Delta
t_{\textstyle{{}_{\textstyle{{}_{2200}}}}}\;\left(\,\frac{\chi_{\textstyle{{}_{\textstyle{{}_{2200}}}}}}{\chi_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}}\,\right)^{\textstyle{{}^{\alpha+1}}}\;\;\;.$
(126)

When the despinning is still going on and $\,\dot{\theta}\gg n\,$, the
corresponding phase lags are:

$\displaystyle\epsilon_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}\;\equiv\;\Delta
t_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}\;\omega_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}\,=\,-~{}\Delta
t_{\textstyle{{}_{\textstyle{{}_{2200}}}}}\;\chi_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}\;\left(\,\frac{\chi_{\textstyle{{}_{\textstyle{{}_{2200}}}}}}{\chi_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}}\,\right)^{\textstyle{{}^{\alpha+1}}}~{}=~{}-~{}\epsilon_{\textstyle{{}_{\textstyle{{}_{2200}}}}}\;\left(\,\frac{\chi_{\textstyle{{}_{\textstyle{{}_{2200}}}}}}{\chi_{\textstyle{{}_{\textstyle{{}_{lmpq}}}}}}\,\right)^{\textstyle{{}^{\alpha}}}~{}~{},\quad\quad$
(127)

which helps us to cast the secular part of the torque into the following
convenient form: 191919 For $\,\dot{\theta}\gg 2n\,$, all the modes
$\,\omega_{\textstyle{{}_{\textstyle{{}_{220q}}}}}\,$ are negative, so
$\,\omega_{\textstyle{{}_{\textstyle{{}_{220q}}}}}\,=\,-\,\chi_{\textstyle{{}_{\textstyle{{}_{220q}}}}}\,$.
Then, keeping in mind that $\,n/\dot{\theta}\ll 1\,$, we process (127), for
$q=1,$ like  $\displaystyle-\,\Delta
t_{\textstyle{{}_{\textstyle{{}_{2200}}}}}\,\chi_{\textstyle{{}_{\textstyle{{}_{2200}}}}}~{}\left(\,\frac{\chi_{\textstyle{{}_{\textstyle{{}_{2200}}}}}}{\chi_{\textstyle{{}_{\textstyle{{}_{2201}}}}}}\,\right)^{\textstyle{{}^{\alpha}}}=\,-\,\Delta
t_{\textstyle{{}_{\textstyle{{}_{2200}}}}}\,2\,|n-\dot{\theta}|\,\left(\frac{2|n-\dot{\theta}|}{|-2\dot{\theta}+3n|}\,\right)^{\textstyle{{}^{\alpha}}}=\,-\,\Delta
t_{\textstyle{{}_{\textstyle{{}_{2200}}}}}\,2\,(\dot{\theta}-n)\,\left[1\,+\,\frac{\alpha}{2}\,\frac{n}{\stackrel{{\scriptstyle\bf\centerdot}}{{\theta\,}}}\,+\,O(\,(n/\dot{\theta})^{2}\,)\right]~{}~{}~{},$
and similarly for other $q\neq 0$, and then plug the results into (124). This
renders us (128).

${}^{\textstyle{{}^{\tiny{(Q\,>\,10)\,}}}}\overline{\cal
T}_{\textstyle{{}_{{}_{\textstyle{{}_{l=2}}}}}}\;=\;{\cal{Z}}\left[\;\,-\;\,\dot{\theta}\,\left(1\,+\,\frac{15}{2}\,e^{2}\,+\,\frac{105}{4}\,e^{4}\,+\,O(e^{6})\right)\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+\;\left.n\left(1+\left(\frac{15}{2}-6\alpha\right)e^{2}+\left(\frac{105}{4}-\frac{363}{8}\alpha\right)e^{4}+O(e^{6})\right)\right]+O({\it
i}^{2}/Q)+O(Q^{-3})+O(\alpha e^{2}Q^{-1}n/\dot{\theta})\;\;\;~{}~{}~{}~{}$
(128a) $\displaystyle\approx\;{\cal
Z}\;\left[\,-\,\dot{\theta}\,\left(1\,+\,\frac{15}{2}\,e^{2}\right)+\,n\,\left(1\,+\,\left(\frac{15}{2}\,-\,6\,\alpha\right)\,e^{2}\,\right)\,\right]\;\;\;,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(128b)

where the overall factor reads as:

$\displaystyle{\cal
Z}~{}=~{}\frac{3\,G\,M_{sec}^{\textstyle{{}^{\,2}}}\;\,k_{2}\;\Delta
t_{\textstyle{{}_{2200}}}}{R}\,~{}\frac{R^{\textstyle{{}^{6}}}}{a^{6}}~{}=~{}\frac{3\,n^{2}\,M_{sec}^{\textstyle{{}^{\,2}}}\;\,k_{2}\;\Delta
t_{\textstyle{{}_{2200}}}}{(M_{prim}\,+\,M_{sec})}~{}\,\frac{R^{\textstyle{{}^{5}}}}{a^{3}}\,=\,\frac{3\,n\,M_{sec}^{\textstyle{{}^{\,2}}}~{}\,k_{2}}{Q_{\textstyle{{}_{2200}}}~{}(M_{prim}\,+\,M_{sec})}~{}\,\frac{R^{\textstyle{{}^{5}}}}{a^{3}}~{}\,\frac{n~{}~{}}{\chi_{\textstyle{{}_{2200}}}}~{}~{},\quad\quad$
(129)

$M_{prim}\,$ and $\,M_{sec}\,$ being the primary’s and secondary’s
masses.202020 To arrive at the right-hand side of (129), we recalled that
$~{}\chi_{\textstyle{{}_{lmpq}}}\,\Delta
t_{\textstyle{{}_{lmpq}}}\,=\,|\epsilon_{\textstyle{{}_{lmpq}}}|\,$ and that
$~{}Q_{\textstyle{{}_{lmpq}}}^{-1}\,=\,|\epsilon_{\textstyle{{}_{lmpq}}}|\,+\,O(\epsilon^{3})\,=\,|\epsilon_{\textstyle{{}_{lmpq}}}|\,+\,O(Q^{-3})\,$,
according to formula (12). Dividing (129) by the primary’s principal moment of
inertia $\,\xi\,M_{primary}\,R^{2}\,$, we obtain the contribution that this
component of the torque brings into the angular deceleration rate
$\;\ddot{\theta}\;$:

$\displaystyle\ddot{\theta}~{}=~{}{\cal{K}}~{}\left\\{\,-\,\stackrel{{\scriptstyle\;\centerdot}}{{\theta}}\,\left[~{}1\,+\,\frac{15}{2}\,e^{2}\,+\,\frac{105}{4}\,e^{4}\,+\,O(e^{6})~{}\right]\right.~{}+~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.n\left[~{}1+\left(\frac{15}{2}-6\alpha\right)e^{2}+\left(\frac{105}{4}-\frac{363}{8}\alpha\right)e^{4}+O(e^{6})~{}\right]\,\right\\}+O({\it
i}^{2}/Q)+O(Q^{-3})+O(\alpha
e^{2}Q^{-1}n/\dot{\theta})\;~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (130a)
$\displaystyle\approx~{}{\cal
K}\;\left[\,-\,\stackrel{{\scriptstyle\;\centerdot}}{{\theta}}\,\left(1\,+\,\frac{15}{2}\,e^{2}\right)+\,n\,\left(1\,+\,\left(\frac{15}{2}\,-\,6\,\alpha\right)\,e^{2}\,\right)\,\right]~{}~{}~{},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(130b)

the factor $\,{\cal{K}}\,$ being given by

$\displaystyle{\cal K}\,\equiv\,\frac{\cal
Z}{\xi\,M_{prim}\,R^{2}}\,=\,\frac{3\,n^{2}\,M_{sec}^{\textstyle{{}^{\,2}}}\;\,k_{2}\;{\Delta
t_{\textstyle{{}_{2200}}}}}{\xi\;M_{prim}\;(M_{prim}\,+\,M_{sec})}\;\frac{R^{\textstyle{{}^{3}}}}{a^{3}}\,=\,\frac{3\,n\,M_{sec}^{\textstyle{{}^{\,2}}}\;\,k_{2}}{\xi\;Q_{\textstyle{{}_{\textstyle{{}_{2200}}}}}\;M_{prim}\;(M_{prim}\,+\,M_{sec})}\;\frac{R^{\textstyle{{}^{3}}}}{a^{3}}\;\,\frac{n~{}~{}}{\chi_{\textstyle{{}_{\textstyle{{}_{2200}}}}}}\;\;\;,~{}\quad~{}$
(131)

where $\,\xi\,$ is a multiplier emerging in the expression
$\,\xi\,M_{primary}\,R^{2}\,$ for the primary’s principal moment of inertia
($\,\xi=2/5\,$ for a homogeneous sphere).

In the special case of $\,\alpha\,=\,-\,1\;$, the above expressions enjoy
agreement with the appropriate result stemming from the corrected MacDonald
model – except that our (128) and (130) contain $\,\Delta
t_{\textstyle{{}_{2200}}}\,$, $\,Q_{\textstyle{{}_{2200}}}\,$, and
$\,\chi_{\textstyle{{}_{2200}}}\,$ instead of $\,\Delta t\,$, $\,Q\,$, and
$\,\chi\,$ standing in formulae (44 - 47) from Williams & Efroimsky (2012).
Formula (130b) tells us that the secular part of the tidal torque vanishes for

$\displaystyle\stackrel{{\scriptstyle\;\centerdot}}{{\theta}}\,-\;n\;=\;-\;6\;n\;e^{2}\;\alpha\;\;\;,$
(132)

which coincides, for $\;\alpha=-1\;$, with the result obtained in Rodríguez et
al. (2008, eqn. 2.4) and Williams & Efroimsky (2012, eqn. 49). This
coincidence however should not be taken at its face value, because it is
_occasional_ or, possibly better to say, exceptional.

Formulae (128 \- 129) were obtained by insertion of the expressions for the
eccentricity functions and the phase lags into (124), and by assuming that
$\,n\ll|\dot{\theta}|\,$. The latter caveat is a crucial element, not to be
overlooked by the users of formulae (128 \- 129) and of their corollary (130
\- 131) for the tidal deceleration rate.

The case of $\,\alpha=-1\,$ is special, in that it permits derivation of (128
\- 132) without assuming
that $\,n\ll|\stackrel{{\scriptstyle\;\bf\centerdot}}{{\theta}}|~{}$. However
for $\,\alpha>-1\,$ the condition
$\,n\ll|\stackrel{{\scriptstyle\;\bf\centerdot}}{{\theta}}|\,$ remains
mandatory, so formulae (128 \- 131) become inapplicable when
$\,\stackrel{{\scriptstyle\;\centerdot}}{{\theta}}\,$ reduces to values of
about several $\,n\,$.

Although formulae (128a) and (130a) contain an absolute error $\,O(\alpha
e^{2}Q^{-1}n/\dot{\theta})\,$, this does not mean that for
$\,\stackrel{{\scriptstyle\;\centerdot}}{{\theta}}\,$ comparable to $\,n\,$
the absolute error becomes $\,O(\alpha e^{2}Q^{-1})\,$ and the relative one
becomes $\,O(\alpha e^{2})\,$. In reality, for
$\,\stackrel{{\scriptstyle\;\centerdot}}{{\theta}}\,$ comparable to $\,n\,$,
the entire approximation falls apart, because formulae (126 \- 127) were
derived from expression (125), which is valid for $\,Q\gg 1\,$ only (unless
$\,\alpha=\,-\,1\,$). So these formulae become inapplicable in the vicinity of
a commensurability. By ignoring this limitation, one can easily encounter
unphysical paradoxes.212121 For example, in the case of $\,\alpha>-1\,$,
formulae (125 \- 127) render infinite values for $\,\Delta
t_{\textstyle{{}_{lmpq}}}\,$ and $\,\epsilon_{\textstyle{{}_{lmpq}}}\,$ on
crossing the commensurability, i.e., when $\,\omega_{\textstyle{{}_{lmpq}}}\,$
goes through zero.

Thence, in all situations, except for the unrealistic rheology
$\,\alpha=-1\,$, limitations of the approximation (128 \- 131) should be kept
in mind. This approximation remains acceptable for $\,n\ll|\dot{\theta}|\,$,
but becomes misleading on approach to the physically-interesting resonances.

###  9.2 An oversight in a popular formula

The form in which our approximation (128 \- 131) is cast may appear awkward.
The formula for the despinning rate $\,\ddot{\theta\,}\,$ is written as a
function of $\,\dot{\theta}\,$ and $\,n\,$, multiplied by the overall factor
$\,{\cal{K}}\,$. This form would be reasonable, were $\,{\cal{K}}\,$ a
constant. That this is not the case can be seen from the presence of the
multiplier $\,\frac{\textstyle
n}{\textstyle\chi_{\textstyle{{}_{2200}}}}\,=\,\frac{\textstyle n}{\textstyle
2\,|\stackrel{{\scriptstyle\bf\centerdot}}{{\theta}}\,-\,n|}\,$ on the right-
hand side of (131).

Still, when written in this form, our result is easy to juxtapose with an
appropriate formula from Correia & Laskar (2004, 2009). There, the expression
for the despinning rate looks similar to ours, up to an important detail: the
overall factor is a constant, because it lacks the said multiplier
$\frac{\textstyle n}{\textstyle\chi_{\textstyle{{}_{2200}}}}$. The multiplier
was lost in those two papers, because the quality factor was introduced there
as $\,1/(n\,\Delta t)\,$, see the line after formula (9) in Correia & Laskar
(2009). In reality, the quality factor $Q$ should, of course, be a function of
the tidal frequency $\,\chi\,$, because $Q$ serves the purpose of describing
the tidal damping at this frequency. Had the quality factor been taken as
$\,1/(\chi\,\Delta t)\,$, it would render the corrected MacDonald model
($\,\alpha=\,-\,1\,$), and the missing multiplied would be there. Being
unphysical,222222 To be exact, the model is unphysical everywhere except in
the closest vicinity of the resonance – see formulae (92 \- 94). the model is
mathematically convenient, because it enables one to write down the secular
part of the torque as one expression, avoiding the expansion into a series
(Williams & Efroimsky 2012). The model was pioneered by Singer (1968) and
employed by Mignard (1979, 1980), Hut (1981) and other authors.

Interestingly, in the special case of the 3:2 spin-orbit resonance, we have
$\,\chi=n\,$. Still, the difference between $\,\chi\,$ and $\,n\,$ in the
vicinity of the resonance may alter the probability of entrapment of Mercury
into this rotation mode. The difference between $\,\chi\,$ and $\,n\,$ becomes
even more considerable near the other resonances of interest. So the
probabilities of entrapment into those resonances must be recalculated.

##  10 Conclusions

The goal of this paper was to lay the ground for a reliable model of tidal
entrapment into spin-orbital resonances. To this end, we approached the tidal
theory from the first principles of solid-state mechanics. Starting from the
expression for the material’s compliance in the time domain, we derived the
frequency-dependence of the Fourier components of the tidal torque. The other
torque, one caused by the triaxiality of the rotator, is not a part of this
study and will be addressed elsewhere.

  * •

We base our work on the Andrade rheological model, providing arguments in
favour of its applicability to the Earth’s mantle, and therefore, very likely,
to other terrestrial planets and moons. The model is also likely to apply to
the icy moons (Castillo-Rogez et al. 2011). We have reformulated the model in
terms of a characteristic anelastic timescale $\tau_{{}_{A}}$ (the Andrade
time). The ratio of the Andrade time to the viscoelastic Maxwell time,
$\zeta=\tau_{{}_{A}}/\tau_{{}_{M}}$, serves as a dimensionless free parameter
of the rheological model.

The parameters $\tau_{{}_{A}}$, $\tau_{{}_{M}}$, $\zeta$ cannot be regarded
constant, though their values may be changing very slowly over vast bands of
frequency. The shapes of these frequency-dependencies may depend on the
dominating dissipation mechanisms and, thereby, on the magnitude of the load,
as different damping mechanisms get activated under different loads.

The main question here is whether, in the low-frequency limit, anelasticity
becomes much weaker than viscosity. (That would imply an increase of
$\,\tau_{{}_{A}}\,$ and $\,\zeta\,$ as the tidal frequency $\,\chi\,$ goes
down.) The study of ices under weak loads, with friction caused mainly by
lattice diffusion (Castillo-Rogez et al. 2011, Castillo-Rogez & Choukroun
2010) has not shown such a decline of anelasticity. However, Karato & Spetzler
(1990) point out that it should be happening in the Earth’s mantle, where the
loads are much higher and damping is caused mainly by unpinning of
dislocations. According to Ibid., in the Earth, the decrease of the role of
anelasticity happens abruptly as the frequency falls below the threshold
$\,\chi_{\textstyle{{}_{0}}}\sim$ 1 yr${}^{-1}\,$. We then may expect a
similar switch in the other terrestrial planets and the Moon, though there the
threshold may be different as it is sensitive to the temperature of the
mantle. The question, though, remains if this statement is valid also for the
small bodies, in which the tidal stresses are weaker and dissipation is
dominated by lattice diffusion.

  * •

By direct calculation, we have derived the frequency dependencies of the
factors $\,k_{\textstyle{{}_{l}}}\,\sin\epsilon_{\textstyle{{}_{l}}}\,$
emerging in the tidal theory. Naturally, the obtained dependencies of these
factors upon the tidal frequency $\,\chi_{\textstyle{{}_{lmpq}}}\,$ (or, to
put it more exactly, upon the tidal mode $\,\omega_{\textstyle{{}_{lmpq}}}\,$)
mimic the frequency-dependence of the imaginary part of the complex
compliance. They scale as $\,\sim\chi^{-\alpha}\,$ with $\,0<\alpha<1\,$, at
higher frequencies; and look as $\,\sim\chi^{-1}\,$ at lower frequencies.
However in the zero-frequency limit the factors
$\,k_{\textstyle{{}_{l}}}\,\sin\epsilon_{\textstyle{{}_{l}}}\,$ demonstrate a
behavior inherent in the tidal lagging and absent in the case of lagging in a
sample: in a close vicinity of the zero frequency, these factors (and the
appropriate components of the tidal torque) become linear in the frequency.
This way, $\,k_{\textstyle{{}_{l}}}\,\sin\epsilon_{\textstyle{{}_{l}}}\,$
first reaches a finite maximum, then decreases continuously to nil as the
frequency approaches to zero, and then changes its sign. So the resonances are
crossed continuously, with neither the tidal torque nor the tidal force
diverging there. For example, the leading term of the torque vanishes at the
synchronous orbit.

This continuous traversing of resonances was pointed out in a heuristic manner
by Efroimsky & Williams (2009). Now we have derived this result directly from
the expression for the compliance of the material of the rotating body. Our
derivation, however, has a problem in it: the frequency, below which the
factors $\,k_{\textstyle{{}_{l}}}\,\sin\epsilon_{\textstyle{{}_{l}}}\,$ and
the appropriate components of the torque change their frequency-dependence to
linear, is implausibly low (lower than $\,10^{-10}\,$Hz, if we take our
formulae literally). The reason for this mishap is that in our formulae we
kept using the known value of the Maxwell time $\,\tau_{{}_{M}}\,$ all the way
down to the zero frequency. Possible changes of the viscosity and,
accordingly, of the Maxwell time in the zero-frequency limit may broaden this
region of linear dependence.

  * •

We have offered an explanation of the unexpected frequency-dependence of
dissipation in the Moon, discovered by LLR. The main point of our explanation
is that the LLR measures the tidal dissipation whose frequency-dependence is
different from that of the seismic dissipation. Specifically, the “wrong” sign
of the exponent in the power dissipation law may indicate that the frequencies
at which tidal friction was observed were below the frequency at which the
lunar $~{}k_{2}\,\sin\epsilon_{2}~{}$ has its peak. Taken the relatively high
frequencies of observation (corresponding to periods of order month to year),
this explanation can be accepted only if the lunar mantle has a low mean
viscosity. This may be the case, taken the presumably high concentration of
the partial melt in the low mantle.

  * •

We have developed a detailed formalism for the tidal torque, and have singled
out its oscillating component.

The studies of entrapment into spin-orbit resonances, performed in the past,
took into account neither the afore-mentioned complicated frequency-dependence
of the torque in the vicinity of a resonance, nor the oscillating part of the
torque. We have written down a concise and practical expression for the
oscillating part, and have raised the question whether it may play a role in
the entrapment and libration dynamics.

##  11 Acknowledgments

This paper stems largely from my prior research carried out in collaboration
with James G. Williams and discussed on numerous occasions with Sylvio Ferraz
Mello, to both of whom I am thankful profoundly. I am grateful to Benoît
Noyelles, Julie Castillo-Rogez, Veronique Dehant, Shun-ichiro Karato, Valéry
Lainey, Valeri Makarov, Francis Nimmo, Stan Peale, and Tim Van Hoolst for
numerous enlightening exchanges and consultations on the theory of tides.

I gladly acknowledge the help and inspiration which I obtained from reading
the unpublished preprint by the late Vladimir Churkin (1998). In Appendix C, I
present several key results from his preprint.

I sincerely appreciate the support from my colleagues at the US Naval
Observatory, especially from John Bangert.

Appendix.

##  Appendix A Continuum mechanics.
A celestial-mechanician’s survival kit

Appendix A offers an extremely short introduction into continuum mechanics.
The standard material, which normally occupies large chapters in books, is
compressed into several pages.

Subsection A.1 presents the necessary terms and definitions. Subsections A.2
explains the basic concepts employed in the theory of stationary deformation,
while subsection A.3 explains extension of these methods to creep. These
subsections also demonstrate the great benefits stemming from the isotropy and
incompressibility assumptions. Subsection A.4 introduces viscosity, while
subsection A.5 offers an example of how elasticity, viscosity get combined
with hereditary reaction, into one expression. Subsection A.6 renders several
simple examples.

###  A.1 Glossary

We start out with a brief guide elucidating the main terms employed in
continuum mechanics.

  * •

Rheology is the science of deformation and flow.

  * •

Elasticity: This is the most trivial reaction – instantaneous, linear, and
fully reversible after stressing is turned off.

  * •

Anelasticity: While still linear, this kind of deformation is not necessarily
instantaneous, and can demonstrate “memory”, both under loading and when the
load is removed. Importantly, the term anelasticity always implies
reversibility: though with delay, the original shape is restored. Thus an
anelastic material can assume two equilibrium states: one is the unstressed
state, the other being the long-term relaxed state. Anelasticity is
characterised by the difference in strain between the two states. It is also
characterised by a relaxation time between these states, and by its inverse –
the frequency at which relaxation is most effective. The Hohenemser-Prager
model, also called SAS (Standard Anelastic Solid), renders an example of
anelastic behaviour.

Anelasticity is an example of but not synonym to hereditary reaction. The
latter includes also those kinds of delayed deformation, which are
irreversible.

  * •

Inelasticity: This term embraces all kinds of irreversible deformation, i.e.,
deformation which stays, fully or partially, after the load is removed.

  * •

Unelasticity ( = Nonelasticity): These terms are very broad, in that they
embrace any behaviour which is not elastic. In the literature, these terms are
employed both for recoverable (anelastic) and unrecoverable (inelastic)
deformations.

  * •

Plasticity: Some authors simply state that plastic deformation is deformation
which is irreversible – a very broad and therefore useless definition which
makes plasticity sound like a synonym to inelasticity.

More precisely, plastic is a stepwise behaviour: no strain is observed until
the stress $\,\sigma\,$ reaches a threshold value $\,\sigma_{{}_{Y}}\,$
(called yield strength), while a steady flow begins as the stress reaches the
said threshold. Plasticity can be either perfect (when deformation is going on
without any increase in load) or with hardening (when increasingly higher
stresses are needed to sustain the flow). It is always irreversible.

In real life, plasticity shows itself in combination with elasticity or/and
viscosity. Models describing these types of behaviour are called
elastoplastic, viscoplastic, and viscoelastoplastic. They are all inelastic,
in that they describe unrecoverable changes of shape.

  * •

Viscosity: Another example of inelastic, i.e., irreversible behaviour. A
viscous stress is proportional to the time derivative of the viscous strain.

  * •

Viscoelasticity: The term is customarily applied to all situations where both
viscous and elastic (but not plastic) reactions are observed. One may then
expect that the equations interrelating the viscoelastic stress to the strain
would contain only the viscosity coefficients and the elastic moduli. However
this is not necessarily true, as some other empirical constants may show up.
For example, the Andrade model (81) contains an elastic term, a viscous term,
and an extra term responsible for a hereditary reaction (the “memory”).
Despite the presence of that extra term, the Andrade creep is still regarded
viscoelastic. So it should be understood that viscoelasticity is, generally,
more than just viscosity combined with elasticity. One might christen such
deformations “viscoelastohereditary”, but such a term would sound
awkward.232323 Sometimes the term elastohereditary is used, but not
viscoelastohereditary or elastoviscohereditary.

On many occasions, complex viscoelastic models can be illustrated with an
infinite set of viscous and elastic elements. These serve to interpret the
hereditary terms as manifestations of viscosity and elasticity only. While
illustrative, these schemes with dashpots and springs have their limitations
and should not be taken too literally. In some (not all) situations, the
hereditary terms may be interpreted as time-dependent additions to the steady-
state viscosity coefficient, the Andrade model being an example of such
situation.

  * •

Viscoplasticity: These are all models wherein both viscosity and plasticity
are present in some combination. In these situations, higher stresses have to
be applied to increase the deformation rate. Just as in the case of
viscoelasticity, viscoplastic models may, in principle, incorporate extra
terms standing for hereditary reaction.

  * •

Elastoviscoplasticity ( = Viscoelastoplasticity): The same as above, though
with elasticity present.

  * •

Hereditary reaction: While the term is self-explanatory, it would be good to
limit its use to effects other than viscosity. In expression (46) for the
stress through strain, the distinction between the viscous and hereditary
reactions is clear: while the viscous part of the stress is rendered
instantaneously by the delta-function term of the kernel, the hereditary
reaction is obtained through integration. In expression (40) for the strain
through stress, though, the viscous part shows up, under the integral, in sum
with the other delayed terms – see, for example, the Andrade model (81). This
is one reason for which we shall use the term hereditary reaction in
application to delayed behaviour different from the pure viscosity. Another
reason is that viscous flow is always irreversible, while a hereditary
reaction may be either irreversible (inelastic) or reversible (anelastic).

  * •

Creep: Widely accepted is the convention that this term signifies a slow-rate
deformation under loads below the yield strength $\,\sigma_{{}_{Y}}\,$.

Numerous authors, though, use the oxymoron plastic creep, thereby extending
the applicability realm of the word creep to any slow deformation.

Here we shall understand creep in the former sense, i.e., with no plasticity
involved.

It would be important to distinguish between viscoelastic deformations, on the
one hand, and viscoplastic (or, properly speaking, viscoelastoplastic)
deformations on the other hand. Plasticity shows itself at larger stresses and
is, typically, nonlinear. It comes into play when the linearity assertion
fails. For most minerals, this happens when the strain approaches the
threshold of $\,10^{-6}\,$. Although it is possible that this threshold is
transcended in some satellites (for example, in the deeper layers of the
Moon), we do not address plasticity in this paper.

###  A.2 Stationary linear deformation of isotropic incompressible media

In the linear approximation, the tensor of elastic stress,
$\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,$, is proportional to the
differences in displacement of the neighbouring elements of the medium. These
differences are components of the tensor gradient $\,\nabla\otimes{\bf{u}}\,$,
where ${\bf{u}}\,$ is the displacement vector.

The tensor gradient can be decomposed, in an invariant way, into its symmetric
and antisymmetric parts:

$\displaystyle\nabla\otimes{\bf{u}}\;=\;\frac{\textstyle 1}{\textstyle
2}\,\left[\,\,\left(\nabla\otimes{\bf{u}}\right)\,+\,\left(\nabla\otimes{\bf{u}}\right)^{{{}^{T}}}\,\,\right]\,+~{}\frac{\textstyle
1}{\textstyle
2}\,\left[\,\,\left(\nabla\otimes{\bf{u}}\right)\,-\,\left(\nabla\otimes{\bf{u}}\right)^{{{}^{T}}}\,\,\right]~{}~{}~{}.$
(133)

The decomposition being invariant, the two parts should contribute into the
stress independently, at least in the linear approximation. However, as well
known (Landau & Lifshitz 1986), the antisymmetric part of (133) describes the
displacement of the medium as a whole and thus brings nothing into the stress.
This is why the linear dependence is normally written as

$\displaystyle\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,=\,{\mathbb{B}}\,\,{\mathbb{U}}~{}~{}~{},$
(134)

where $\,{\mathbb{B}}\,$ is a four-dimensional tensor having
$\,3^{4}\,=\,81\,$ components, while the strain tensor

$\displaystyle{\mathbb{U}}\,\equiv\,\frac{\textstyle 1}{\textstyle
2}\,\left[\,\left(\nabla\otimes{\bf{u}}\right)\,+\,\left(\nabla\otimes{\bf{u}}\right)^{{{}^{T}}}\,\right]$
(135)

is the symmetric part of the tensor gradient. Its components are related to
the displacement vector $\,{\bf{u}}~{}$ through
$~{}u_{\alpha\beta}\,\equiv\,\frac{\textstyle 1}{\textstyle
2}\left(\frac{\textstyle{\partial u_{\alpha}}}{\textstyle{\partial
x_{\beta}}}+\frac{\textstyle{\partial u_{\beta}}}{\textstyle{\partial
x_{\alpha}}}\right)\,$.

Although the matrix $\,{\mathbb{B}}\,$ is comprised of 81 empirical constants,
in isotropic materials the description can be reduced to two constants only.
To see this, recall that the expansion of the strain into a part with trace
and a traceless part,
$\,{\mathbb{U}}\,=\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}\;\mbox{Sp}\,{\mathbb{U}}\,+\,\left(\,{\mathbb{U}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}\;\mbox{Sp}\,{\mathbb{U}}\,\right)\,$,
is invariant. Here the trace of $\,{\mathbb{U}}\,$ is denoted with
$\,\mbox{Sp}\,{\mathbb{U}}\,\equiv\,u_{\alpha\alpha}\,$, summation over
repeated indices being implied. The notation $\,{\mathbb{I}}\,$ stands for the
unity matrix consisting of elements $\,\delta_{\gamma\nu}\,$.

In an isotropic medium, the elastic stress must be invariantly expandable into
parts proportional to the afore-mentioned parts of the strain. The first part
of the stress is proportional, with an empirical coefficient $\,3\,K\,$, to
the trace part
$\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}\;\mbox{Sp}\,{\mathbb{U}}\,$
of the strain. The second part of the stress will be proportional, with an
empirical coefficient $\,2\,\mu\,$, to the traceless part
$\,\left(\,{\mathbb{U}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,\right)\,$
of the strain:

$\displaystyle\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}~{}=~{}K~{}{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,+\,2\,\mu\left(\,{\mathbb{U}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}\;\mbox{Sp}\,{\mathbb{U}}\,\right)~{}=~{}-~{}p~{}{\mathbb{I}}\,+\,2\,\mu\left(\,{\mathbb{U}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}\;\mbox{Sp}\,{\mathbb{U}}\,\right)$
(136a) or, in Cartesian coordinates:
$\displaystyle\sigma_{\gamma\nu}~{}=~{}\,K~{}\delta_{\gamma\nu}\,u_{\alpha\alpha}\,+\,2\,\mu\,\left(\,u_{\gamma\nu}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,\delta_{\gamma\nu}\,u_{\alpha\alpha}\,\right)~{}=~{}-~{}p~{}\delta_{\gamma\nu}\,+\,2\,\mu\,\left(\,u_{\gamma\nu}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,\delta_{\gamma\nu}\,u_{\alpha\alpha}\,\right)~{}~{}~{},\quad\quad$
(136b)

where

$\displaystyle
p\,\equiv\,-\,\frac{1}{3}\;\mbox{Sp}\,{\mathbb{S}}\,=\,-\,K\,\mbox{Sp}\,{\mathbb{U}}$
(137)

is the hydrostatic pressure. Thus the elastic stress gets decomposed, in an
invariant way, as:

$\displaystyle\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}~{}=~{}\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,+\;\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}~{}~{}~{},$
(138)

where the volumetric elastic stress is given by

$\displaystyle\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\;\,\equiv\;K~{}{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}~{}=~{}{\mathbb{I}}~{}\frac{1}{3}~{}\mbox{Sp}\,{\mathbb{S}}\,=\,-\,p\,{\mathbb{I}}~{}~{}~{},$
(139)

while the deviatoric elastic stress is:

$\displaystyle\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}~{}\equiv~{}2\,\mu\left(\,{\mathbb{U}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}\;\mbox{Sp}\,{\mathbb{U}}\,\right)~{}~{}~{}.$
(140)

Inverse to (136a \- 136b) are the following expressions for the strain tensor:

$\displaystyle{\mathbb{U}}~{}=~{}\frac{1}{9\,K}~{}{\mathbb{I}}~{}\mbox{Sp}\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,+\,\frac{1}{2\,\mu}\,\left(\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}~{}\mbox{Sp}\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,\right)$
(141a) and $\displaystyle
u_{\gamma\nu}~{}=~{}\frac{1}{9\,K}~{}\delta_{\gamma\nu}~{}\stackrel{{\scriptstyle(e)}}{{\sigma}}_{\alpha\alpha}~{}+~{}\frac{1}{2\,\mu}\,\left(\,\stackrel{{\scriptstyle(e)}}{{\sigma}}_{\gamma\nu}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,\delta_{\gamma\nu}\,\stackrel{{\scriptstyle(e)}}{{\sigma}}_{\alpha\alpha}\,\right)~{}~{}~{},\quad$
(141b)

where the term with trace, $\,\frac{\textstyle 1}{\textstyle
9\,K}\;{\mathbb{I}}\;\mbox{Sp}\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,$,
is called the volumetric strain, while the traceless term,
$\,\frac{\textstyle{1}}{\textstyle{2\,\mu}}\,\left(\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}\;\mbox{Sp}\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,\right)\,$,
is named the deviatoric strain. The quantity $\,K\,$ is called the bulk
modulus, while $\,\mu\,$ is called the shear modulus.

Expressions (136) trivially entail the following interrelation between traces:

$\displaystyle\mbox{Sp}\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,=\,3\,K\,\mbox{Sp}\,{\mathbb{U}}~{}\quad~{}\mbox{or,
in terms of
components:}\quad~{}\stackrel{{\scriptstyle(e)}}{{\sigma}}_{\alpha\alpha}\,=\,3\,K\,u_{\alpha\alpha}~{}~{}~{}.$
(142)

As demonstrated in many books (e.g., in Landau & Lifshitz 1986), the trace of
the strain is equal to the relative variation of the volume, experienced by
the material subject to deformation:
$~{}u_{\alpha\alpha}\,=\,\nabla\cdot{\bf{u}}\,=\,\frac{\textstyle{dV\,^{\prime}-dV}}{\textstyle{dV}}~{}$,
where $\,{\bf{u}}\,$ is the displacement vector. In the no-compressibility
approximation, the trace of the strain and, according to (142), that of the
stress become zero. Then, in the said approximation, the hydrostatic pressure
(137) and the volumetric elastic stress (139) become nil, and all we are left
with is the deviatoric elastic stress (and, accordingly, the deviatoric part
of the strain). Formulae (136) and (141) get simplified to

$\displaystyle\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}~{}=~{}2\,\mu\,{\mathbb{U}}~{}~{},\quad~{}~{}\mbox{which
is the same
as}~{}\quad\stackrel{{\scriptstyle(e)}}{{\sigma}}_{\gamma\nu}~{}=~{}2\,\mu\,u_{\gamma\nu}~{}~{}~{},\,\quad$
(143)

and to

$\displaystyle
2\;{\mathbb{U}}~{}=~{}J\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}~{}~{},\quad~{}~{}\mbox{which
is the same as}~{}\quad
2\,u_{\gamma\nu}~{}=~{}J\,\stackrel{{\scriptstyle(e)}}{{\sigma}}_{\gamma\nu}~{}~{}~{},\quad$
(144)

the quantity $\,J\,\equiv\,1/\mu\,$ being called the compliance of the
material.

###  A.3 Evolving linear deformation of isotropic incompressible isotropic
media. Hereditary reaction

Equations (134 \- 144) were written for static deformation, so each of these
equations can be assumed to connect the strain and the elastic stress taken at
the same instant of time (for a static deformation their values stay constant
anyway).

Extension of this machinery is needed when one wants to describe evolving
deformation of materials with “memory”. Thence the four-dimensional tensor
$\,{\mathbb{B}}\,$ becomes a linear operator $\,\tilde{\mathbb{B}}\,$ acting
on the strain tensor function as a whole. To render the value of the stress at
time $\,t\,$, the operator will “consume” as arguments all the values of
strain over the interval
$\;t\,^{\prime}\,\in\,\left(\right.-\infty\,,\,t\left.\right]\;$:

$\displaystyle\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}(t)\,=\,\tilde{\mathbb{B}}(t)\,\,{\mathbb{U}}~{}~{}~{}.$
(145)

Thus $\,\tilde{\mathbb{B}}\,$ will be an integral operator, with integration
going from $\,t\,^{\prime}\,=\,-\,\infty\,$ through $\,t\,^{\prime}\,=\,t\,$.

In the static case, the linearity guaranteed elasticity, i.e., the ability of
the body to regain its shape after the loading is turned off: no stress yields
no strain. In a more general situation of materials with “memory”, this
ability is no longer retained, as the material may demonstrate creep. This is
why, in this section, the stress is called hereditary and is denoted with
$\,\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}\,$.

Just as in the stationary case, we wish the properties of the medium to remain
isotropic. As the decomposition of the strain into the trace and traceless
parts remains invariant at each moment of time, these two parts will,
separately, generate the trace and traceless parts of the hereditary stress in
an isotropic medium. This means that, in such media, the four-dimensional
tensor operator $\,\tilde{\mathbb{B}}\,$ gets reduced to two scalar linear
operators $\,\tilde{K}\,$ and $\,\tilde{\mu}\,$:

$\displaystyle\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}~{}=~{}\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,+\;\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}~{}=~{}\tilde{K}~{}{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,+\,2\,\tilde{\mu}\left(\,{\mathbb{U}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}\;\mbox{Sp}\,{\mathbb{U}}\,\right)~{}~{}~{},$
(146)

where both $\,\tilde{K}\,$ and $\,\tilde{\mu}\,$ are integral operators acting
on the tensor function $\,u_{\gamma\nu}(t\,^{\prime})\,$ as a whole, i.e.,
with integration going from $\,t\,^{\prime}\,=\,-\,\infty\,$ through
$\,t\,^{\prime}\,=\,t\,$.

If we also assume that, under evolving load, the medium stays incompressible,
the trace of the strain, $\,u_{\alpha\alpha}\,$, will stay zero. An operator
generalisation of (142) now reads:
$~{}\sigma_{\alpha\alpha}(t)\,=\,3\,\tilde{K}(t)\,u_{\alpha\alpha}~{}$. Under
a reasonable assumption of $\,\sigma_{\alpha\alpha}\,$ being nil in the
distant past, this integral operator renders $\,\sigma_{\alpha\alpha}\,=\,0\,$
at all times. This way, in a medium that is both isotropic and incompressible,
we have:

$\displaystyle{\mathbb{U}}~{}=~{}{\mathbb{U}}_{\textstyle{{}_{deviatoric}}}~{}\quad~{}\mbox{and,
~{}accordingly:}~{}\quad~{}\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}~{}=~{}\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\quad.\quad\quad$
(147)

Then the time-dependent analogues to formulae (143) and (144) will be:

$\displaystyle\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}(t)~{}=~{}2\,\tilde{\mu}(t)\,{\mathbb{U}}$
(148)

and

$\displaystyle
2\,{\mathbb{U}}(t)~{}=~{}\hat{J}(t)\,\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}~{}~{}~{},$
(149)

where the compliance $\,\hat{J}\,$, too, has been promoted to operatorship and
crowned with a caret.

Formula (148) tells us that in a medium, which is both isotropic and
incompressible, relation between the stress and strain tensors can be
described with one scalar integral operator $\,\tilde{\mu}\,$ only, the
complementary operator $\,\hat{J}\,$ being its inverse. (Here the adjective
“scalar” does not imply multiplication with a scalar. It means that the
operator preserves its functional form under a change of coordinates.)

Below we shall bring into the picture also the viscous component of the
stress, a component related to the strain through a four-dimensional tensor
whose $\,3^{4}=81\,$ components are differential operators. In that case too,
the isotropy of the medium will enable us to reduce the 81-component tensor
operator to two differential operators transforming as scalars. Besides, the
incompressibility of the medium makes the viscous stress traceless. Thus it
will turn out that, in an isotropic and incompressible medium, the viscous
component of the stress can be described by only one scalar differential
operator – much like the elastic and hereditary parts of the stress. (Once
again, “scalar” means: indifferent to coordinate transformations.)

Eventually, the elastic, hereditary, and viscous deformations will be united
under the auspices of a general viscoelastic formalism. In an isotropic
medium, this combined formalism will be reduced to two integro-differential
operators only. In a medium which is both isotropic and incompressible, the
formalism will be reduced to only one scalar integro-differential operator.

###  A.4 The viscous stress

While the elastic stress $\,\stackrel{{\scriptstyle{{(e)}}}}{{\mathbb{S}}}\,$
is linear in the strain, the viscous stress
$\,\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\,$ is linear in the first
derivatives of the components of the velocity with respect to the coordinates:

$\displaystyle\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\,=\,{\mathbb{A}}\,(\nabla\otimes{\bf{v}})\,$
(150)

where $\,{\mathbb{A}}\,$ is the so-called viscosity tensor,
$\,\nabla\otimes{\bf{v}}\,$ is the tensor gradient of the velocity. The
velocity of a fluid parcel relative to its average position is connected to
the displacement vector $\,{\bf u}\,$ through $\,{\bf{v}}\,=\,d{\bf u}/dt\,$.

The tensor gradient of the velocity can be expanded, in an invariant way, into
its antisymmetric and symmetric parts:

$\displaystyle\nabla\otimes{\bf{v}}\;=\;\Omega\;+\;{\mathbb{E}}~{}~{}~{},$
(151)

where the antisymmetric part is furnished by the vorticity tensor

$\displaystyle\Omega\;\equiv\;\frac{1}{2}~{}\left[\,(\nabla\otimes{\bf{v}})~{}-\;(\nabla\otimes{\bf{v}})^{{}^{T}}\,\right]~{}~{}~{},$
(152)

while the symmetric part is given by the rate-of-shear tensor

$\displaystyle{\mathbb{E}}\;\equiv\;\frac{1}{2}~{}\left[\,(\nabla\otimes{\bf{v}})\,+\,(\nabla\otimes{\bf{v}})^{{}^{T}}\,\right]~{}~{}~{}.$
(153)

The latter is obviously related to the strain tensor through

$\displaystyle{\mathbb{E}}\;=\;\frac{\partial~{}}{\partial
t}\;{\mathbb{U}}\;~{}~{}.$ (154)

It can be demonstrated (e.g., Landau & Lifshitz 1987) that the antisymmetric
vorticity tensor describes the rotation of the medium as a whole242424 This is
why this tensor’s components coincide with those of the angular velocity
$\vec{\omega}$ of the body. For example, $\;\Omega_{21}\,=\,\frac{\textstyle
1}{\textstyle 2}\,\left(\,\frac{\textstyle\partial v_{2}}{\textstyle\partial
x_{1}}\,-\,\frac{\textstyle\partial v_{1}}{\textstyle\partial
x_{2}}\,\right)\;$ coincides with $\,\omega_{3}\,$. and therefore contributes
nothing to the stress.252525Since expansion (151) of the tensor gradient into
the vorticity and rate-of-shear tensors is invariant, then so is the
conclusion about the irrelevance of the vorticity tensor for the stress
picture. For this reason, the viscous stress can be written as

$\displaystyle\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\,=\,{\mathbb{A}}\,{\mathbb{E}}\,=\,{\mathbb{A}}\;\frac{\partial~{}}{\partial
t}\;{\mathbb{U}}\;~{}~{}.$ (155)

The matrix $\,{\mathbb{A}}\,$ is four-dimensional and contains $\,3^{4}=81\,$
components. Just as the matrix $\,{\mathbb{B}}\,$ emerging in expression (134)
for the elastic stress, the matrix $\,{\mathbb{A}}\,$ can be reduced, in an
isotropic medium, to only two empirical constants. To see this, mind that the
rate-of-shear tensor can be decomposed, in an invariant manner, into two
parts:

$\displaystyle{\mathbb{E}}~{}=~{}\frac{1}{3}~{}{\mathbb{I}}~{}\nabla\cdot{\bf{v}}~{}+~{}\left(\,{\mathbb{E}}~{}-~{}\frac{1}{3}~{}{\mathbb{I}}~{}\nabla\cdot{\bf{v}}\,\right)~{}~{}~{}.$
(156)

where the rate-of-expansion tensor

$\displaystyle\frac{\textstyle 1}{\textstyle
3}~{}{\mathbb{I}}~{}\nabla\cdot{\bf{v}}$ (157)

is diagonal and has a trace, while the combination

$\displaystyle{\mathbb{E}}~{}-~{}\frac{1}{3}~{}{\mathbb{I}}~{}\nabla\cdot{\bf{v}}\;=\;\frac{1}{2}~{}\left[\,(\nabla\otimes{\bf{v}})\,+\,(\nabla\otimes{\bf{v}})^{{}^{T}}\,\right]~{}-~{}\frac{1}{3}~{}{\mathbb{I}}~{}\nabla\cdot{\bf{v}}$
(158)

is symmetric and traceless. These two parts contribute linearly proportional
inputs into the stress. The first input is proportional, with an empirical
coefficient $~{}3\,\zeta~{}$, to the rate-of-expansion term, while the second
input into the stress is proportional, with an empirical coefficient
$~{}2\,\eta~{}$, to the symmetric traceless combination:

$\displaystyle\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\,=\,3\,\zeta\;\frac{\textstyle
1}{\textstyle
3}~{}{\mathbb{I}}~{}\nabla\cdot{\bf{v}}~{}+~{}2\,\eta\,\left(\,{\mathbb{E}}\;-\;\frac{\textstyle
1}{\textstyle
3}~{}{\mathbb{I}}~{}\nabla\cdot{\bf{v}}\,\right)~{}=~{}\zeta\,\frac{\textstyle\partial~{}}{\textstyle\partial
t}\,\left(\,{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,\right)~{}+~{}2\,\eta\,\frac{\textstyle\partial~{}}{\textstyle\partial
t}\,\left(\,{\mathbb{U}}\;-\;\frac{\textstyle 1}{\textstyle
3}~{}{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,\right)~{}~{}~{}.~{}\quad$ (159)

Here we recalled that
$\,\nabla\cdot{\bf{v}}\,=\,\frac{\textstyle\partial~{}}{\textstyle\partial
t}\,\nabla\cdot{\bf{u}}\,=\,\frac{\textstyle\partial~{}}{\textstyle\partial
t}\,u_{\alpha\alpha}\,=\,\frac{\textstyle\partial~{}}{\textstyle\partial
t}\,\mbox{Sp}\,{\mathbb{U}}\,$. Since $\,\mbox{Sp}\,{\mathbb{U}}\,$ is equal
to the volume variation
$~{}\frac{\textstyle{dV^{\prime}-dV}}{\textstyle{dV}}~{}$ experienced by the
material, we see that the first term in (159) is volumetric, the second being
deviatoric.

The quantity $\,\eta\,$ is called the first viscosity or the shear viscosity.
The quantity $\,\zeta\,$ is named the second viscosity or the bulk viscosity.

###  A.5 An example of approach to viscoelastic behaviour

In this subsection, we shall consider one possible approach to description of
viscoelastic regimes. As we mentioned in subsection A.1, the term
viscoelasticity covers not only combinations of elasticity and viscosity, but
can also include other forms of delayed reaction. So the term viscoelastic is
customarily used as a substitution for too long a term viscoelastohereditary.

One possible approach would be to assume that the elastic, hereditary, and
viscous stresses simply sum up, and that each of them is related the the same
strain $\,{\mathbb{U}}\;$: an

$\displaystyle\stackrel{{\scriptstyle(total)}}{{\mathbb{S}}}\,=~{}\stackrel{{\scriptstyle{{(e)}}}}{{\mathbb{S}}}\,+\,\stackrel{{\scriptstyle{{{(h)}}}}}{{\mathbb{S}}}\,+\,\stackrel{{\scriptstyle{{{(v)}}}}}{{\mathbb{S}}}~{}=~{}\left(\,{\mathbb{B}}~{}+~{}\tilde{\mathbb{B}}~{}+~{}{\mathbb{A}}\,\frac{\partial~{}}{\partial
t}\,\right)\;{\mathbb{U}}~{}~{}~{},$ (160a) or simply
$\displaystyle\stackrel{{\scriptstyle(total)}}{{\mathbb{S}}}\;=\;\hat{\mathbb{B}}\,{\mathbb{U}}~{}\quad,\quad\quad\mbox{where}\quad\hat{\mathbb{B}}\,\equiv\,{\mathbb{B}}~{}+~{}\tilde{\mathbb{B}}~{}+~{}{\mathbb{A}}\,\frac{\partial~{}}{\partial
t}~{}~{}~{},$ (160b)

where the three operators – the integral operator $\,\tilde{\mathbb{B}}\,$,
the differential operator
$\,{\mathbb{A}}\,\frac{\textstyle\partial~{}}{\textstyle\partial t}\,$, and
the operator of multiplication by matrix $\,{\mathbb{B}}\,$ – comprise an
integro-differential operator $\,\hat{\mathbb{B}}\,$.

In an isotropic medium, each of the three matrices, $\,\tilde{\mathbb{B}}\,$,
$\,{\mathbb{A}}\,\frac{\textstyle\partial~{}}{\textstyle\partial t}\,$, and
$\,{\mathbb{B}}\,$, includes two terms only. This happens because in such a
medium each of the three parts of the stress gets decomposed invariantly into
its deviatoric and volumetric components: The elastic stress becomes:

$\displaystyle\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\;=\;\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,+\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,=\;3\,K\,\left(\frac{\textstyle
1}{\textstyle
3}~{}{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,\right)~{}+~{}2\,\mu\,\left(\,{\mathbb{U}}\;-\;\frac{\textstyle
1}{\textstyle 3}~{}{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,\right)\quad,\quad$
(161)

with $\,K\,$ and $\,\mu\,$ being the bulk elastic modulus and the shear
elastic modulus, correspondingly, $\;{\mathbb{I}}\,$ standing for the unity
matrix, and Sp denoting the trace of a matrix:
$~{}\mbox{Sp}\,{\mathbb{U}}\,\equiv\,\sum_{i}U_{ii}\;$.

The hereditary stress becomes:

$\displaystyle\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}~{}=~{}\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,+~{}\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}~{}=\;3\,\tilde{K}~{}\left(\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,\right)~{}+~{}2\,\tilde{\mu}\,\left(\,{\mathbb{U}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,\right)~{}~{}~{},~{}~{}$
(162)

where $\,\tilde{K}\,$ and $\,\tilde{\mu}\,$ are the bulk-modulus operator and
the shear-modulus operator, accordingly.

The viscous stress acquires the form:

$\displaystyle\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\;=\;\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,+\,\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,=\;3~{}\zeta\,\frac{\textstyle\partial~{}}{\textstyle\partial
t}\,\left(\frac{\textstyle 1}{\textstyle
3}~{}{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,\right)~{}+~{}2\,\eta\,\frac{\textstyle\partial~{}}{\textstyle\partial
t}\,\left(\,{\mathbb{U}}\;-\;\frac{\textstyle 1}{\textstyle
3}~{}{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,\right)\quad,\quad$ (163)

the quantities $\zeta$ and $\eta$ being termed as the bulk viscosity and the
shear viscosity, correspondingly

The term $~{}\frac{\textstyle 1}{\textstyle
3}~{}{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}~{}$ is called the volumetric part
of the strain, while $~{}{\mathbb{U}}\,-\,\frac{\textstyle 1}{\textstyle
3}~{}{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}~{}$ is called the deviatoric part.
Accordingly, in expressions (161 \- 163) for the stresses, the pure-trace
terms are called volumetric, the other term being named deviatoric.

The total stress, too, can now be split into the total volumetric and the
total deviatoric parts:

$\displaystyle\stackrel{{\scriptstyle(total)}}{{\mathbb{S}}}$ $\displaystyle=$
$\displaystyle\overbrace{\left(\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,+\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,\right)}^{\stackrel{{\scriptstyle\textstyle\stackrel{{\scriptstyle{{(e)}}}}{{\mathbb{S}}}}}{{}}}\,+\,\overbrace{\left(\,\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,+\,\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,\right)}^{\stackrel{{\scriptstyle\textstyle\stackrel{{\scriptstyle{{(v)}}}}{{\mathbb{S}}}}}{{}}}\,+\,\overbrace{\left(\,\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,+\,\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,\right)}^{\stackrel{{\scriptstyle\textstyle\stackrel{{\scriptstyle{{(h)}}}}{{\mathbb{S}}}}}{{}}}$
(164a) $\displaystyle=$
$\displaystyle\overbrace{\left(\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,+\,\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,+\,\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}_{\textstyle{{}_{volumetric}}}\,\right)}^{\stackrel{{\scriptstyle\textstyle{{{\mathbb{S}}_{\textstyle{{}_{volumetric}}}}}}}{{}}}\,+\,\overbrace{\left(\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,+\,\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,+\,\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,\right)}^{\stackrel{{\scriptstyle\textstyle{\mathbb{S}}_{\textstyle{{}_{deviatoric}}}}}{{}}}$
$\displaystyle=$
$\displaystyle\left(3\,K\,+\,3\,\tilde{K}\,+\,3\,\zeta\,\frac{\partial\,}{\partial
t}\right)\left(\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\right)\,+\,\left(2\,\mu\,+\,2\,\tilde{\mu}\,+\,2\,\eta\,\frac{\partial\,}{\partial
t}\right)\left(\,{\mathbb{U}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}\;\mbox{Sp}\,{\mathbb{U}}\,\right)~{}\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle=$ $\displaystyle
3~{}\hat{K}~{}\left(\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}~{}\mbox{Sp}\,{\mathbb{U}}\,\right)~{}+~{}2~{}\hat{\mu}~{}\left(\,{\mathbb{U}}\,-\,\frac{\textstyle{1}}{\textstyle{3}}\,{\mathbb{I}}\;\mbox{Sp}\,{\mathbb{U}}\,\right)~{}~{}~{},$
(164b)

where

$\displaystyle\hat{K}\;\equiv\;K\;+\;\tilde{K}\;+\;\zeta\;\frac{\partial~{}}{\partial
t}~{}\quad\quad\mbox{and}\quad\quad\hat{\mu}\;\equiv\;\mu\;+\;\tilde{\mu}\;+\;\eta\;\frac{\partial~{}}{\partial
t}~{}~{}~{}.\quad\quad\quad$ (165)

As expected, a total linear deformation of an isotropic material can be
described with two integro-differential operators, one acting on the
volumetric strain, another on the deviatoric strain.

If an isotropic medium is also incompressible, the relative change of the
volume vanishes: $\,\mbox{Sp}\,{\mathbb{U}}=0\,$. Accordingly, the volumetric
part of the strain becomes nil, and so do the volumetric parts of the elastic,
hereditary, and viscous stresses. For such media, we end up with a simple
relation which includes only deviators:

$\displaystyle\stackrel{{\scriptstyle(total)}}{{\mathbb{S}}}\,=\,{\mathbb{S}}_{\textstyle{{}_{deviatoric}}}\,=\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,+\,\stackrel{{\scriptstyle(h)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,+\,\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}_{\textstyle{{}_{deviatoric}}}\,=\,2\,{\mu}~{}{\mathbb{U}}\,+\,2\,\tilde{\mu}~{}{\mathbb{U}}\,+\,2\,\eta\,\frac{\partial\,}{\partial
t}~{}{\mathbb{U}}\quad\quad$ (166)

or simply:

$\displaystyle\stackrel{{\scriptstyle(total)}}{{\mathbb{S}}}\,=\,{\mathbb{S}}_{\textstyle{{}_{deviatoric}}}\,=\,2\,\hat{\mu}~{}{\mathbb{U}}~{}~{}~{},$
(167)

where $\,{\mathbb{U}}\,$ contains only a deviatoric part, while

$\displaystyle\hat{\mu}\,\equiv\,\mu\,+\,\tilde{\mu}\,+\,\eta\,\frac{\partial\,}{\partial
t}$ (168)

is the total, integro-differential operator, which is mapping the preceding
history and the present rate of change of the strain to the present value of
the stress.

It should be reiterated that the above approach is based on the assertion that
the elastic, viscous, and hereditary stresses sum up, and that all three are
related to the same total strain. A simple example of this approach, called
the Kelvin-Voigt model, is rendered below in subsection A.6.3.

A different approach would be to assume that the strain consists of three
distinct parts – elastic, hereditary, and viscous – and that these components
are related to the same overall stress. A simple example of this treatment,
termed the Maxwell model, is set out in subsection A.6.4. A more complex
example of this approach is furnished by the Andrade model presented in
subsection 5.3. Another way of combining elasticity and viscosity (with no
hereditary reaction involved) is implemented by the Hohenemser-Prager (SAS)
model explained in subsection A.6.5 below.

###  A.6 Examples of viscoelastic behaviour with no hereditary reaction

####  A.6.1 Elastic deformation

The truly simplest example of deformation is elastic:

$\displaystyle\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,=\,2\,\mu\,{\mathbb{U}}~{}\quad,\quad\quad{\mathbb{U}}~{}=~{}J\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}~{}~{}~{},$
(169)

where $\,\mu\,$ and $\,J\,$ are the unrelaxed rigidity and compliance:

$\displaystyle\mu\,=\,\mu(0)~{}~{}~{},\quad~{}J\,=\,J(0)~{}~{}~{},\quad~{}\mu\;=\;{1}/{J}~{}~{}~{}.$
(170)

In the frequency domain, this relation assumes the same form as it would in
the time domain:

$\displaystyle\bar{\sigma}_{\gamma\nu}(\chi)\,=\,2\,\mu\,\bar{u}_{\gamma\nu}(\chi)~{}\quad,\quad\quad
2\,\bar{u}_{\gamma\nu}(\chi)\,=\,J\,\bar{\sigma}_{\gamma\nu}(\chi)~{}~{}~{}.$
(171)

####  A.6.2 Viscous deformation

The next example is that of a purely viscous behaviour:

$\displaystyle\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\,=\,2\,\eta\,\frac{\partial\,}{\partial
t}~{}{\mathbb{U}}~{}~{}~{}.$ (172)

It is straightforward from (172) and (27) that, in this regime, the Fourier
components of the stress262626 Although we no longer spell it out, the word
stress everywhere means: deviatoric stress, as we agreed to consider the
medium incompressible. are connected to those of the strain through

$\displaystyle\bar{\sigma}_{\gamma\nu}(\chi)\,=\,2\,\bar{\mu}(\chi)\;\bar{u}_{\gamma\nu}(\chi)~{}\quad,\quad\quad
2\,\bar{u}_{\gamma\nu}(\chi)\,=\,\bar{J}(\chi)\;\bar{\sigma}_{\gamma\nu}(\chi)~{}~{}~{},$
(173)

where the complex rigidity and the complex compliance are given by

$\displaystyle\bar{\mu}\,=\,{\it
i}\,\eta\,\chi\quad,\quad\quad\bar{J}\,=\,-\,\frac{{\it
i}}{\eta\,\chi}~{}~{}~{}.$ (174)

####  A.6.3 Viscoelastic deformation: a Kelvin-Voigt material

The Kelvin-Voigt model, also called the Voigt model, can be represented with a
purely viscous damper and a purely elastic spring connected in parallel.
Subject to the same elongation, these elements have their forces summed up.
This illustrates the situation where the total, viscoelastic stress consists
of a purely viscous and a purely elastic inputs called into being by the same
strain:

$\displaystyle{\mathbb{S}}\;=\;\stackrel{{\scriptstyle(ve)}}{{\mathbb{S}}}\;=\;\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\,+\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}~{},\,\quad\mbox{while}\,\quad{\mathbb{U}}\,=\;\stackrel{{\scriptstyle(v)}}{{\mathbb{U}}}\,=\;\stackrel{{\scriptstyle(e)}}{{\mathbb{U}}}~{}~{}~{}.$
(175)

Then the total stress reads:

$\displaystyle{\mathbb{S}}\,=\,\left(2\,{\mu}\,+\,2\,\eta\,\frac{\partial\,}{\partial
t}\right)~{}{\mathbb{U}}~{}~{}~{},$ (176a) which is often presented in the
form of
$\displaystyle{\mathbb{S}}\,=\,2\,\mu\,\left(\,{\mathbb{U}}\,+~{}\tau_{{}_{V}}\stackrel{{\scriptstyle\centerdot}}{{\mathbb{U}}}\,\right)~{}~{}~{},$
(176b)

with the so-called _Voigt time_ defined as

$\displaystyle\tau_{{}_{V}}\;\equiv\;{\eta}/{\mu}~{}~{}~{}.$ (177)

Comparing (176) with (46), we understand that the kernel of the rigidity
operator for the Kelvin-Voigt model can be written down as

$\displaystyle\mu(t\,-\,t\,^{\prime})\;=\;\mu\;+\;\eta\;\delta(t\,-\,t\,^{\prime})~{}~{}~{}.$
(178)

Suppose the strain is varying in time as

$\displaystyle{u}_{\gamma\nu}(t)\;=\;\frac{\sigma_{0}}{2\,\mu}\;\left[\,1\;-\;\exp\left(\,-\,\frac{t\,-\,t_{0}}{\tau_{{{}_{V}}}}\,\right)\,\right]~{}\Theta(t\,-\,t_{0})~{}~{}~{},$
(179)

so that

$\displaystyle\stackrel{{\scriptstyle\bf\centerdot}}{{u}}_{\gamma\nu}(t)\;=\;\frac{\sigma_{0}}{2\,\mu\,}\;\frac{1}{\tau_{{{}_{V}}}}\;\exp\left(\,-\,\frac{t\,-\,t_{0}}{\tau_{{{}_{V}}}}\,\right)\,~{}\Theta(t\,-\,t_{0})~{}~{}~{}.$
(180)

Then insertion of (178) and (180) into (LABEL:permitted_2) or, equivalently,
insertion of (179) into (176a) demonstrates that this strain results from a
stress272727 For example, plugging of (178) and (180) into (LABEL:permitted_2)
leads to:  $\displaystyle{}\sigma(t)$ $\displaystyle=$
$\displaystyle\int^{t\,^{\prime}=\,t}_{t\,^{\prime}=\,-\,\infty}\left[\,\mu\;+\;\eta\;\delta(t\,-\,t\,^{\prime})\,\right]\;\frac{\sigma_{0}}{\mu}\;\frac{\textstyle
1}{\textstyle\tau_{{{}_{V}}}}\;\exp\left(\,-\,\frac{t\,^{\prime}\,-\,t_{0}}{\tau_{{{}_{V}}}}\,\right)\,~{}\Theta(t\,^{\prime}\,-\,t_{0})\,dt\,^{\prime}$
(184) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{c}\textstyle\sigma_{0}~{}\left[\;\int^{t\,^{\prime}=\,t}_{t\,^{\prime}=\,t_{0}}\exp\left(\,-\,\frac{\textstyle
t\,^{\prime}\,-\,t_{0}}{\textstyle\tau_{{{}_{V}}}}\,\right)\,\frac{\textstyle
dt\,^{\prime}}{\textstyle\tau_{{{}_{V}}}}\;+\;\exp\left(\,-\,\frac{\textstyle
t\,-\,t_{0}}{\textstyle\tau_{{{}_{V}}}}\,\right)\right]~{}~{}\quad\mbox{for}\quad
t\,\geq\,t_{0}{}{}{}{}\\\ {}\hfil\\\
0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad~{}\,\mbox{for}\quad
t\,<\,t_{0}~{}~{}~{},\end{array}\right.$ which is simply
$\,\;\sigma_{0}\,\Theta(t\,-\,t_{0})\;$.

$\displaystyle\sigma_{\gamma\nu}(t)\,=\,{\sigma_{0}}\;\Theta(t\,-\,t_{0})~{}~{}~{}.$
(185)

It would however be a mistake to deduce from this that the compliance function
is equal to $~{}\mu^{-1}\,\left[\,1\;-\;\exp\left(\,-\,\frac{\textstyle
t\,-\,t\,^{\prime}}{\textstyle\tau_{{{}_{V}}}}\,\right)\,\right]~{}$, even
though such a misstatement is sometimes made in the literature. This
expression furnishes the compliance function only in the special situation of
a Heaviside-step stress (185), but not in the general case.

As can be easily shown from (27), in the frequency domain model (176) reads as
(173), except that the complex rigidity and the complex compliance are now
given by

$\displaystyle\bar{\mu}\,=\,{\mu}\,\left(\,1\,+\,{\it
i}\,\chi\,\tau_{{}_{V}}\,\right)\,\quad,\quad\quad\bar{J}\,=\,\frac{J}{1\,+\,{\it
i}\,\chi\,\tau_{{}_{V}}}~{}~{}~{}.$ (186)

Recall that, for brevity, here and everywhere we write $\,\mu\,$ and $\,J\,$
instead of $\,{\mu}(0)\,$ and $\,J(0)\,$.

The Kelvin-Voigt material becomes elastic in the low-frequency limit, and
viscous in the high-frequency limit.

####  A.6.4 Viscoelastic deformation: a Maxwell material

The Maxwell model can be represented with a viscous damper and an elastic
spring connected in series. Experiencing the same force, these elements have
their elongations summed up. This example illustrates the situation where the
total, viscoelastic strain consists of a purely viscous and a purely elastic
contributions generated by the same stress $\,{\mathbb{S}}\,$:

$\displaystyle{\mathbb{U}}\,=\;\stackrel{{\scriptstyle(v)}}{{\mathbb{U}}}\,+\,\stackrel{{\scriptstyle(e)}}{{\mathbb{U}}}~{},\,\quad\mbox{where}\,\quad\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,=\,2\,\mu\,\stackrel{{\scriptstyle(e)}}{{\mathbb{U}}}~{}~{}~{}~{}\mbox{and}\quad~{}\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\,=\,2\,\eta\,\frac{\partial\,}{\partial
t}~{}\stackrel{{\scriptstyle(v)}}{{\mathbb{U}}}~{}~{}~{}.$ (187)

Since in the Maxwell regime both contributions to the strain are generated by
the same stress

$\displaystyle{\mathbb{S}}\;=\;\stackrel{{\scriptstyle(ve)}}{{\mathbb{S}}}\;=\;\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\,=\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}~{}~{}~{},$
(188)

formula (187) can be written down as

$\displaystyle\stackrel{{\scriptstyle\centerdot}}{{\mathbb{U}}}\,=\,\frac{1}{2\,\mu}\;\stackrel{{\scriptstyle\centerdot}}{{\mathbb{S}}}\,+\,\frac{1}{2\,\eta}\;{\mathbb{S}}~{}~{}~{}$
(189a) or, in a more conventional form:
$\displaystyle\stackrel{{\scriptstyle\centerdot}}{{\mathbb{S}}}\,+\;\frac{1\;}{\tau_{{}_{M}}}\,{\mathbb{S}}~{}=~{}2\,\mu\,\stackrel{{\scriptstyle\centerdot}}{{\mathbb{U}}}~{}~{}~{},$
(189b)

with the so-called _Maxwell time_ introduced as

$\displaystyle\tau_{{}_{M}}\;\equiv\;{\eta}/{\mu}~{}~{}~{}.$ (190)

Although formally the Maxwell time is given by an expression mimicking the
definition of the Voigt time, the meaning of these times is different.

Comparing (189) with the general expression (43) for the compliance operator,
we see that, for the Maxwell model, the compliance operator in the time domain
assumes the form:

$\displaystyle
J(t\,-\,t\,^{\prime})\,=\,\left[\,J\,+\,\left(t\;-\;t\,^{\prime}\right)\;\frac{1}{\eta}\,\right]\;\Theta(t\,-\,t\,^{\prime})~{}~{}~{},$
(191)

where $\,J\,\equiv\,1/\mu\,$. In the frequency domain, (189) can be written
down as (173), with the complex rigidity and compliance given by

$\displaystyle\bar{\mu}(\chi)\,=\,{\mu}\;\frac{{\it
i}\,\chi\,\tau_{M}}{1\,+\,{\it
i}\chi\tau_{M}}\,\quad,\quad\quad\bar{J}(\chi)\,=\,J\,\left(\,1\,-\,\frac{{\it
i}}{\chi\,\tau_{{}_{M}}}\,\right)~{}=~{}J~{}-~{}\frac{{\it
i}}{\chi\,\eta}~{}~{}~{}.$ (192)

Clearly, such a body becomes elastic in the high-frequency limit, and becomes
viscous at low frequencies (the latter circumstance making the Maxwell model
attractive to seismologists).

####  A.6.5 Viscoelastic deformation: the Hohenemser-Prager (SAS) model

An attempt to combine the Kelvin-Voigt and Maxwell models leads to the
Hohenemser-Prager model, also known as the Standard Anelastic Solid (SAS):

$\displaystyle{\tau_{{}_{M}}}{\bf\dot{\mathbb{S}}}\,+\;\,{\mathbb{S}}~{}=~{}2\,\mu\,\left(\,{\mathbb{U}}\,+\,\tau_{{}_{V}}\,{\bf\dot{\mathbb{U}}}\,\right)~{}~{}~{},$
(193)

In the limit of $\,\tau_{{}_{M}}\rightarrow 0\,$, this model approaches the
one of Kelvin and Voigt (and $\,\tau_{{}_{V}}\,$ acquires the meaning of the
Voigt time).

A transition from the SAS to Maxwell model, however, can be achieved only
through re-definition of parameters. One should set: $\;2\,\mu\rightarrow 0\;$
and $\;\tau_{{}_{V}}\rightarrow\infty\;$, along with
$\,2\mu\tau_{{}_{V}}\rightarrow 2\eta\,$. Then (193) will become (189), with
$\,\tau_{{}_{M}}\,$ playing the role of the Maxwell time.

In the frequency domain, (193) can be put into the form of (173), the complex
rigidity and the complex compliance being expressed through the parameters as

$\displaystyle\bar{\mu}\,=\,\mu\;\frac{\textstyle
1+i\tau_{{}_{V}}\chi}{\textstyle{1+i\tau_{{}_{M}}\chi}}\,\quad,\quad\quad\bar{J}\,=\,J\;\frac{\textstyle
1+i\tau_{{}_{M}}\chi}{\textstyle{1+i\tau_{{}_{V}}\chi}}~{}~{}~{}.$ (194)

This entails:
$~{}\tan\delta\,\equiv\,{\textstyle{{\cal{I}}{\it{m}}[\bar{\mu}]}}/{\textstyle{{\cal{R}}{\it{e}}[\bar{\mu}]}}\,=\,\frac{\textstyle{(\tau_{{}_{V}}-\tau_{{}_{M}})\,\chi}}{\textstyle{1\,+\,\tau_{{}_{V}}\tau_{{}_{M}}\chi^{2}}}\;$,
whence it is easy to show that the tangent is related to its maximal value
through

$\tan\delta\;=\;2\;\left[\tan\delta\right]_{max}\;\frac{\tau\;\chi}{1\,+\,\tau^{2}\,\chi^{2}}\,\quad,\quad\quad\mbox{where}\quad\tau\,\equiv\,\sqrt{\tau_{{}_{M}}\tau_{{}_{V}}}~{}~{}~{}.$

This is the so-called Debye peak, which is indeed observed in some materials.

To prove that the SAS solid is indeed anelastic, one has to make sure that a
Heaviside step stress $\,\Theta(t^{\prime})\,$ entails a strain proportional
to $\,1-\exp(-\Gamma t)\,$, and to demonstrate that a predeformed sample
subject to stress $\,\Theta(-t^{\prime})\,$ regains it shape as
$\,\exp(-\Gamma t)\,$, where the relaxation constant $\,\Gamma\,$ is positive.
In subsection C.3 we shall do this for a SAS sphere.

##  Appendix B Interconnection between the quality factor and the phase lag

The power $\,P\,$ exerted by a tide-raising secondary on its primary can be
written as

$\displaystyle
P\;=\;-\;\int\,\rho\;\mbox{\boldmath$\vec{\boldmath{\,V}}$}\;\cdot\;\nabla
W\;d^{3}x$ (195)

$\rho\,,\;\mbox{\boldmath$\vec{\boldmath{\,V}}$}\,$, and $\,W\,$ signifying
the density, velocity, and tidal potential in the small volume $~{}d^{3}x~{}$
of the primary. The mass-conservation law
$~{}\nabla\cdot(\rho\mbox{\boldmath$\vec{\boldmath{\,V}}$})\,+\frac{\textstyle\partial\rho}{\textstyle\partial
t}\,=\,0\,~{}$ enables one to shape the dot-product into the form of

$\displaystyle\rho\,\mbox{\boldmath$\vec{\boldmath{\,V}}$}\cdot\nabla
W\,=\,\nabla\cdot(\rho\,\mbox{\boldmath$\vec{\boldmath{\,V}}$}\,W)\,-\,\rho\,W\,\nabla\cdot\mbox{\boldmath$\vec{\boldmath{\,V}}$}\,-\,\mbox{\boldmath$\vec{\boldmath{\,V}}$}\,W\,\nabla\rho\;\;~{}.\;\;\;\;$
(196)

Under the realistic assumption of the primary’s incompressibility, the term
with $\,\nabla\cdot\mbox{\boldmath$\vec{\boldmath{\,V}}$}\,$ may be omitted.
To get rid of the term with $\,\nabla\rho\,$, one has to accept a much
stronger approximation of the primary being homogeneous. Then the power will
be rendered by

$\displaystyle
P\;=\;-\;\int\,\nabla\,\cdot\,(\rho\;\mbox{\boldmath$\vec{\boldmath{\,V}}$}\;W)\,d^{3}x\;=\;-\;\int\,\rho\;W\;\mbox{\boldmath$\vec{\boldmath{\,V}}$}\,\cdot\,{\vec{\bf{n}}}\;\,dS\;\;\;,$
(197)

${\vec{\bf{n}}}\,$ being the outward normal and $\,dS\,$ being an element of
the surface area of the primary. This expression for the power (pioneered,
probably, by Goldreich 1963) enables one to calculate the work through radial
displacements only, in neglect of horizontal motion.

Denoting the radial elevation with $\,\zeta\,$, we can write the power per
unit mass, $~{}{\cal P}\equiv P/M~{}$, as:

$\displaystyle{\cal P}\;=\;\left(-\,\frac{\partial W}{\partial
r}\right)\;\mbox{\boldmath$\vec{\boldmath{\,V}}$}\cdot{\vec{\bf{n}}}\;=\;\left(-\,\frac{\partial
W}{\partial r}\right)\frac{d\zeta}{dt}\;\;\;.$ (198)

A harmonic external potential

$\displaystyle
W~{}=~{}W_{0}~{}\cos(\,\omega_{\textstyle{{}_{lmpq}}}\,t\,)~{}~{}~{},$ (199)

applied at a point of the primary’s surface, will elevate this point by

$\displaystyle\zeta~{}=~{}h_{2}~{}\frac{W_{o}}{\mbox{g}}~{}\cos(\omega_{\textstyle{{}_{lmpq}}}\,t~{}-~{}\epsilon_{\textstyle{{}_{lmpq}}})~{}=~{}h_{2}\;\frac{W_{o}}{\mbox{g}}~{}\cos(\omega_{\textstyle{{}_{lmpq}}}\,t\;-\;\omega_{\textstyle{{}_{lmpq}}}\,\Delta
t_{\textstyle{{}_{lmpq}}})~{}~{}~{},$ (200)

with g being the surface gravity acceleration, and $\,h_{2}\,$ denoting the
Love number.

In formula (200), $\,\omega_{\textstyle{{}_{lmpq}}}\,$ is one of the modes
(105) showing up in the Darwin-Kaula expansion (103). The quantity
$\,\epsilon_{\textstyle{{}_{lmpq}}}\,=\,\omega_{\textstyle{{}_{lmpq}}}\,\Delta
t_{\textstyle{{}_{lmpq}}}\,$ is the corresponding phase lag, while $\,\Delta
t_{\textstyle{{}_{lmpq}}}\,$ is the positively defined time lag at this mode.
Although the tidal modes $\,\omega_{\textstyle{{}_{lmpq}}}\,$ can assume any
sign, both the potential $\,W\,$ and elevation $\,\zeta\,$ can be expressed
via the positively defined forcing frequency
$~{}\chi_{\textstyle{{}_{lmpq}}}\,=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|~{}$
and the absolute value of the phase lag:

$\displaystyle W~{}=~{}W_{0}~{}\cos(\chi\,t)~{}~{}~{},$ (201)
$\displaystyle\zeta~{}=~{}h_{2}~{}\frac{W_{o}}{\mbox{g}}~{}\cos(\chi\,t~{}-~{}|\,\epsilon\,|)~{}~{}~{},$
(202)

subscripts $\,lmpq\,$ being dropped here and hereafter for brevity.

The vertical velocity of the considered element of the primary’s surface will
be

$\displaystyle\frac{d\zeta}{dt}\;=\;-\;h_{2}\;\chi\;\frac{W_{o}}{\mbox{g}}\;\sin(\chi
t\;-\;|\epsilon|)\;=\;-\;h_{2}\;\chi\;\frac{W_{o}}{\mbox{g}}\;\left(\sin\chi
t\;\cos|\epsilon|\;-\;\cos\chi t\;\sin|\epsilon|\right)\;\;.\;\;$ (203)

Introducing the notation $~{}A\,=\,h_{2}\,\frac{\textstyle
W_{0}}{\textstyle\mbox{g}}\,\frac{\textstyle\partial W_{0}}{\textstyle\partial
r}~{}$, we write the power per unit mass as

$\displaystyle{\cal P}\;=\;\left(-\,\frac{\partial W}{\partial
r}\right)\frac{d\zeta}{dt}\;=\;A~{}\chi~{}\cos(\chi\,t)~{}\sin(\chi
t\;-\;|\epsilon|)\;\;\;,$ (204)

and write the work $\,w\,$ per unit mass, performed over a time interval
$\,(t_{0}\,,\;t)\,$, as:

$\displaystyle
w|^{\textstyle{{}^{~{}t}}}_{\textstyle{{}_{~{}t_{0}}}}=\int_{t_{0}}^{t}{\cal
P}~{}dt=A\int^{\chi t}_{\chi t_{0}}\cos(\chi\,t)\,\sin(\chi
t-|\epsilon|)d(\chi\,t)=A\,\cos|\epsilon|\int_{\chi t_{0}}^{\chi t}\cos
z\,\sin z\,dz-A\,\sin|\epsilon|\int_{\chi t_{0}}^{\chi t}\cos^{2}z\,dz$
$\displaystyle=~{}-~{}\frac{A}{4}~{}{\LARGE\left[\right.}\,\cos(2\chi
t-|\epsilon|)\,+\,2\;\chi\;t\;\sin|\epsilon|~{}{\LARGE\left.\right]}^{\textstyle{{}^{~{}t}}}_{\textstyle{{}_{~{}t_{0}}}}~{}~{}~{}.$
(205)

Being cyclic, the first term in (205) renders the elastic energy stored in the
body. The second term, being linear in time, furnishes the energy damped. This
clear interpretation of the two terms was offered by Stan Peale [2011,
personal communication].

The work over a time period $\,T\,=\,2\pi/\chi\,$ is equal to the energy
dissipated over the period:

$\displaystyle w|^{\textstyle{{}^{~{}t=T}}}_{\textstyle{{}_{~{}t=0}}}=\Delta
E_{\textstyle{{}_{cycle}}}\;=\;-\;A\;\pi\;\sin|\epsilon|~{}~{}~{}.$ (206)

It can be shown that the peak work is obtained over the time span from
$\,\pi\,$ to $\,|\epsilon|\,$ and assumes the value

$\displaystyle
E_{\textstyle{{}_{peak}}}^{\textstyle{{}^{(work)}}}\,=\;~{}\frac{A}{2}~{}\left[\,\cos|\epsilon|~{}-~{}\sin|\epsilon|~{}\left(\,\frac{\pi}{2}\,-\,|\epsilon|\,\right)\,\right]~{}~{}~{},$
(207)

whence the appropriate quality factor is given by:

$\displaystyle Q^{-1}_{\textstyle{{}_{work}}}\,=~{}\frac{~{}-~{}\Delta
E_{\textstyle{{}_{cycle}}}}{2\,\pi\,E_{\textstyle{{}_{peak}}}^{\textstyle{{}^{(work)}}}}~{}=~{}\frac{\tan|\epsilon|}{1\;-\;\left(\,\frac{\textstyle\pi}{\textstyle
2}\,-\,|\epsilon|\,\right)~{}\tan|\epsilon|}~{}~{}~{}.$ (208)

To calculate the peak energy stored in the body, we would note that the first
term in (205) is maximal when taken over the span from
$\,\chi\,t\,=\,\pi/4\,+\,|\epsilon|/2\,$ through
$\,\chi\,t\,=\,3\pi/4\,+\,|\epsilon|/2~{}$:

$\displaystyle
E_{\textstyle{{}_{peak}}}^{\textstyle{{}^{(energy)}}}\,=\;~{}\frac{A}{2}~{}~{}~{},$
(209)

and the corresponding quality factor is:

$\displaystyle Q^{-1}_{\textstyle{{}_{energy}}}\,=~{}\frac{~{}-~{}\Delta
E_{\textstyle{{}_{cycle}}}}{2\,\pi\,E_{\textstyle{{}_{peak}}}^{\textstyle{{}^{(energy)}}}}~{}=~{}\sin|\epsilon|~{}~{}~{}.$
(210)

Goldreich (1963) suggested to employ the span $\,\chi\,t\,=\,(0\,,\;\pi/4)\,$.
The absolute value of the resulting power, denoted in Ibid. as $\,E^{*}\,$, is
equal to

$\displaystyle E^{*}\,=\;~{}\frac{A}{2}~{}\cos|\epsilon|~{}~{}~{}$ (211)

and is not the peak value of the energy stored nor of the work performed.
Goldreich (1963) however employed it to define a quality factor, which we
shall term $~{}Q_{\textstyle{{}_{Goldreich}}}~{}$. This factor is introduced
via

$\displaystyle Q^{-1}_{\textstyle{{}_{Goldreich}}}\,=~{}\frac{~{}-~{}\Delta
E_{\textstyle{{}_{cycle}}}}{2\,\pi\,E^{*}}~{}=~{}\tan|\epsilon|~{}~{}~{}.$
(212)

In our opinion, the quality factor $\,Q_{\textstyle{{}_{energy}}}\,$ defined
through (210) is preferable, because the expansion of tides contains terms
proportional to
$\,k_{l}(\chi_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\chi_{\textstyle{{}_{lmpq}}})~{}\,$.
Since the long-established tradition suggests to substitute $\,\sin\epsilon\,$
with $\,1/Q\,$, it is advisable to define the $\,Q\,$ exactly as (210), and
also to call it $\,Q_{l}\,$, to distinguish it from the seismic quality factor
(Efroimsky 2012).

##  Appendix C Tidal response of a homogeneous viscoelastic sphere (Churkin
1998)

This section presents some results from the unpublished preprint by Churkin
(1998). We took the liberty of upgrading the notations282828 Churkin (1998)
employed the notation $\,k_{\it l}(\tau)\,$ for what we call
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{k}}_{\textstyle{{}_{l}}}(\tau)\,$.
Our notations are more convenient in that they amplify the close analogy
between the Love functions and the compliance function. and correcting some
minor oversights.

###  C.1 A homogeneous Kelvin-Voigt spherical body

In combination with the Correspondence Principle, the formulae from subsection
A.6.3 furnish the following expression for the complex Love numbers of a
Kelvin-Voigt body:

$\displaystyle\bar{k}_{\textstyle{{}_{l}}}(\chi)\;=\;\frac{3}{2(l-1)}\;\,\frac{1}{\textstyle
1\;+\;A_{\textstyle{{}_{l}}}\,\left(\,1\;+\;\;\tau_{\textstyle{{}_{V}}}\;{\it
i}\;\chi\,\right)\;}~{}~{}~{},\quad$ (213)

It then can be demonstrated, with aid of (61), that the time-derivative of the
corresponding Love function is

$\displaystyle\stackrel{{\scriptstyle\bf\centerdot}}{{k}}_{\textstyle{{}_{l}}}(\tau)~{}=~{}\left\\{\begin{array}[]{c}\frac{\textstyle
3}{\textstyle 2(l-1)}~{}\,\frac{\textstyle 1}{\textstyle
A_{\textstyle{{}_{\textstyle{{}_{l}}}}}\;\tau_{\textstyle{{}_{V}}}}\;\exp(\textstyle\;-\,\tau\,\zeta_{\textstyle{{}_{\textstyle{{}_{l}}}}}\,)\;\Theta(\tau)\quad~{}~{}\quad\mbox{for}\quad\tau_{\textstyle{{}_{V}}}\,>\,0{}{}{}{}\\\
{}\hfil\\\ \frac{\textstyle 3}{\textstyle 2(l-1)}~{}\,\frac{\textstyle
1}{\textstyle
1\;+\;A_{\textstyle{{}_{\textstyle{{}_{l}}}}}}\;\,\delta(\tau)\quad\quad\quad\quad\quad\quad\quad~{}\,\mbox{for}\quad\tau_{\textstyle{{}_{V}}}\,=\,0~{}~{}~{},\end{array}\right.$
(217)

while the Love function itself has the form of

$\displaystyle{k}_{\textstyle{{}_{l}}}(\tau)$ $\displaystyle=$
$\displaystyle\frac{3}{2(l-1)}~{}\,\frac{1}{A_{\textstyle{{}_{\textstyle{{}_{l}}}}}\;\zeta_{\textstyle{{}_{\textstyle{{}_{l}}}}}\;\tau_{\textstyle{{}_{V}}}}\;\left[\;1\;-\;\exp(\textstyle\;-\,\tau\,\zeta_{\textstyle{{}_{\textstyle{{}_{l}}}}}\,)\;\right]\;\Theta(\tau)$
$\displaystyle=$
$\displaystyle\frac{3}{2(l-1)}~{}\,\frac{1}{1\;+\;A_{\textstyle{{}_{\textstyle{{}_{l}}}}}}\;\left[\;1\;-\;\exp(\textstyle\;-\,\tau\,\zeta_{\textstyle{{}_{\textstyle{{}_{l}}}}}\,)\;\right]\;\Theta(\tau)\;\;\;,\quad\;$

where

$\displaystyle\zeta_{\textstyle{{}_{\textstyle{{}_{l}}}}}\;\equiv\;\frac{1\;+\;A_{\textstyle{{}_{\textstyle{{}_{l}}}}}}{A_{\textstyle{{}_{\textstyle{{}_{l}}}}}\;\tau_{\textstyle{{}_{V}}}}~{}~{}.$
(219)

Formulae (217) may look confusing, in that
$\,\exp(\textstyle\;-\,\tau\,\zeta_{\textstyle{{}_{\textstyle{{}_{l}}}}}\,)\,$
simply vanishes in the elastic limit, i.e., when
$~{}\tau_{\textstyle{{}_{V}}}\rightarrow 0~{}$ and
$~{}\zeta_{\textstyle{{}_{l}}}\rightarrow\infty~{}$. We however should not be
misled by this mathematical artefact stemming from the nonanaliticity of the
exponent function. Instead, we should keep in mind that a physical meaning is
attributed not to the Love functions or their derivatives but to the results
of the Love operator’s action on realistic disturbances. For example, a
Heaviside step potential

$\displaystyle
W_{\it{l}}(\mbox{{\boldmath$\vec{R}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*}\,,\;t\,^{\prime})\;=\;W\;\Theta(t\,^{\prime})$
(220)

applied to a homogeneous Kelvin-Voigt spherical body will furnish, through
relation (60), the following response of the potential:

$\displaystyle U_{\it l}(\mbox{{\boldmath$\vec{r}$}},\,t)$ $\displaystyle=$
$\displaystyle\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{t\,^{\prime}=\,-\infty}^{\,t\,^{\prime}=\,t}\stackrel{{\scriptstyle\centerdot}}{{k}}_{\it
l}(t-t\,^{\prime})\;W\;\Theta(t\,^{\prime})\,dt\,^{\prime}\,=\;\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{t\,^{\prime}=\,0}^{\,t\,^{\prime}=\,t}\stackrel{{\scriptstyle\centerdot}}{{k}}_{\it
l}(t-t\,^{\prime})\;W\;dt\,^{\prime}\,\quad\quad\quad\quad~{}$
$\displaystyle=$ $\displaystyle W\;\left(\frac{R}{r}\right)^{{\it
l}+1}\,\int_{\tau\,=\,0}^{\,\tau\,=\,t}\stackrel{{\scriptstyle\centerdot}}{{k}}_{\it
l}(\tau)\;d\tau\;=\;\frac{3}{2(l-1)}~{}\,\frac{\;1\;-\;\exp(\textstyle\;-\,t\,\zeta_{\textstyle{{}_{\textstyle{{}_{l}}}}}\,)\;}{1\;+\;A_{\textstyle{{}_{l}}}}\;\left(\frac{R}{r}\right)^{{\it
l}+1}W~{}~{}~{}.\quad\quad\quad\quad\quad$

In the elastic limit, this becomes:

$\displaystyle\tau_{\textstyle{{}_{V}}}\rightarrow
0\quad\Longrightarrow\quad\zeta_{\textstyle{{}_{l}}}\rightarrow\infty\quad\Longrightarrow\quad
U_{\it
l}(\,\mbox{{\boldmath$\vec{r}$}}\,,\,\;t\,)\;\rightarrow\;\frac{3}{2(l-1)}~{}\,\frac{\;1\;}{1\;+\;A_{\textstyle{{}_{l}}}}\;\left(\frac{R}{r}\right)^{{\it
l}+1}W~{}~{}~{},\quad$ (222)

which reproduces the case described by the static Love number
$\;k_{\textstyle{{}_{l}}}\,=\,\frac{\textstyle 3}{\textstyle
2}~{}\,\frac{\textstyle\;1\;}{\textstyle 1\;+\;A_{\textstyle{{}_{l}}}}\;\,$.

An alternative way of getting (222) would be to employ formulae (LABEL:KV2)
and (59a).

###  C.2 A homogeneous Maxwell spherical body

Using the formulae presented in the Appendix A.6.4, and relying upon the
Correspondence Principle, we write down the complex Love numbers for a Maxwell
material as

$\displaystyle\bar{k}_{\textstyle{{}_{l}}}(\chi)\;=\;\frac{3}{2(l-1)}\;\frac{1}{\;\textstyle
1\;+\;\frac{\textstyle A_{\textstyle{{}_{l}}}\;\tau_{\textstyle{{}_{M}}}\;{\it
i}\;\chi}{\textstyle 1\;+\;\tau_{\textstyle{{}_{M}}}\;{\it
i}\;\chi\;}\;}\;=\;\frac{3}{2(l-1)}\;\frac{1}{1\;+\;A_{\textstyle{{}_{l}}}}\;\left[\;1\;+\;\frac{A_{\textstyle{{}_{l}}}}{\textstyle
1\;+\;\left(\,1\;+\;A_{\textstyle{{}_{l}}}\,\right)\;\tau_{\textstyle{{}_{M}}}\;{\it
i}\;\chi\;}\;\right]~{}~{}~{},\quad$ (223)

which corresponds, via (61), to

$\displaystyle\stackrel{{\scriptstyle\bf\centerdot}}{{k}}_{\textstyle{{}_{l}}}(\tau)~{}=~{}\frac{3}{2(l-1)}~{}\,\frac{\;\;\delta(\tau)\;+\;{A_{\textstyle{{}_{l}}}}\;\gamma_{\textstyle{{}_{l}}}\;\exp(\textstyle\;-\,\tau\,\gamma_{\textstyle{{}_{\textstyle{{}_{l}}}}}\,)\;\Theta(\tau)\;}{\textstyle
1~{}+~{}\textstyle A_{\textstyle{{}_{l}}}}$ (224)

and

$\displaystyle{k}_{\textstyle{{}_{l}}}(\tau)~{}=~{}\frac{3}{2(l-1)}~{}\,\frac{\;\;1\;+\;{A_{\textstyle{{}_{l}}}}\;\left[\,1\;-\;\exp(\textstyle\;-\,\tau\,\gamma_{\textstyle{{}_{\textstyle{{}_{l}}}}}\,)\;\right]\;}{\textstyle
1~{}+~{}\textstyle A_{\textstyle{{}_{l}}}}\;\,\Theta(\tau)~{}~{}~{},$ (225)

where

$\displaystyle\gamma_{\textstyle{{}_{l}}}\,\equiv\;\frac{1}{\textstyle\left(\,1\;+\;A_{\textstyle{{}_{l}}}\,\right)\;\tau_{\textstyle{{}_{M}}}\;}~{}~{}~{}.$
(226)

A Heaviside step potential

$\displaystyle
W_{\it{l}}(\mbox{{\boldmath$\vec{R}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*}\,,\;t\,^{\prime})\;=\;W\;\Theta(t\,^{\prime})$
(227)

will, according to formula (60), render the following response:

$\displaystyle U_{\it
l}(\mbox{{\boldmath$\vec{r}$}},\,t)\;=\;\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{t\,^{\prime}=\,-\infty}^{\,t\,^{\prime}=\,t}\stackrel{{\scriptstyle\centerdot}}{{k}}_{\it
l}(t-t\,^{\prime})\;W\;\Theta(t\,^{\prime})\,dt\,^{\prime}\,=\;\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{t\,^{\prime}=\,0}^{\,t\,^{\prime}=\,t}\stackrel{{\scriptstyle\centerdot}}{{k}}_{\it
l}(t-t\,^{\prime})\;W\;dt\,^{\prime}\,\quad\quad\quad\quad\quad~{}$ (228)
$\displaystyle=W\;\left(\frac{R}{r}\right)^{{\it
l}+1}\,\int_{\tau\,=\,0}^{\,\tau\,=\,t}\stackrel{{\scriptstyle\centerdot}}{{k}}_{\it
l}(\tau)\;d\tau\;=\;\frac{3}{2(l-1)}~{}\,\frac{\;1\;+\;A_{\textstyle{{}_{l}}}\;\left[1\;-\;\exp(\textstyle\;-\,t\,\gamma_{\textstyle{{}_{\textstyle{{}_{l}}}}}\,)\;\right]\;}{1\;+\;A_{\textstyle{{}_{l}}}}\;\left(\frac{R}{r}\right)^{{\it
l}+1}W~{}\Theta(t)~{}.\quad~{}\quad~{}$

In the elastic limit, we obtain:

$\displaystyle\tau_{\textstyle{{}_{M}}}\rightarrow\infty\quad\Longrightarrow\quad\gamma_{\textstyle{{}_{l}}}\rightarrow
0\quad\Longrightarrow\quad U_{\it
l}(\,\mbox{{\boldmath$\vec{r}$}}\,,\,\;t\,)\;\rightarrow\;\frac{3}{2(l-1)}~{}\,\frac{\;1\;}{1\;+\;A_{\textstyle{{}_{l}}}}\;\left(\frac{R}{r}\right)^{{\it
l}+1}W~{}~{}~{},\quad$ (229)

which corresponds to the situation described by the static Love number
$\;k_{\textstyle{{}_{l}}}\,=\,\frac{\textstyle 3}{\textstyle
2(l-1)}~{}\,\frac{\textstyle\;1\;}{\textstyle
1\;+\;A_{\textstyle{{}_{l}}}}\;\,$.

###  C.3 A homogeneous Hohenemser-Prager (SAS) spherical body

The Correspondence Principle, along with the formulae from subsection A.6.5,
yields the following expression for the complex Love numbers of a Hohenemser-
Prager (SAS) spherical body:

$\displaystyle\bar{k}_{\textstyle{{}_{l}}}(\chi)\;=\;\frac{3}{2\,(l-1)}\;\,\frac{1}{\;1\;+\;A_{\textstyle{{}_{l}}}\;\frac{\textstyle
1\,+\,{\it i}\,\chi\,\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}}{\textstyle
1\,+\,{\it i}\,\chi\,\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}}\;}~{}~{}~{}.$
(230)

Combined with (61), this entails:

$\displaystyle\stackrel{{\scriptstyle\bf\centerdot}}{{k}}_{\textstyle{{}_{l}}}(\tau)\;=\;\frac{3}{2\,(l-1)}\;\,\frac{1}{\;1\;+\;A_{\textstyle{{}_{l}}}\;\,\frac{\textstyle\,\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}}{\,\textstyle\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}}\;}\;\left[\;\delta(\tau)\;+\;\frac{A_{\textstyle{{}_{l}}}}{\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}}\,\;\frac{\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}\,-\,\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}}{~{}\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}\,+\,A_{\textstyle{{}_{l}}}\,\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}}~{}\,\exp\left(\,-\,\frac{\,1\;+\;A_{\textstyle{{}_{l}}}\,}{\,\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}\;+\;A_{\textstyle{{}_{l}}}\;\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}\,}\;\tau\,\right)\;\right]\quad\quad\quad$
(231)

and

$\displaystyle{k}_{\textstyle{{}_{l}}}(\tau)\;=\;\frac{3}{2\,(l-1)}\;\,\frac{\textstyle\;1~{}-~{}\frac{\textstyle
A_{\textstyle{{}_{l}}}}{\textstyle\,\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}}\,\;\frac{\textstyle\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}\,-\,\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}}{\textstyle~{}1\,+\,A_{\textstyle{{}_{l}}}\,}\left[\;1-\;~{}\,\exp\left(\,-\,\frac{\textstyle\,1\;+\;A_{\textstyle{{}_{l}}}\,}{\textstyle\,\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}\;+\;A_{\textstyle{{}_{l}}}\;\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}\,}\;\tau\,\right)\;\right]\;}{\textstyle\;1\;+\;A_{\textstyle{{}_{l}}}\;\,\frac{\textstyle\,\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}}{\,\textstyle\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}}\;}\;~{}\Theta(\tau)\quad\quad\quad$
(232)

A Heaviside step potential

$\displaystyle
W_{\it{l}}(\mbox{{\boldmath$\vec{R}$}}\,,\;\mbox{{\boldmath$\vec{r}$}}^{\;*}\,,\;t\,^{\prime})\;=\;W\;\Theta(t\,^{\prime})$
(233)

applied to a SAS spherical body will then result in the following variation of
its potential:

$\displaystyle U_{\it
l}(\mbox{{\boldmath$\vec{r}$}},\,t)~{}=\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{t\,^{\prime}=\,-\infty}^{\,t\,^{\prime}=\,t}\stackrel{{\scriptstyle\centerdot}}{{k}}_{\it
l}(t-t\,^{\prime})\,W\,\Theta(t\,^{\prime})\,dt\,^{\prime}=\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{t\,^{\prime}=\,0}^{\,t\,^{\prime}=\,t}\stackrel{{\scriptstyle\centerdot}}{{k}}_{\it
l}(t-t\,^{\prime})\,W\,dt\,^{\prime}=W\left(\frac{R}{r}\right)^{{\it
l}+1}\,\int_{\tau\,=\,0}^{\,\tau\,=\,t}\stackrel{{\scriptstyle\centerdot}}{{k}}_{\it
l}(\tau)\,d\tau\quad\quad$
$\displaystyle=\frac{3}{2(l-1)}\left(\frac{1}{{1+A_{\textstyle{{}_{l}}}\,\frac{\textstyle\tau_{{\textstyle{{}_{\textstyle{{}_{V}}}}}}}{\textstyle\tau_{{\textstyle{{}_{\textstyle{{}_{M}}}}}}}\,}}+\frac{A_{\textstyle{{}_{l}}}}{1+A_{\textstyle{{}_{l}}}}\,\frac{\tau_{{\textstyle{{}_{\textstyle{{}_{V}}}}}}\,-\,\tau_{{\textstyle{{}_{\textstyle{{}_{M}}}}}}}{\tau_{{\textstyle{{}_{\textstyle{{}_{M}}}}}}+A_{\textstyle{{}_{l}}}\,\tau_{{\textstyle{{}_{\textstyle{{}_{V}}}}}}}\right)\left[1-\exp\left(-\,\frac{\textstyle\,1\,+\,A_{\textstyle{{}_{l}}}\,}{\textstyle\,\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}+A_{\textstyle{{}_{l}}}\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}\,}\;t\right)\right]\left(\frac{R}{r}\right)^{{\it
l}+1}W~{}\Theta(t)~{}.~{}~{}\quad$ (234)

Within this model, the elastic limit is achieved by setting
$\,\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}\,=\,\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}\,$,
whence we obtain the case described by the static Love number
$\;k_{\textstyle{{}_{l}}}\,=\,\frac{\textstyle 3}{\textstyle
2}~{}\,\frac{\textstyle\;1\;}{\textstyle 1\;+\;A_{\textstyle{{}_{l}}}}\;\,$.
Interestingly, the elastic regime is achieved even when these times are not
zero. Their being equal to one another turns out to be sufficient.

Repeating the above calculation for tidal disturbance
$\,W\,\Theta(-t\,^{\prime})\,$, we shall see that, after the tidal
perturbation is removed, a tidally prestressed sphere regains its shape, the
stress relaxing at a rate proportional to
$~{}\exp\left(-\,~{}\frac{\textstyle\,1\,+\,A_{\textstyle{{}_{l}}}\,}{\textstyle\,\tau_{\textstyle{{}_{\textstyle{{}_{M}}}}}+A_{\textstyle{{}_{l}}}\tau_{\textstyle{{}_{\textstyle{{}_{V}}}}}\,}\,~{}t\right)~{}$.

##  Appendix D The correspondence principle
(elastic-viscoelastic analogy)

###  D.1 The correspondence principle, for nonrotating bodies

While the static Love numbers depend on the static rigidity $\,\mu\,$ through
(3), it is not immediately clear if a similar formula interconnects also
$\,\bar{k}_{\it l}(\chi)\,$ with $\,\bar{\mu}(\chi)\,$. To understand why and
when the relation should hold, recall that formulae (3) originate from the
solution of a boundary-value problem for a system incorporating two equations:

$\displaystyle\sigma_{\textstyle{{}_{\beta\nu}}}$ $\displaystyle=$
$\displaystyle 2\;\mu\;u_{\textstyle{{}_{\beta\nu}}}~{}~{}~{},$ (235a)
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial\sigma_{\textstyle{{}_{\beta\nu}}}}{\partial
x_{\textstyle{{}_{\nu}}}}\;-\;\frac{\partial p}{\partial
x_{\textstyle{{}_{\beta}}}}\;-\;\rho\;\frac{\partial(W\,+\,U)}{\partial
x_{\textstyle{{}_{\beta}}}}~{}~{}~{},$ (235b)

the latter being simply the equation of equilibrium written for a _static_
viscoelastic medium, in neglect of compressibility and heat conductivity. The
notations $\,\sigma_{\textstyle{{}_{\beta\nu}}}\,$ and
$\,u_{\textstyle{{}_{\beta\nu}}}\,$ stand for the deviatoric stress and
strain, $~{}p\,\equiv\,-\,\frac{\textstyle 1}{\textstyle
3}\,\mbox{Sp}\,{\mathbb{S}}~{}$ is the pressure (set to be nil in
incompressible media), while $\,W\,$ and $\,U\,$ are the perturbing and
perturbed potentials. By solving the system, one arrives at the static
relation $\,U_{\it l}=k_{\it l}\,W_{\it l}\,$, with the customary static Love
numbers $\,k_{\it l}\,$ expressed via $\,\rho\,$, $\,R\,$, and $\,\mu\,$ by
(3).

Now let us write equation like (235a \- 235b) for time-dependent deformation
of a nonrotating body:

$\displaystyle{\mathbb{S}}$ $\displaystyle=$ $\displaystyle
2~{}\hat{\mu}~{}{\mathbb{U}}~{}~{}~{},$ (236a)
$\displaystyle\rho~{}{\bf\ddot{u}}$ $\displaystyle=$
$\displaystyle\nabla{\mathbb{S}}~{}-~{}\nabla
p~{}-~{}\nabla(W\,+\,U)~{}~{}~{}$ (236b)

or, in terms of components:

$\displaystyle\sigma_{\textstyle{{}_{\beta\nu}}}$ $\displaystyle=$
$\displaystyle 2~{}\hat{\mu}~{}u_{\textstyle{{}_{\beta\nu}}}~{}~{}~{},$ (237a)
$\displaystyle\rho\,\ddot{u}_{\textstyle{{}_{\beta}}}$ $\displaystyle=$
$\displaystyle\frac{\partial\sigma_{\textstyle{{}_{\beta\nu}}}}{\partial
x_{\textstyle{{}_{\nu}}}}~{}-~{}\frac{\partial p}{\partial
x_{\textstyle{{}_{\beta}}}}\;-\;\rho\;\frac{\partial(W\,+\,U)}{\partial
x_{\textstyle{{}_{\beta}}}}~{}~{}~{}.$ (237b)

In the frequency domain, this will look:

$\displaystyle\bar{\sigma}_{\textstyle{{}_{\beta\nu}}}(\chi)$ $\displaystyle=$
$\displaystyle
2~{}\bar{\mu}(\chi)~{}\bar{u}_{\textstyle{{}_{\beta\nu}}}(\chi)~{}~{}~{},$
(238a)
$\displaystyle\rho\,\chi^{2}\,\bar{u}_{\textstyle{{}_{\beta\nu}}}(\chi)$
$\displaystyle=$
$\displaystyle\frac{\partial\bar{\sigma}_{\textstyle{{}_{\beta\nu}}}(\chi)}{\partial
x_{\textstyle{{}_{\nu}}}}~{}-~{}\frac{\partial\bar{p}(\chi)}{\partial
x_{\textstyle{{}_{\beta}}}}\;-\;\rho\;\frac{\partial\left[\bar{W}(\chi)\,+\,\bar{U}(\chi)\right]}{\partial
x_{\textstyle{{}_{\beta}}}}~{}~{}~{},$ (238b)

where a bar denotes a spectral component for all functions except $\,\mu~{}$ –
recall that $\,\bar{\mu}\,$ is a spectral component not of the kernel
$\,\mu(\tau)\,$ but of its time-derivative $\,{\bf{\dot{\mu}}}(\tau)\,$.

Unless the frequencies are extremely high, we can neglect the body-fixed
acceleration term $\,\chi^{2}\,\bar{u}_{\textstyle{{}_{\beta\nu}}}(\chi)\,$ in
the second equation, in which case our system of equations for the spectral
components will mimic (235). Thus we arrive at the so-called _correspondence
principle_ (also known as the elastic-viscoelastic analogy), which maps a
solution of a linear viscoelastic boundary-value problem to a solution of a
corresponding elastic problem with the same initial and boundary conditions.
As a result, the algebraic equations for the Fourier (or Laplace) components
of the strain and stress in the viscoelastic case mimic the equations
connecting the strain and stress in the appropriate elastic problem. So the
viscoelastic operational moduli $\,\bar{\mu}(\chi)\,$ or $\,\bar{J}(\chi)\,$
obey the same algebraic relations as the elastic parameters $\,\mu\,$ or
$\,J\,$.

In the literature, there is no consensus on the authorship of this principle.
For example, Haddad (1995) mistakenly attributes it to several authors who
published in the 1950s and 1960s. In reality, the principle was pioneered
almost a century earlier by Darwin (1879), for isotropic incompressible media.
The principle was extended to more general types of media by Biot (1954,
1958), who also pointed out some limitations of this principle.

###  D.2 The correspondence principle, for rotating bodies

Consider a body of mass $\,M_{prim}\,$, which is spinning at a rate
$\vec{\omega}$ and is also performing some orbital motion (for example, is
orbiting, with its partner of mass $\,M_{sec}\,$, around their mutual centre
of mass). Relative to some inertial coordinate system, the centre of mass of
the body is located at $\,\mbox{{\boldmath$\vec{x}$}}_{{}_{CM}}\,$, while a
small parcel of its material is positioned at $\vec{x}$ . Relative to the
centre of mass of the body, the parcel is located at
$\,\mbox{{\boldmath$\vec{r}$}}\,=\,\mbox{{\boldmath$\vec{x}$}}\,-\,\mbox{{\boldmath$\vec{x}$}}_{{}_{CM}}\,$.
The body being deformable, we can decompose $\vec{r}$ into its average value,
$\,\mbox{{\boldmath$\vec{r}$}}_{0}\,$, and an instantaneous displacement
$\vec{u}$ :

$\displaystyle\left.\begin{array}[]{c}\mbox{{\boldmath$\vec{x}$}}\;=\;\mbox{{\boldmath$\vec{x}$}}_{{}_{CM}}\,+\;\mbox{{\boldmath$\vec{r}$}}{}{}\\\
{}\hfil\\\
\mbox{{\boldmath$\vec{r}$}}\;=\;\mbox{{\boldmath$\vec{r}$}}_{0}\,+\;\mbox{{\boldmath$\vec{u}$}}{}{}{}{}\end{array}\right\\}~{}\quad~{}\Longrightarrow\quad~{}\quad\mbox{{\boldmath$\vec{x}$}}\;=\;\mbox{{\boldmath$\vec{x}$}}_{{}_{CM}}\,+\;\mbox{{\boldmath$\vec{r}$}}_{0}\,+\;\mbox{{\boldmath$\vec{u}$}}~{}~{}.$
(242)

Denote with $D/Dt$ the time-derivative in the inertial frame. The symbol
$d/dt$ and its synonym, overdot, will be reserved for the time-derivative in
the body frame, so $\,{d\mbox{{\boldmath$\vec{r}$}}_{0}}/{dt}\,=\,0\,$. Then

$\displaystyle\frac{D\mbox{{\boldmath$\vec{r}$}}}{Dt}\;=\;\frac{d\mbox{{\boldmath$\vec{r}$}}}{dt}~{}+~{}\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\mbox{{\boldmath$\vec{r}$}}\quad~{}\quad\mbox{and}\quad~{}\quad\frac{D^{2}\mbox{{\boldmath$\vec{r}$}}}{Dt^{2}}\;=\;\frac{d^{2}\mbox{{\boldmath$\vec{r}$}}}{dt^{2}}\;+\;2\;\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\frac{d\mbox{{\boldmath$\vec{r}$}}}{dt}\;+\;\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\left(\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\mbox{{\boldmath$\vec{r}$}}\right)\;+\;\mbox{{\boldmath$\dot{\vec{\omega}}$}}\,\times\,\mbox{{\boldmath$\vec{r}$}}~{}~{}.\quad\quad$
(243)

Together, the above formulae result in

$\displaystyle\frac{D^{2}\mbox{{\boldmath$\vec{x}$}}}{Dt^{2}}\;=\;\frac{D^{2}\mbox{{\boldmath$\vec{x}$}}_{{}_{CM}}}{Dt^{2}}\;+\;\frac{D^{2}\mbox{{\boldmath$\vec{r}$}}}{Dt^{2}}$
$\displaystyle=$
$\displaystyle\frac{D^{2}\mbox{{\boldmath$\vec{x}$}}_{{}_{CM}}}{Dt^{2}}\;+\;\frac{d^{2}\mbox{{\boldmath$\vec{r}$}}}{dt^{2}}\;+\;2\;\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\frac{d\mbox{{\boldmath$\vec{r}$}}}{dt}\;+\;\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\left(\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\mbox{{\boldmath$\vec{r}$}}\right)\;+\;\mbox{{\boldmath$\dot{\vec{\omega}}$}}\,\times\,\mbox{{\boldmath$\vec{r}$}}$
(244) $\displaystyle=$
$\displaystyle\frac{D^{2}\mbox{{\boldmath$\vec{x}$}}_{{}_{CM}}}{Dt^{2}}\;+\;\frac{d^{2}\mbox{{\boldmath$\vec{u}$}}}{dt^{2}}\;+\;2\;\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\frac{d\mbox{{\boldmath$\vec{u}$}}}{dt}\;+\;\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\left(\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\mbox{{\boldmath$\vec{r}$}}\right)\;+\;\mbox{{\boldmath$\dot{\vec{\omega}}$}}\,\times\,\mbox{{\boldmath$\vec{r}$}}~{}~{}.\quad\quad\quad$

The equation of motion for a small parcel of the body’s material will read as

$\displaystyle\rho\;\frac{D^{2}\mbox{{\boldmath$\vec{x}$}}}{Dt^{2}}\;=\;\nabla{\mathbb{S}}\;-\;\nabla
p\;+\;\mbox{\boldmath$\vec{\boldmath{F}}$}_{self}\;+\;\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}~{}~{}~{},$
(245)

where $\,\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}\,$ is the exterior gravity
force per unit volume, while $\,\mbox{\boldmath$\vec{\boldmath{F}}$}_{self}\,$
is the “interior” gravity force per unit volume, i.e., the self-force
wherewith the rest of the body is acting upon the selected parcel of medium.
Insertion of (244) in (245) furnishes:

$\displaystyle\rho\,\left[\,\frac{D^{2}\mbox{{\boldmath$\vec{x}$}}_{{}_{CM}}}{Dt^{2}}\,+\,\stackrel{{\scriptstyle\bf{\centerdot\,\centerdot}}}{{\textbf{\mbox{\boldmath$\vec{\boldmath
u}$}}}}\,+\,2\,\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\stackrel{{\scriptstyle\bf{\centerdot}}}{{\textbf{\mbox{\boldmath$\vec{\boldmath
u}$}}}}\,+\,\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\left(\mbox{{\boldmath$\vec{\omega}$}}\,\times\,\mbox{{\boldmath$\vec{r}$}}\right)\,+\,\mbox{{\boldmath$\dot{\vec{\omega}}$}}\,\times\,\mbox{{\boldmath$\vec{r}$}}\,\right]\,=\,\nabla{\mathbb{S}}\,-\,\nabla
p\,+\,\mbox{\boldmath$\vec{\boldmath{F}}$}_{self}\,+\,\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}~{}~{}.\quad\quad$
(246)

At the same time, for the primary body as a whole, we can write:

$\displaystyle
M_{prim}\;\frac{D^{2}\mbox{{\boldmath$\vec{x}$}}_{{}_{CM}}}{Dt^{2}}\;=\;\int_{V}\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}\;d^{3}\mbox{{\boldmath$\vec{r}$}}~{}~{},$
(247)

the integration being carried out over the volume $\,V\,$ of the primary.
(Recall that $\,\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}\,$ is a force per
unit volume.) Combined together, the above two equations will result in

$\displaystyle\rho\left[\,\stackrel{{\scriptstyle\bf{\centerdot\,\centerdot}}}{{\textbf{\mbox{\boldmath$\vec{\boldmath
u}$}}}}+\,2\,\mbox{{\boldmath$\vec{\omega}$}}\times\stackrel{{\scriptstyle\bf{\centerdot}}}{{\textbf{\mbox{\boldmath$\vec{\boldmath
u}$}}}}+\,\mbox{{\boldmath$\vec{\omega}$}}\times\left(\mbox{{\boldmath$\vec{\omega}$}}\times\mbox{{\boldmath$\vec{r}$}}\right)+\mbox{{\boldmath$\dot{\vec{\omega}}$}}\times\mbox{{\boldmath$\vec{r}$}}\,\right]\,=\,\nabla{\mathbb{S}}\,-\,\nabla
p\,+\,\mbox{\boldmath$\vec{\boldmath{F}}$}_{self}\,+\,\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}\,-\,\frac{\rho~{}}{\,M_{prim}\,}\int_{V}\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}\;d^{3}\mbox{{\boldmath$\vec{r}$}}~{}~{}.\quad~{}$
(248)

For a spherically-symmetrical (not necessarily radially-homogeneous) body, the
integral on the right-hand side clearly removes the Newtonian part of the
force, leaving the harmonics intact:

$\displaystyle\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}\,-\,\frac{\rho~{}}{\,M_{prim}\,}\int_{V}\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}~{}d^{3}\mbox{{\boldmath$\vec{r}$}}~{}=~{}\rho~{}\sum_{l=2}^{\infty}\nabla
W_{l}~{}~{},$ (249)

where the harmonics are given by

$\displaystyle
W_{l}(\mbox{{\boldmath$\vec{r}$}},\,\mbox{{\boldmath$\vec{r}$}}^{*})\;=\;-\;\frac{G\,M_{sec}}{r^{*}}\;\left(\frac{r}{r^{*}}\right)^{l}\,P_{l}(\cos\gamma)~{}~{}~{},$
(250)

$\mbox{{\boldmath$\vec{r}$}}^{\,*}$ being the vector pointing from the centre
of mass of the primary to that of the secondary, and $\,\gamma\,$ being the
angular separation between $\vec{r}$ and
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,$, subtended at the centre of mass of
the primary.

In reality, a tiny extra force $\,{\cal F}\,$, the tidal force per unit
volume, is left over due to the body being slightly distorted:

$\displaystyle\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}\,-\,\frac{\rho~{}}{\,M_{prim}\,}\int_{V}\mbox{\boldmath$\vec{\boldmath{F}}$}_{ext}~{}d^{3}\mbox{{\boldmath$\vec{r}$}}~{}=~{}\rho~{}\sum_{l=2}^{\infty}\nabla
W_{l}~{}+~{}{\cal F}~{}~{}.$ (251)

Here $\,{\cal F}\,$ is the density multiplied by the average tidal
acceleration experienced by the body as a whole. In neglect of $\,{\cal F}\,$,
we arrive at

$\displaystyle\rho\left[\,\stackrel{{\scriptstyle\bf{\centerdot\,\centerdot}}}{{\textbf{\mbox{\boldmath$\vec{\boldmath
u}$}}}}+\,2\,\mbox{{\boldmath$\vec{\omega}$}}\times\stackrel{{\scriptstyle\bf{\centerdot}}}{{\textbf{\mbox{\boldmath$\vec{\boldmath
u}$}}}}+\,\mbox{{\boldmath$\vec{\omega}$}}\times\left(\mbox{{\boldmath$\vec{\omega}$}}\times\mbox{{\boldmath$\vec{r}$}}\right)+\mbox{{\boldmath$\dot{\vec{\omega}}$}}\times\mbox{{\boldmath$\vec{r}$}}\,\right]\,=\,\nabla{\mathbb{S}}\,-\,\nabla
p\,-\,\rho~{}\sum_{l=2}^{\infty}\nabla(U_{l}\,+\;W_{l})~{}~{}.\quad~{}$ (252)

Here, to each disturbing term of the exterior potential, $\,W_{l}\,$,
corresponds a term $\,U_{l}\,$ of the self-potential, the self-force thus
being expanded into
$\,\mbox{\boldmath$\vec{\boldmath{F}}$}_{self}\,=\,-\,\sum_{l=2}^{\infty}\nabla
U_{l}\,$.

Equation (252) could as well have been derived in the body frame, where it
would have assumed the same form.

Denoting the tidal frequency with $\,\chi\,$, we see that the terms on the
left-hand side have the order of $\,\rho\,\chi^{2}\,u~{}$,
$~{}\rho\,\omega\,\chi\,u~{}$, $~{}\rho\,\omega^{2}\,r\,$, and
$~{}\rho\,\dot{\chi}\,\omega\,r~{}$, correspondingly. In realistic situations,
the first two terms, thus, can be neglected, and we end up with

$\displaystyle 0\,=\,\nabla{\mathbb{S}}\,-\,\nabla
p\,-~{}\rho~{}\sum_{l=2}^{\infty}\nabla(U_{l}\,+\;W_{l})~{}-~{}\rho~{}\mbox{{\boldmath$\vec{\omega}$}}\times\left(\mbox{{\boldmath$\vec{\omega}$}}\times\mbox{{\boldmath$\vec{r}$}}\right)~{}-~{}\rho\,\mbox{{\boldmath$\dot{\vec{\omega}}$}}\times\mbox{{\boldmath$\vec{r}$}}~{}~{},\quad~{}$
(253)

the term $\;-\,\nabla p\,$ vanishing in an incompressible media.

###  D.3 The centripetal term and the zero-degree Love number

The centripetal term in (253) can be split into a purely radial part and a
part that can be incorporated into the $\,W_{2}\,$ term of the tide-raising
potential, as was suggested by Love (1909, 1911). Introducing the colatitude
$\,\phi\,^{\prime}\,$ through
$~{}\cos\phi\,^{\prime}~{}=~{}\frac{\mbox{{\boldmath$\vec{\omega}$}}}{\,|\mbox{{\boldmath$\vec{\omega}$}}|\,}\cdot\frac{\mbox{{\boldmath$\vec{r}$}}}{\,|\mbox{{\boldmath$\vec{r}$}}|\,}~{}$,
we can write down the evident equality

$\displaystyle\mbox{{\boldmath$\vec{\omega}$}}\times\left(\mbox{{\boldmath$\vec{\omega}$}}\times\mbox{{\boldmath$\vec{r}$}}\right)~{}=~{}\mbox{{\boldmath$\vec{\omega}$}}~{}(\mbox{{\boldmath$\vec{\omega}$}}\cdot\mbox{{\boldmath$\vec{r}$}})~{}-~{}\mbox{{\boldmath$\vec{r}$}}~{}\mbox{{\boldmath$\vec{\omega}$}}^{\,2}=~{}\nabla\left[\,\frac{1}{2}\,\left(\mbox{{\boldmath$\vec{\omega}$}}\cdot\mbox{{\boldmath$\vec{r}$}}\right)^{2}\,-~{}\frac{1}{2}\,\mbox{{\boldmath$\vec{\omega}$}}^{\,2}\,\mbox{{\boldmath$\vec{r}$}}^{\,2}\,\right]~{}=~{}\nabla\left[~{}~{}\frac{1}{2}\,\mbox{{\boldmath$\vec{\omega}$}}^{\,2}\,\mbox{{\boldmath$\vec{r}$}}^{\,2}\,\left(\,\cos^{2}\phi\,^{\prime}~{}-~{}1\,\right)~{}~{}\right]~{}~{}~{}.\quad\quad\quad\quad\quad\quad$

The definition $~{}P_{2}(\cos\phi\,^{\prime})\,=~{}\frac{\textstyle
1}{\textstyle 2}\left(3\,\cos^{2}\phi\,^{\prime}\,-~{}1\right)~{}$ easily
renders: $~{}\cos^{2}\phi\,^{\prime}~{}=~{}\frac{\textstyle 2}{\textstyle
3}~{}P_{2}(\cos\phi\,^{\prime})~{}+~{}\frac{\textstyle 1}{\textstyle 3}~{}$,
whence:

$\displaystyle\mbox{{\boldmath$\vec{\omega}$}}\times\left(\mbox{{\boldmath$\vec{\omega}$}}\times\mbox{{\boldmath$\vec{r}$}}\right)~{}=~{}\nabla\left[~{}~{}\frac{1}{3}~{}\mbox{{\boldmath$\vec{\omega}$}}^{\,2}\,\mbox{{\boldmath$\vec{r}$}}^{\,2}\,\left[\,P_{2}\left(\cos\phi\,^{\prime}\right)~{}-~{}1~{}\right]~{}~{}\right]~{}~{}~{}.$
(254)

We see that the centripetal force splits into a second-harmonic and purely-
radial parts:

$\displaystyle-~{}\rho~{}\mbox{{\boldmath$\vec{\omega}$}}\times\left(\mbox{{\boldmath$\vec{\omega}$}}\times\mbox{{\boldmath$\vec{r}$}}\right)~{}=~{}-~{}\nabla\left[~{}\frac{\rho}{3}~{}\mbox{{\boldmath$\vec{\omega}$}}^{\,2}\,\mbox{{\boldmath$\vec{r}$}}^{\,2}\,P_{2}\left(\cos\phi\,^{\prime}\right)~{}\right]~{}+~{}\nabla\;\left[~{}\frac{\rho}{3}~{}\mbox{{\boldmath$\vec{\omega}$}}^{\,2}\,\mbox{{\boldmath$\vec{r}$}}^{\,2}~{}\right]~{}~{}~{},$
(255)

where we assume the body to be homogeneous. The second-harmonic part can be
incorporated into the external potential. The response to this part will be
proportional to the degree-2 Love number $\,k_{2}\,$.

The purely radial part of the centripetal potential generates a radial
deformation. This part of the potential is often ignored, the associated
deformation being tacitly included into the equilibrium shape of the body.
Compared to the main terms of the equation of motion, this radial term is of
the order of $\,10^{-3}\,$ for the Earth, and is smaller for most other
bodies. As the rotation variations of the Earth are of the order of
$\,10^{-5}\,$, this term leads to a tiny change in the geopotential and to an
associated displacement of the order of a micrometer. 292929 Tim Van Hoolst,
private communication.

However, for other rotators the situation may be different. For example, in
Phobos, whose libration magnitude is large (about 1 degree), the radial term
may cause an equipotential-surface variation of about 10 cm. This magnitude is
large enough to be observed by future missions and should be studied in more
detail.303030 Tim Van Hoolst, private communication. The emergence of the
purely radial deformation gives birth to the zero-degree Love number (Dahlen
1976, Matsuyama & Bills 2010). Using Dahlen’s results, Yoder (1982, eqns 21 -
22) demonstrated that the contribution of the radial part of the centripetal
potential to the change in mean motion of Phobos is about 3%, which is smaller
than the uncertainty in our knowledge of Phobos’ $\,k_{2}/Q\,$. It should be
mentioned, however, that the calculations by Dahlen (1976) and Matsuyama &
Bills (2010) were performed for steady (or slowly changing) rotation, and not
for libration. This means that Yoder’s application of Dahlen’s result to
Phobos requires extra justification.

What is important for us here is that the radial term does not interfere with
the calculation of the Love number. Being independent of the longitude, this
term generates no tidal torque either, provided the obliquity is neglected.

###  D.4 The toroidal term

The inertial term
$~{}~{}-~{}\rho~{}\mbox{{\boldmath$\dot{\vec{\omega}}$}}\times\mbox{{\boldmath$\vec{r}$}}\,$
in the equation of motion (253) can be cast into the form

$\displaystyle-~{}\rho~{}\mbox{{\boldmath$\dot{\vec{\omega}}$}}\times\mbox{{\boldmath$\vec{r}$}}\;=\;\rho~{}\mbox{{\boldmath$\vec{r}$}}\times\nabla(\mbox{{\boldmath$\dot{\vec{\omega}}$}}\cdot\mbox{{\boldmath$\vec{r}$}})~{}~{}~{},$
(256)

whence we see that this term is of a toroidal type. Being almost nil for a
despinning primary, this force becomes important for a librating object.

In spherically-symmetric bodies, the toroidal force (256) generates toroidal
deformation only. This deformation produces neither radial uplifts nor
variations of the gravitational potential. Hence its presence does not
influence the expressions for the Love numbers associated with vertical
displacement ($\,h_{\textstyle{{}_{l}}}\,$) or the potential
($\,k_{\textstyle{{}_{l}}}\,$). As this deformation yields no change in the
gravitational potential of the tidally-perturbed body, there is no tidal
torque associated with this deformation. Being divergence-free, this
deformation entails no contraction or expansion either, i.e., it is purely
shear. Still, this deformation contributes to dissipation. Besides, since the
toroidal forcing results in the toroidal deformation, it can, in principle, be
associated with a “toroidal” Love number.

To estimate the dissipation caused by the toroidal rotational force, Yoder
(1982) introduced an equivalent effective torque. He pointed out that this
force becomes important when the magnitude of the physical libration is
comparable to that of the optical libration. According to Ibid., the toroidal
force contributes to the change of the mean motion of Phobos about 1.6%, which
is less than the input from the purely radial part.

##  Appendix E The Andrade and Maxwell models at different frequencies

###  E.1 Response of a sample obeying the Andrade model

Within the Andrade model, the tangent of the phase lag demonstrates the so-
called “elbow dependence”. At high frequencies, the tangent of the lag obeys a
power law with an exponent equal to $\,-\alpha\,$, where
$\,0\,<\,\alpha\,<\,1\,$. At low frequencies, the tangent of the lag once
again obeys a power law, this time though with an exponent
$~{}-\,(1-\alpha)\,$. This model fits well the behaviour of ices, metals,
silicate rocks, and many other materials.

However the applicability of the Andrade law may depend upon the intensity of
the load and, accordingly, upon the damping mechanisms involved. Situations
are known, when, at low frequencies, anelasticity becomes much less efficient
than viscosity. In these cases, the Andrade model approaches, at low
frequencies, the Maxwell model.

####  E.1.1 The high-frequency band

At high frequencies, expression (89b) gets simplified. In the numerator, the
term with $\,z^{-\alpha}\,$ dominates: $~{}z^{-\alpha}\,\gg\,z^{-1}\,\zeta\,$,
which is equivalent to
$\,z\,\gg\,\zeta^{\textstyle{{}^{\textstyle{\frac{1}{1-\alpha}}}}}\,$. In the
denominator, the constant term dominates: $~{}1\,\gg\,z^{-\alpha}\,$, or
simply: $\,z\,\gg\,1\,$. To know which of the two conditions,
$\,z\,\gg\,\zeta^{\textstyle{{}^{\textstyle{\frac{1}{1-\alpha}}}}}\,$ or
$\,z\,\gg\,1\,$, is stronger, we recall that at high frequencies anelasticity
beats viscosity. So the $\,\alpha-$term in (85) is large enough. In other
words, the Andrade timescale $\,\tau_{{}_{A}}\,$ should be smaller (or, at
least, not much higher) than the viscoelastic time $\,\tau_{{}_{M}}\,$.
Accordingly, at high frequencies, $\,\zeta\,$ is smaller (or, at least, not
much higher) than unity. Hence, within the high-frequency band, either the
condition $\,z\,\gg\,1\,$ is stronger than
$\,z\,\gg\,\zeta^{\textstyle{{}^{\textstyle{\frac{1}{1-\alpha}}}}}\,$ or the
two conditions are about equivalent. This, along with (90) and (91) enables us
to write:

$\displaystyle\tan\delta~{}\,\approx\,~{}(\chi\,\tau_{{}_{A}})^{\textstyle{{}^{-\alpha}}}\sin\left(\frac{\alpha\,\pi}{2}\right)~{}\Gamma(\alpha+1)~{}\quad~{}\quad~{}\mbox{for}\quad\chi\;\gg\;\tau^{-1}_{{}_{A}}\,=\,\tau_{{}_{M}}^{-1}\zeta^{-1}~{}~{}~{}.~{}~{}\,~{}$
(257)

The tangent being small, the expression for $\,\sin\delta\,$ looks identical:

$\displaystyle\sin\delta\,~{}\approx\,~{}(\chi\,\tau_{{}_{A}})^{\textstyle{{}^{-\alpha}}}~{}\sin\left(\frac{\alpha\,\pi}{2}\right)~{}\Gamma(\alpha+1)~{}\quad~{}\quad~{}\mbox{for}~{}~{}~{}\chi\;\gg\;\tau^{-1}_{{}_{A}}\,=\,\tau_{{}_{M}}^{-1}\zeta^{-1}~{}~{}~{}.~{}~{}\,~{}$
(258)

####  E.1.2 The intermediate region

In the intermediate region, the behaviour of the phase lag $\,\delta\,$
depends upon the frequency-dependence of $\,\zeta\,$. For example, if there
happens to exist an interval of frequencies over which the conditions
$\,1\,\gg\,z\,\gg\,\zeta^{\textstyle{{}^{\textstyle{\frac{1}{1-\alpha}}}}}\,$
are obeyed, then over this interval we shall have: $\,1\ll\,z^{-\alpha}\,$ and
$\,z^{-\alpha}\gg\,\zeta z^{-1}\,$. Applying these inequalities to (89b), we
see that over such an interval of frequencies $\,\tan\delta\,$ will behave as
$\,z^{-2\alpha}\,\tan\left(\frac{\textstyle\alpha\,\pi}{\textstyle
2}\right)\,$.

####  E.1.3 The low-frequency band

At low frequencies, the term $\,z^{-1}\,\zeta\,$ becomes leading in the
numerator of (89b): $\,z^{-\alpha}\,\ll\,z^{-1}\,\zeta\,$, which requires
$\,z\,\ll\,\zeta^{\textstyle{{}^{\textstyle{\frac{1}{1-\alpha}}}}}\,$. In the
denominator, the term with $\,z^{-\alpha}\,$ becomes the largest:
$\,1\,\ll\,z^{-\alpha}\,$, whence $\,z\,\ll\,1\,$. Since at low frequencies
the viscous term in (85) is larger than the anelastic term, we expect that for
these frequencies $\,\zeta\,$ is larger (at least, not much smaller) than
unity. Thence the condition $\,z\,\ll\,1\,$ becomes sufficient. Its fulfilment
ensures the fulfilment of
$\,z\,\ll\,\zeta^{\textstyle{{}^{\textstyle{\frac{1}{1-\alpha}}}}}\,$. Thus we
state:

$\displaystyle\tan\delta~{}\approx\,~{}(\chi\,\tau_{{}_{A}})^{\textstyle{{}^{-(1-\alpha)}}}\frac{\zeta}{~{}\cos\left(\frac{\textstyle\alpha\,\pi}{\textstyle
2}\right)~{}\Gamma(\alpha+1)}\quad\quad\quad\mbox{for}\quad\quad\chi~{}\ll~{}\tau^{-1}_{{}_{A}}\,=~{}\tau_{{}_{M}}^{-1}\,\zeta^{-1}~{}~{}~{}.~{}~{}\quad\quad$
(259)

The appropriate expression for $\,\sin\delta\,$ will be:

$\displaystyle\sin\delta~{}\approx\,~{}1~{}-~{}O\left(~{}(\chi\,\tau_{{}_{A}})^{{{2(1-\alpha)}}}\,\zeta^{-2}\,\right)\quad\,\quad\quad\quad\,\mbox{for}\quad\quad\chi~{}\ll\;\tau_{{}_{A}}^{-1}\,=~{}\tau_{{}_{M}}^{-1}\,\zeta^{-1}~{}~{}~{},~{}~{}$
(260)

It would be important to emphasise that the threshold
$\,\tau^{-1}_{{}_{A}}\,=~{}\tau_{{}_{M}}^{-1}\,\zeta^{-1}\,$ standing in (257)
and (258) is different from the threshold
$\,\tau^{-1}_{{}_{A}}\,=~{}\tau_{{}_{M}}^{-1}\,\zeta^{-1}\,$ showing up in
(259) and (260), even though these two thresholds are given by the same
expression. The reason for this is that the timescales $\,\tau_{{}_{A}}\,$ and
$\,\tau_{{}_{M}}\,$ are not fixed constants. While the Maxwell time is likely
to be a very slow function of the frequency, the Andrade time may undergo a
faster change over the transitional region: $\,\tau_{{}_{A}}\,$ must be larger
than $\,\tau_{{}_{M}}\,$ at low frequencies (so anelasticity yields to
viscosity), and must become shorter than or of the order of
$\,\tau_{{}_{M}}\,$ at high frequencies (so anelasticity becomes stronger).
This way, the threshold $\,\tau^{-1}_{{}_{A}}\,$ standing in (259 \- 260) is
lower than the threshold $\,\tau^{-1}_{{}_{A}}\,$ standing in (257 \- 258).
The gap between these thresholds is the region intermediate between the two
pronounced power laws (257) and (259).

####  E.1.4 The low-frequency band: a special case, the Maxwell model

Suppose that, below some threshold $\,\chi_{\textstyle{{}_{0}}}\,$,
anelasticity quickly becomes much less efficient than viscosity. This would
imply a steep increase of $\,\zeta\,$ (equivalently, of $\,\tau_{{}_{A}}\,$)
at low frequencies. Then, in (89b), we shall have: $\,1\gg\,z^{-\alpha}\,$ and
$\,z^{-\alpha}\ll\,\zeta z^{-1}\,$. This means that, for frequencies below
$\,\chi_{\textstyle{{}_{0}}}\,$, the tangent will behave as

$\displaystyle\tan\delta~{}\,\approx\,~{}z^{-1}\,\zeta~{}=~{}(\chi\,\tau_{{}_{M}})^{-1}~{}\quad~{}\quad~{}\mbox{for}\quad\quad\chi\,\ll\,\chi_{\textstyle{{}_{0}}}~{}~{}~{}.~{}~{}\,~{}$
(261)

the well-known viscous scaling law for the lag.

The study of ices and minerals under weak loads (Castillo-Rogez et al. 2011,
Castillo-Rogez 2011) has not shown such an abrupt vanishing of anelasticity.
However, Karato & Spetzler (1990) point out that this should be happening in
the Earth’s mantle, where the loads are much higher and anelasticity is caused
by unpinning of dislocations.

###  E.2 The behaviour of
$\,|k_{\textstyle{{}_{l}}}(\chi)|\;\sin\epsilon_{\textstyle{{}_{l}}}(\chi)=-\,{\cal{I}}{\it{m}}\left[\,\bar{k}_{\textstyle{{}_{l}}}(\chi)\,\right]\,$
within the Andrade and Maxwell models

As we explained in subsection 4.1, products
$\,~{}k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{{\it{l}}mpq}}})\,\sin\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{{\it{l}}mpq}}})\,\;$
enter the $\;{\it l}mpq\;$ term of the Darwin-Kaula series for the tidal
potential, force, and torque. Hence the importance to know the behaviour of
these products as functions of the tidal frequency
$~{}\chi_{\textstyle{{}_{{\it{l}}mpq}}}~{}$.

####  E.2.1 Prefatory algebra

It ensues from (4.3) that

$\displaystyle\bar{k}_{\textstyle{{}_{l}}}(\chi)\;=\;\frac{3}{2\,(l-1)}\;\frac{\left(\;{\cal{R}}{\it{e}}\left[\bar{J}(\chi)\right]\;\right)^{2}\;+\;\left(\;{\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]\;\right)^{2}\;+\;A_{\textstyle{{}_{l}}}\;J\;{\cal{R}}{\it{e}}\left[\bar{J}(\chi)\right]\;+i\;A_{\textstyle{{}_{l}}}\;J\;{\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]}{\left(\;{\cal{R}}{\it{e}}\left[\bar{J}(\chi)\right]\;+\;A_{\textstyle{{}_{l}}}\;J\;\right)^{2}\;+\;\left(\;{\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]\;\right)^{2}}~{}~{}~{},~{}~{}~{}$
(262)

whence

$\displaystyle|\bar{k}_{\textstyle{{}_{l}}}(\chi)|\;\sin\epsilon_{\textstyle{{}_{l}}}(\chi)\;=\;-\;{\cal{I}}{\it{m}}\left[\bar{k}_{\textstyle{{}_{l}}}(\chi)\right]\;=\;\frac{3}{2\,(l-1)}\;\frac{-\;A_{\textstyle{{}_{l}}}\;J\;{\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]}{\left(\;{\cal{R}}{\it{e}}\left[\bar{J}(\chi)\right]\;+\;A_{\textstyle{{}_{l}}}\;J\;\right)^{2}\;+\;\left(\;{\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]\;\right)^{2}}~{}~{}~{},~{}~{}~{}~{}~{}$
(263)

$J\,=\,J(0)\,\equiv\,1/\mu\,=\,1/\mu(0)\;$ being the unrelaxed compliance (the
inverse of the unrelaxed shear modulus $\,\mu\,$). For an Andrade material,
the compliance $\;\bar{J}\;$ in the frequency domain is rendered by (86). Its
imaginary and real parts are given by (87 \- 88). It is then easier to rewrite
(263) as

$\displaystyle|\bar{k}_{\textstyle{{}_{l}}}(\chi)|\;\sin\epsilon_{\textstyle{{}_{l}}}(\chi)\;=\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle\frac{3~{}A_{\textstyle{{}_{l}}}}{2\,(l-1)}~{}\frac{\zeta~{}z^{-1}~{}+~{}z^{-\alpha}~{}\sin\left(\frac{\textstyle\alpha\,\pi}{\textstyle
2}\right)~{}\Gamma(\alpha\,+\,1)}{\left[A_{\textstyle{{}_{l}}}+1+z^{-\alpha}\,\cos\left(\frac{\textstyle\alpha\,\pi}{\textstyle
2}\right)\,\Gamma(\alpha+1)\right]^{2}+\,\left[\zeta\,z^{-1}+\,z^{-\alpha}\,\sin\left(\frac{\textstyle\alpha\,\pi}{\textstyle
2}\right)\,\Gamma(1+\alpha)\right]^{2}}~{}~{}~{},\quad\quad$ (264)

where

$\displaystyle
z~{}\equiv~{}\chi~{}\tau_{{}_{A}}\,=~{}\chi~{}\tau_{{}_{M}}\,\zeta$ (265)

and

$\displaystyle\zeta~{}\equiv~{}\frac{\tau_{{}_{A}}}{\tau_{{}_{M}}}~{}~{}~{}.$
(266)

For $\,\beta\,\rightarrow\,0\,$, i.e., for
$\,\tau_{{}_{A}}\,\rightarrow\,\infty\,$, (264) coincides with the appropriate
expression for a spherical Maxwell body.

####  E.2.2 The high-frequency band

Within the upper band, the term with $\,z^{-\alpha}\,$ dominates the
numerator, while $\,A_{\textstyle{{}_{l}}}\,$ dominates the denominator. The
domination of $\,z^{-\alpha}\,$ in the numerator requires that
$\,z\gg\zeta^{\textstyle{{}^{\,{\textstyle\frac{1}{1-\alpha}}}}}\,$, which is
the same as
$\,\chi\gg\tau_{{}_{M}}^{-1}\,\zeta^{\textstyle{{}^{\textstyle\,\frac{\alpha}{1-\alpha}}}}\,$.
The domination of $\,A_{\textstyle{{}_{l}}}\,$ in the denominator requires:
$\,z\gg\,A^{\textstyle{{}^{\,-1/\alpha}}}\,$, which is the same as
$\,\chi\gg\,\tau_{{}_{M}}^{-1}\,\zeta^{-1}\,A^{\textstyle{{}^{\,-\,1/\alpha}}}\,$.
It also demands that $\,\zeta\,z^{-1}\ll A_{\textstyle{{}_{l}}}\,$, which is:
$\,\chi\gg\tau^{-1}_{{}_{M}}\,A^{-1}_{\textstyle{{}_{l}}}\,$.

For realistic values of $\,A_{\textstyle{{}_{l}}}\,$ (say, $10^{3}$) and
$\,\alpha\,$ (say, 0.25), we have:
$\,A^{\textstyle{{}^{\,-\,1/\alpha}}}\sim\,10^{-12}\,$. At high frequencies,
anelasticity beats viscosity, so $\,\zeta\,$ is less than unity (or, at least,
is not much larger than unity). On these grounds, the requirement
$\,\chi\gg\tau_{{}_{M}}^{-1}\,\zeta^{\textstyle{{}^{\textstyle\,\frac{\alpha}{1-\alpha}}}}\,$
is the strongest here. Its fulfilment guarantees that of both
$\,\chi\gg\,\tau_{{}_{M}}^{-1}\,\zeta^{-1}\,A^{\textstyle{{}^{\,-\,1/\alpha}}}\,$
and $\,\chi\gg\tau_{{}_{M}}/A_{\textstyle{{}_{l}}}\,$. Thus we have:

$\displaystyle|\bar{k}_{\textstyle{{}_{l}}}(\chi)|\,\sin\epsilon_{\textstyle{{}_{l}}}(\chi)$
$\displaystyle\approx$