#
${{~{}~{}~{}~{}~{}~{}~{}~{}~{}}^{{}^{{}^{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}The~{}Astronomical~{}Journal\,~{}150~{}:~{}98\,~{}(2015)}}}}$
Tidal Evolution of Asteroidal Binaries.
Ruled by Viscosity. Ignorant of Rigidity.

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

###### Abstract

This is a pilot paper serving as a launching pad for study of orbital and spin
evolution of binary asteroids. The rate of tidal evolution of asteroidal
binaries is defined by the dynamical Love numbers $\,k_{l}\,$ divided by
quality factors $\,Q\,$. Common in the literature is the (oftentimes
illegitimate) approximation of the dynamical Love numbers with their static
counterparts. Since the static Love numbers are, approximately, proportional
to the inverse rigidity, this renders a popular fallacy that the tidal
evolution rate is determined by the product of the rigidity by the quality
factor: $\,k_{l}/Q\propto 1/(\mu Q)\,$. In reality, the dynamical Love numbers
depend on the tidal frequency and all rheological parameters of the tidally
perturbed body (not just rigidity). We demonstrate that in asteroidal binaries
the rigidity of their components plays virtually no role in tidal friction and
tidal lagging, and thereby has almost no influence on the intensity of tidal
interactions (tidal torques, tidal dissipation, tidally induced changes of the
orbit). A key quantity that overwhelmingly determines the tidal evolution is a
product of the effective viscosity $\,\eta\,$ by the tidal frequency
$\,\chi\,$. The functional form of the torque’s dependence on this product
depends on who wins in the competition between viscosity and self-gravitation.
Hence a quantitative criterion, to distinguish between two regimes. For higher
values of $\,\eta\chi\,$, we get $\,k_{l}/Q\propto 1/(\eta\chi)\;$; while for
lower values we obtain $\,k_{l}/Q\propto\eta\chi\,$. Our study rests on an
assumption that asteroids can be treated as Maxwell bodies. Applicable to
rigid rocks at low frequencies, this approximation is used here also for
rubble piles, due to the lack of a better model. In the future, as we learn
more about mechanics of granular mixtures in a weak gravity field, we may have
to amend the tidal theory with other rheological parameters, ones that do not
show up in the description of viscoelastic bodies. This line of study provides
a tool to exploring the orbital history of asteroidal pairs, as well as of
their final spin states.

##  1 Dynamics of binary asteroids

Numerous binary asteroids have been discovered by now (see, e.g., Pravec &
Harris 2007, Taylor & Margot 2011, Jacobson & Scheeres 2011, Margot et al.
2015). Analysis of their orbital and rotational dynamics will give us a key to
understanding these asteroid’s internal properties from observation of their
orbits and spin.

Spin-orbit evolution of a binary is implemented through tidal torques that the
companions exert upon one another. Torques due to the permanent figure also
participate in this interaction and play a role in tidal capture into spin-
orbit resonances (e.g., Noyelles et al. 2014, Batygin & Morbidelli 2015). In
this paper, however, we shall concentrate on tides.

###  1.1 Difficulties in modeling of binary asteroids’ tidal dynamics

At first glance, the strategy and methods for exploring binary asteroids’
dynamics should be the same as those employed in the studies of planets and
moons. In reality, we are facing a more complicated case, for three main
reasons:

  * 1\.

Many asteroidal pairs are very close, so calculation of their tidal
interaction requires inclusion of terms higher than quadrupole (Mathis & Le
Poncin-Lafitte 2009, Compère & Lemaître 2014, Taylor & Margot 2010).

  * 2\.

Tidal reaction of a connected (not rubble) small body is overwhelmingly
defined by rheology and only to a minuscule degree by self-gravitation (which
is not the case for planets – see, e.g., Efroimsky 2012 b).

  * 3\.

Many asteroids, however, are believed to be loose aggregates. Their rheology
is largely defined by microgravity, and may differ from the rheology of solids
not only in parameters’ values. A constitutive equation for granular matter in
a weak gravity field may, in principle, contain parameters other than rigidity
or viscosity.

###  1.2 Tidal torques

When an extended celestial body is deformed by the potential of an exterior
perturber, the distortion results in an additional tidal potential produced by
the extended body itself. The latter potential acts back on the perturber and
changes its orbital motion.

In each point of the extended body, the exterior perturbing potential changes
in time and usually contains an infinite set of Fourier harmonics. We shall
assume the reaction linear, so the Fourier spectrum of the reaction (i.e., of
the shape changes and of the resulting tidal potential) mimics the spectrum of
the perturbing potential.

As ever in physics, the reaction lags behind action. Consequently, one may
expect that the Fourier modes of the tidal elevation and tidal potential
should always lag behind the corresponding modes of the perturbing potential.
This point, however, needs attention: falling behind in time does not  fix the
sign of a geometric lag. 111 Causality requires reaction to fall behind
action, in time . So each Fourier mode of the additional tidal potential falls
behind the appropriate mode of the perturbing potential, by a time lag
$\,\Delta t\,$ which is mode-dependent but is always positive definite. This,
however, does not imply that the phase  lags should always be positive.
Indeed, the phase lag at a certain Fourier mode is a product of this mode by
the appropriate time lag (see the equation 75 below), so the sign of this lag
always coincides with that of the mode. Likewise, the observable geometric
lags (related to the phase lag through the formula 79 below), share the sign
with the Fourier mode. Accordingly, no matter whether the perturber is below
or above the synchronous orbit, some components of the tide, geometrically,
fall behind the perturber; and some lead. For example, the principal
(semidiurnal) Phobos-raised bulge in Mars falls behind. Of the lesser bulges,
some fall behind and some lead, at various angular rates.

Since the perturber “feels” the tides it itself creates in the extended body,
the perturber experiences an orbital tidal force and an orbital tidal torque.
Then the deformed body experiences a tidal torque of an opposite sign, which
acts to change this body’s spin. This interaction is reciprocal, in that the
deformed body also acts as a perturber upon its companion – and exerts a
despinning torque thereon, and experiences an opposite orbital torque.

###  1.3 A specific problem

In the literature on tidal dynamics, it has long become a convention to
introduce the Love numbers and quality factors as separate entities. More
often than not, authors postulate some frequency-dependence of the quality
factors, treating the Love numbers as constant parameters (e.g., Mignard 1979,
1980, 1981; Correia & Laskar 2004; Efroimsky & Lainey 2007; Cheng et al.
2014). Above that, the dynamical Love numbers are very often identified with
their static counterparts (Goldreich & Peale 1968, Peale & Cassen 1978, Murray
& Dermott 1999, Taylor & Margot 2010).

The static Love numbers are, approximately, proportional to the inverse
rigidity. For example, the static quadrupole Love number is given by

$\displaystyle
k^{\textstyle{{}^{(static)}}}_{2}\,=\;\frac{3}{2}\;\,\frac{1}{1\;+\;\frac{\textstyle
19~{}\mu}{\textstyle
2~{}\mbox{g}\,\rho\,R}}~{}\approx~{}\frac{3}{19}\;\frac{\,\textstyle\mbox{g}~{}\rho~{}R\,}{\textstyle\mu}~{}~{}~{},$
(1)

with g, $\,\rho\,$, and $\,R\,$ being the surface gravity, density and radius
of a spherical body. This altogether renders a common misunderstanding that
the tidal evolution rate is determined by the product of the rigidity by the
quality factor: $\,k_{l}/Q\propto 1/(\mu Q)\,$.

A minor issue with this treatment is that the tidal quality factors, as
functions of frequency, are different for different values of the degree
$\,l\,$ (and are different from the seismic quality factor). Negligible at
ordinary frequencies, these differences become critical in the zero-frequency
limit, i.e., at crossing of spin-orbit resonances (Makarov et al. 2012,
Noyelles et al. 2014).

A bigger issue – the one to be dealt with – is that the dynamical Love
numbers, aside from their frequency dependence, depend on all rheological
parameters of the tidally perturbed body (not just rigidity). 222 The Andrade
model serves as an example of rheology containing parameters other than
rigidity and viscosity – see the equations (3.2) in Section 3.2 below. More
examples of the kind can be found, e.g., in Henning et al. (2009). Our goal is
to demonstrate that in asteroidal binaries the rigidity of their components
plays virtually no role in tidal friction and tidal lagging. Consequently, it
has virtually no influence on tidal despinning, or tidal heating, or tidally
induced orbital evolution. As we shall see, a parameter that overwhelmingly
determines the tidal torques (and therefore dissipation, as well as orbit
changes) is a product of the effective viscosity $\,\eta\,$ by the tidal
frequency $\,\chi\,$.

##  2 Notation and formalism

Table 1: Symbol key  Notation | Description
---|---
$\theta$ | the rotation angle of the body
$R$ | the radius of the body
${\cal{T}}^{\rm{{}^{\,(TIDE)}}}_{z}$ | the polar component of the tidal torque acting on the body
$r$ | the instantaneous distance between the body and the perturber
$a$ | the semimajor axis
$e$ | the orbital eccentricity
$i$ | the orbital inclination (obliquity)
$\Omega$ | the longitude of the node
$\omega$ | the argument of the pericentre
${\cal{M}}$ | the mean anomaly
$n$ | the mean motion
$G$ | Newton’s gravitational constant
$\omega_{\textstyle{{}_{lmpq}}}$ | a tidal Fourier mode
$\chi_{\textstyle{{}_{lmpq}}}$ | a forcing tidal frequency
$\chi$ | a shorter notation for $\,\chi_{\textstyle{{}_{lmpq}}}\,$
$\tau_{{}_{M}}$ | the viscoelastic characteristic time (Maxwell time)
$\tau_{{}_{A}}$ | the inelastic characteristic time (Andrade time)
$\eta$ | the viscosity
$\mu$ | the unrelaxed rigidity modulus
$J$ | the unrelaxed compliance
$\alpha$ | the Andrade parameter

Referring the reader to Appendix A for more detail, here we provide basic
formulae to be used. In Table 1 and thereafter, the word body  will mean: the
tidally perturbed body . Its companion will be termed as the perturber .
Evidently, their roles are always interchangeable, and the overall dynamics is
defined by tides in both companions.

###  2.1 Modes and frequencies

Both the perturbing potential and the response (tidal distortion of the
perturbed body and the resulting additional tidal potential) can be expanded
over the Fourier modes

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

parameterised with four integers $\,l$, $m$, $p$, $q\,$ (Kaula 1964). Here
$\,\theta\,$ and $\,{\bf{\dot{\theta\,}}}\,$ are the rotation angle and
rotation rate of the tidally perturbed body, while $\,\omega\,$ and
$\,\Omega\,$ are the perturber’s argument of the pericentre and the longitude
of the node, as seen from the perturbed body. The “anomalistic” mean motion
$\,n\,$ is defined  through $\,n\equiv{\bf{\dot{\cal{M}}}}\,$ (with
$\,{\cal{M}}=\,0\,$ in the pericentre). The modes
$\,\omega_{\textstyle{{}_{lmpq}}}\,$ can assume either sign, while the
physical forcing frequencies in the distorted body are positive definite:

$\displaystyle\chi_{\textstyle{{}_{lmpq}}}\,=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|~{}\approx~{}|\,(l-2p+q)\;n\,-\,m\;\dot{\theta}\,|\,~{},$
(3)

so

$\displaystyle\omega_{\textstyle{{}_{lmpq}}}~{}=~{}\chi_{\textstyle{{}_{lmpq}}}~{}\,\mbox{Sgn}\,(\omega_{\textstyle{{}_{lmpq}}})~{}~{}.$
(4)

###  2.2 The despinning tidal torque

Development of the polar torque can be found, e.g., in Efroimsky (2012 a). The
torque contains both an oscillating and a secular part, the latter looking as

$\displaystyle\langle\,{\cal{T}}_{z}^{\rm{{}^{\,(TIDE)}}}\rangle\;=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad~{}$
(5)
$\displaystyle\,2\,G\,M^{{{\,2}}}\sum_{{\it{l}}=2}^{\infty}\frac{R^{\textstyle{{}^{2l\,+\,1}}}}{a^{\textstyle{{}^{2l\,+\,2}}}}\sum_{m=0}^{l}\frac{(l-m)!}{(l+m)!}\;m\sum_{p=0}^{l}F^{\,2}_{lmp}(i)\sum^{\it\infty}_{q=-\infty}G^{\,2}_{lpq}(e)\;k_{l}(\omega_{\textstyle{{}_{lmpq}}})\;\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,~{}.~{}$

Here $\,G\,$ is Newton’s gravity constant; $\,M\,$ is the mass of the
perturber; $\,a,\,i,\,e\,$ are the semimajor axis, inclination, and
eccentricity; and the angular brackets $\,\langle\,.\,.\,.\,\rangle\,$ denote
time averaging. The notations $\,F_{lmp}(i)\,$ and $\,G_{lpq}(e)\,$ signify
the inclination functions and the eccentricity polynomials. The degree-$l\,$
Love numbers $\,k_{l}(\omega_{lmpq})\,$ and phase lags
$\,\epsilon_{\textstyle{{}_{l}}}(\omega_{lmpq})\,$ are functions of the
Fourier tidal modes (2) – see Appendix A.

In the series (5), some terms are negative (despinning) and some are positive
(spinning-up). Usually, the leading term is despinning and so is the overall
torque. This, however, is not obligatory. For example, when the initial spin
is prograde but slower than the mean motion, or when the initial spin is
retrograde, the tidal torque acts to increase the spin rate in the prograde
sense (Noyelles et al. 2014). Also mind that capture into spin-orbit
resonances takes place due to a combined action of the tidal torque and a
torque caused by the permanent triaxiality. Entrapment into higher than 1:1
spin-orbit resonances is more probable for tighter binaries and higher
eccentricities.

###  2.3 The orbital tidal torque

The orbital tidal torque acting on the perturber is opposite to the torque (5)
wherewith the perturber despins or spins up the tidally distorted body. This
indicates that the products
$\,k_{l}/Q_{l}\,=\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})\,\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$
emerging in (5) must be key players not only in the spin evolution but also in
orbital evolution.

In practice, orbital calculations employ not the orbital torque, but the tidal
potential. As distinct from the torque, the potential carries the factors
$\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})\,\cos\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$.
Recall, however, that the potential is inserted into the Lagrange- or
Delaunay-type planetary equations. There it is differentiated with respect to
orbital elements, and we once again end up with 333 In the planetary
equations, we also get small terms with
$\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})\,\cos\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$.
They emerge because the tidally deformed body is no longer a perfect sphere,
and its mass cannot be assumed to be concentrated at its centre. As a result
of this, the anomalistic mean motion
$\,n\equiv\,\stackrel{{\scriptstyle\bf{\centerdot}}}{{\cal{M\,}}}$ slightly
exceeds the Keplerian mean motion $\,\sqrt{G\,(M+m)/a^{3}\,}\,$. The terms
with
$\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})\,\cos\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$
do not show up in the planetary equations for $\,a\,$, $\,e\,$, $\,i\,$, and
cause only small secular changes in $\,{\cal{M}}\,$, $\,\omega\,$ and
$\,\Omega\,$.
$\;k_{l}(\omega_{\textstyle{{}_{lmpq}}})\;\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$.

Although our paper addresses
$\,k_{l}/Q_{l}\,=\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})\,\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$,
for completeness we also provide in Appendix A.6 an expression for
$\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})\,\cos\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$,
for the Maxwell body and for some other rheologies.

##  3 Formalism to describe rheology

As explained in Appendix A, in an $\,lmpq\,$ term of the torque we can switch
from the Fourier mode $\,\omega_{\textstyle{{}_{lmpq}}}\,$ to the physical
frequency
$\,\chi_{\textstyle{{}_{lmpq}}}\,=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\;\,$:

$\displaystyle
k_{\textstyle{{}_{l}}}(\omega_{{}_{lmpq}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\omega_{{}_{lmpq}})\,=\,k_{\textstyle{{}_{l}}}(\chi_{{}_{lmpq}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi_{{}_{lmpq}})\;\mbox{Sgn}\,\omega_{{}_{lmpq}}\quad.$
(6)

The functional form of the dependence of
$\,k_{\textstyle{{}_{l}}}(\chi)~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi)\,$
upon the frequency $\,\chi=\chi_{\textstyle{{}_{lmpq}}}\,$ is defined by self-
gravitation of the body and by its rheological properties.

It is common in the literature to denote
$\,\sin\epsilon_{\textstyle{{}_{l}}}\,$ with $\,1/Q_{\textstyle{{}_{l}}}\,$,
and to term the so-defined quantities $\,Q_{\textstyle{{}_{l}}}\,$ as the
tidal quality factors:

$\displaystyle\frac{1}{Q_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})}~{}=~{}\sin|\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})|~{}=\sin\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}~{}.$
(7)

In this notation,

$\displaystyle
k_{\textstyle{{}_{l}}}(\omega_{{}_{lmpq}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\omega_{{}_{lmpq}})\;=\,~{}\frac{\textstyle
k_{\textstyle{{}_{l}}}(\chi_{{}_{lmpq}})}{Q_{\textstyle{{}_{l}}}(\chi_{{}_{lmpq}})}~{}\,\mbox{Sgn}\,\omega_{{}_{lmpq}}\quad.$
(8)

###  3.1 Constitutive equation

Rheological properties of a material are encoded in a constitutive equation
interconnecting the present-time deviatoric strain tensor
$\,u_{\gamma\nu}(t)\,$ with the values that have been assumed by the
deviatoric stress $\,{\sigma}_{\gamma\nu}(t\,^{\prime})\,$ over the time
$\,t\,^{\prime}\,\leq\,t\,$. Under linear deformation, the equation has the
form of convolution, in the time domain:

$\displaystyle
2\,u_{\gamma\nu}(t)\,=\,\hat{J}(t)~{}\sigma_{\gamma\nu}\,=\,\int^{t}_{-\infty}\stackrel{{\scriptstyle\;\centerdot}}{{J}}(t-t\,^{\prime})~{}{\sigma}_{\gamma\nu}(t\,^{\prime})\,dt\,^{\prime}~{}~{},~{}~{}~{}$
(9)

and the form of product, in the frequency domain:

$\displaystyle
2\;\bar{u}_{\gamma\nu}(\chi)\,=\;\bar{J}(\chi)\;\bar{\sigma}_{\gamma\nu}(\chi)\;\;.$
(10)

Here $\,\bar{u}_{\gamma\nu}(\chi)\,$ and $\,\bar{\sigma}_{\gamma\nu}(\chi)\,$
are the Fourier images of strain and stress, while the complex compliance
$\,\bar{J}(\chi)\,$ is a Fourier image of the kernel
$\,\dot{J}(t-t\,^{\prime})\,$ of the integral operator (9). See, e.g.,
Efroimsky (2012 a, b) for details.

###  3.2 Solids. The Maxwell model and its generalisation

At low frequencies, deformation of most solids is viscoelastic and obeys the
Maxwell model. This 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 illustrates a situation where
a deviatoric stress $\,{\mathbb{S}}\,$ generates a deviatoric strain
comprising a purely elastic part
$\,\stackrel{{\scriptstyle(e)}}{{\mathbb{U}}}\,$ and a purely viscous part
$\,\stackrel{{\scriptstyle(v)}}{{\mathbb{U}}}\;$:

$\displaystyle{\mathbb{U}}\,=\;\stackrel{{\scriptstyle(e)}}{{\mathbb{U}}}\,+\,\stackrel{{\scriptstyle(v)}}{{\mathbb{U}}}~{},\,\quad\mbox{where}\,\quad\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}\,=\,2\,\mu\,\stackrel{{\scriptstyle(e)}}{{\mathbb{U}}}~{}~{}~{}~{}\mbox{and}\quad~{}\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\,=\,2\,\eta\,\frac{\partial\,}{\partial
t}~{}\stackrel{{\scriptstyle(v)}}{{\mathbb{U}}}~{}~{}~{},$ (11)

$\,\eta\,$ and $\,\mu\,$ denoting the viscosity and unrelaxed rigidity.

As in the Maxwell regime both parts of the strain are generated by the same
stress

$\displaystyle{\mathbb{S}}\;=\;\stackrel{{\scriptstyle(v)}}{{\mathbb{S}}}\,=\,\stackrel{{\scriptstyle(e)}}{{\mathbb{S}}}~{}~{}~{},$
(12)

the formulae (11) can be assembled into one equation:

$\displaystyle\stackrel{{\scriptstyle\centerdot}}{{\mathbb{U}}}\,=\,\frac{1}{2\,\mu}\;\stackrel{{\scriptstyle\centerdot}}{{\mathbb{S}}}\,+\,\frac{1}{2\,\eta}\;{\mathbb{S}}~{}~{}~{}$
(13a) or, equivalently:
$\displaystyle\stackrel{{\scriptstyle\centerdot}}{{\mathbb{S}}}\,+\;\frac{1\;}{\tau_{{}_{M}}}\,{\mathbb{S}}~{}=~{}2\,\mu\,\stackrel{{\scriptstyle\centerdot}}{{\mathbb{U}}}~{}~{}~{},$
(13b)

with the _Maxwell time_ introduced as

$\displaystyle\tau_{{}_{M}}\;\equiv\;\frac{\,\eta\,}{\,\mu\,}~{}~{}~{}.$ (14)

Comparing the equation (13) with the general expression (9) for the compliance
operator, we see that the kernel is a time derivative of the compliance
function

${}^{\textstyle{{}^{(Maxwell)}}}J(t\,-\,t\,^{\prime})\,=\,\left[\,J\,+\,\left(t\;-\;t\,^{\prime}\right)\;\frac{1}{\eta}\,\right]\;\Theta(t\,-\,t\,^{\prime})~{}~{}~{},$
(15)

$\Theta(t\,-\,t\,^{\prime})\,$ being the Heaviside step function, and $\,J\,$
being the unrelaxed compliance:

$\displaystyle J~{}\equiv~{}\frac{\,1\,}{\,\mu\,}\quad.$ (16)

In the frequency domain, the equation (13) can be cast into the form (10),
with the complex compliance given by

${}^{\textstyle{{}^{(Maxwell)}}}{\bar{\mathit{J\,}}}(\chi)~{}=~{}J\,-\,\frac{i}{\eta\chi}~{}=~{}J\,\left(\,1~{}-~{}\frac{i}{\chi\,\tau_{{}_{M}}}\right)\quad,$
(17)

the terms $\,J\,$ and $\,-\,{i}/(\eta\chi)\,$ giving the elastic and viscous
parts of deformation, correspondingly. Such a body becomes elastic at high and
viscous at low frequencies.

It is often convenient to employ the dimensionless versions of the compliance:

${}^{\textstyle{{}^{(Maxwell)}}}{\bar{\cal{J\,}}}(\chi)\,={{}^{\textstyle{{}^{(Maxwell)}}}{\bar{\mathit{J\,}}}(\chi)}\,J^{-1}\,=\,1\,-\,i\,(\chi\,\tau_{{}_{M}})^{-1}\,~{},~{}~{}$
(18)

and of its real and imaginary parts:

$\displaystyle{{}^{\textstyle{{}^{(Maxwell)}}}}\Re(\chi)$ $\displaystyle=$
$\displaystyle 1~{}~{},$ (19a)
$\displaystyle{{}^{\textstyle{{}^{(Maxwell)}}}}\Im(\chi)$ $\displaystyle=$
$\displaystyle-\,(\chi\,\tau_{{}_{M}})^{-1}~{}~{}.\qquad~{}$ (19b)

The Maxwell rheology is a special case of the Andrade rheology that comprises
inputs from elasticity, viscosity, and inelastic processes (mainly,
dislocation unjamming):

$\displaystyle{\bar{\mathit{J\,}}}(\chi)$ $\displaystyle=$ $\displaystyle
J\,+\,\beta\,(i\chi)^{-\alpha}\;\Gamma\,(1+\alpha)\,-\,\frac{i}{\eta\chi}$
(20a) $\displaystyle=$ $\displaystyle
J\,+\,\beta\,(i\chi)^{-\alpha}\;\Gamma\,(1+\alpha)\,-\,i\,J\,(\chi\,\tau_{{}_{M}})^{-1}~{}~{},$
(20b) where $\,\Gamma\,$ is the Gamma function, while $\,\alpha\,$ and
$\,\beta\,$ denote the dimensionless and dimensional Andrade parameters.

While conventional, the expressions (20a \- 20b) for the Andrade compliance
suffer an inconvenient feature, the fractional dimensions of the parameter
$\,\beta\,$. It was therefore suggested in Efroimsky (2012 a, b) to shape the
compliance into a more usable form

$\displaystyle{\bar{\mathit{J\,}}}(\chi)~{}=~{}J\,\left[\,1\,+\,(i\,\chi\,\tau_{{}_{A}})^{-\alpha}\;\Gamma\,(1+\alpha)~{}-~{}i~{}(\chi\,\tau_{{}_{M}})^{-1}\right]\,\;,~{}~{}~{}$
(20c)

with the parameter $\,\tau_{{}_{A}}\,$ introduced via

$\displaystyle\beta\,=\,J~{}\tau_{{}_{A}}^{-\alpha}~{}~{}.$ (21)

and named as the Andrade time . The compliance (20c) is identical to (20a) and
(20b), but is spared of the parameter of fractional dimensions.

In the dimensionless form, the Andrade complex compliance reads as:

$\displaystyle{\bar{\cal{J\,}}}(\chi)\equiv\frac{{\bar{\mathit{J\,}}}(\chi)}{J}~{}=~{}1\,+\,(i\,\chi\,\tau_{{}_{A}})^{-\alpha}\;\Gamma\,(1+\alpha)~{}-~{}i~{}(\chi\,\tau_{{}_{M}})^{-1}\quad,$
(22)

its real and complex parts being

$\displaystyle\Re(\chi)$ $\displaystyle\equiv$ $\displaystyle{\cal R}{\it
e}[\bar{\cal{J}}(\chi)]~{}=~{}\frac{{\cal R}{\it
e}[\bar{J}(\chi)]}{J}~{}=~{}\quad\qquad\qquad\qquad~{}$ (23) $\displaystyle=$
$\displaystyle
1\;+\;(\chi\tau_{{}_{A}})^{-\alpha}\;\cos\left(\,\frac{\alpha\,\pi}{2}\,\right)\;\Gamma(\alpha\,+\,1)~{}~{},\qquad$
$\displaystyle\Im(\chi)$ $\displaystyle\equiv$ $\displaystyle{\cal I}{\it
m}[\bar{\cal{J}}(\chi)]~{}=~{}\frac{{\cal I}{\it
m}[\bar{J}(\chi)]}{J}\quad\qquad\qquad\qquad\qquad\quad~{}$ (24)
$\displaystyle=$
$\displaystyle-\,(\chi\,\tau_{{}_{M}})^{-1}\,-\,(\chi\,\tau_{{}_{A}})^{-\alpha}\;\sin\left(\,\frac{\alpha\,\pi}{2}\,\right)\;\Gamma(\alpha\,+\,1)~{}~{}.\qquad~{}$

At frequencies below some threshold (Karato & Spetzler 1990, Eqn. 17),
dislocation unjamming becomes less efficient a process, and the mantle
response becomes purely viscoelastic. This means that the Andrade time
$\,\tau_{{}_{A}}\,$ rapidly increases (so the Andrade term in the formulae
(3.2) falls off rapidly) when the frequency
$\,\chi=\chi_{\textstyle{{}_{lmpq}}}\,$ goes below the threshold value. As a
result of this, at low frequencies  the complex compliance of solids
approaches that of a Maxwell body, and we obtain the formulae (17 \- 19).

Mind that in this section the rigidity and compliance assume their unrelaxed
values:

$\displaystyle\mu~{}=~{}\mu(0)\qquad\mbox{and}\qquad J~{}=~{}J(0)\quad.$ (25)

###  3.3 Rubble piles

Ample observational data on rotation of asteroids have led to a consensus that
many (possibly, most) asteroids are loose aggregates kept together by gravity
forces. Rheology of such objects should be borrowed from the recent studies on
mechanics of granular matter. Such studies render essentially nonlinear
constitutive equations which contain parameters other than rigidity or
viscosity. We however expect that averaging of such equations over a volume
much larger than a typical size of a granule should furnish linearisable
equations with effective  rigidity and viscosity and, possibly, other
effective rheological parameters. 444 Even if we are aware of the existence
and relevance of some rheological parameter, its exact physical meaning is
inseparable from context, i.e., from a particular way how the parameter enters
the constitutive equation. For example, the same couple of parameters,
rigidity and viscosity, enter the Maxwell and Kelvin-Voigt models in different
ways. Accordingly, the physical meaning of viscosity within these two models
is different. Likewise, different is the meaning of rigidity in these models.
Enough to say, the viscous limit is approached through
$\,\mu\rightarrow\infty\,$, in the Maxwell model, and through
$\,\mu\rightarrow 0\,$, in the Kelvin-Voigt model.
In their often cited elegant work, Goldreich and Sari (2009) calculated the
effective rigidity of a granular aggregate. Although a big step forward, that
project still requires continuation. It is necessary to carry out a similar
estimate for the effective viscosity of such a medium. An even bigger
challenge will be to study how exactly these two (and, possibly, other)
parameters enter the constitutive equation. We leave this part of work for
future.

In this paper, our goal is simply to demonstrate that tidal evolution of an
asteroidal binary is parameterised not only by its components’ rigidity, but
also by other rheological parameters – like the viscosity.

Partial justification of treating rubble as a viscoelastic material comes from
numerical modeling – see, e.g., the work by Walsh et al. (2008) who studied
both monodisperse (same-size) piles and simple bimodal distributions (those
containing two different sizes of particles). It was found that monodisperse
aggregates behave in a manner distant from fluid, while mixing of spheres of
different sizes yields a closer-to-fluid behaviour. Within a somewhat
different model by Tanga et al. (2009), even monodisperse piles demonstrated
hydrodynamical behaviour. At the same time, simulated rubble piles did not
behave like perfect fluids, due to their ability to sustain shear stress.
(Also see the experimental study by Murdoch et al. 2013.) This means that the
aggregates have a finite shear rigidity $\,\mu\,$ and may be regarded
viscoelastic.

##  4 Why we need complex Love numbers,
not just $\,k_{l}\,$ and $\,Q\,$

It has long become common in the astronomical literature to introduce,
separately, a degree-$l\,$ Love number $\,k_{l}\,$ and the corresponding
quality factor $\,Q\,$. A minor flaw in this tradition is that, fundamentally,
the inverse of $\,Q\,$ is the sine of the degree-$l\,$ phase lag:
$\,1/Q\,=\,\sin\epsilon_{l}\,$, so we better equip it with an appropriate
subscript: $\,Q_{l}\,$. A bigger inconsistency is that $\,k_{l}\,$ and
$\,Q_{l}\,$ cannot be introduced independently, because they both are
determined by self-gravitation and rheology of the body. As we shall see
below, this circumstance becomes crucial in description of spin dynamics of
asteroids. The only path to consistent treatment lies in introducing complex
Love numbers.

###  4.1 Static tide. Love numbers

Let a homogeneous spherical incompressible primary of radius $\,R\,$ be
perturbed by a point-like exterior stationary secondary called perturber.
Introduce a spherical coordinate system associated with the primary. The
latitudes $\,\phi,\,\phi^{*}\,$ are reckoned from the primary’s equator, while
the longitudes $\,\lambda,\,\lambda^{*}\,$ are reckoned from a fixed meridian.
The perturber of mass $\,M\,$ is stationarily located in an exterior point
$\,{\mbox{{\boldmath$r$}}}^{~{}*}=(r^{*},\,\lambda^{*},\,\phi^{*})\,$.

Consider an arbitrary exterior point
$\,\mbox{{\boldmath$r$}}=(r,\lambda,\phi)\,$ and the surface point
$\,\mbox{{\boldmath${R}$}}=(R,\lambda,\phi)\,$ located beneath it (so $\,R\leq
r,\,r^{*}\,$). In the surface point ${R}$ , the disturbing potential due to
the perturber located in $\,\mbox{{\boldmath$r$}}^{\,*}\,$ can be written as

$\displaystyle W(\mbox{{\boldmath${R}$}}\,,\,\mbox{{\boldmath$r$}}^{~{}*})$
$\displaystyle=$
$\displaystyle\sum_{{\it{l}}=2}^{\infty}~{}W_{\it{l}}(\mbox{{\boldmath${R}$}}\,,~{}\mbox{{\boldmath$r$}}^{~{}*})\quad,\quad\mbox{with}\quad
W_{\it{l}}(\mbox{{\boldmath${R}$}}\,,~{}\mbox{{\boldmath$r$}}^{~{}*})\propto
P_{\it{l}}(\cos\gamma)\quad,\quad$ (26)

$P_{\it{l}}(\cos\gamma)\,$ being the Legendre polynomials, and $\,\gamma\,$
being the angle between the vectors ${\mbox{{\boldmath$r$}}}^{\,*}$ and ${R}$
originating from the primary’s centre.

Likewise, the additional tidal potential
$\,U(\mbox{{\boldmath$r$}},\,\mbox{{\boldmath$r$}}^{\,*})\,$ in the exterior
point $r$ , due to the primary’s deformation caused by the perturber being in
$\,\mbox{{\boldmath$r$}}^{\,*}\,$, can be expanded as: 555 Since the points
$r$ and ${R}$ are located on the same ray pointing out of the centre of the
body, the angle between $\,\mbox{{\boldmath$r$}}^{\,*}\,$ and $r$ is equal to
the angle $\,\gamma\,$ between $\,\mbox{{\boldmath$r$}}^{\,*}\,$ and ${R}$ .

$\displaystyle U(\mbox{{\boldmath$r$}}\,,\,\mbox{{\boldmath$r$}}^{~{}*})$
$\displaystyle=$
$\displaystyle\sum_{{\it{l}}=2}^{\infty}~{}U_{\it{l}}(\mbox{{\boldmath$r$}}\,,~{}\mbox{{\boldmath$r$}}^{~{}*})\quad,\quad\mbox{with}\quad
U_{\it{l}}(\mbox{{\boldmath$r$}}\,,~{}\mbox{{\boldmath$r$}}^{~{}*})\propto
P_{\it{l}}(\cos\gamma)\quad.$ (27)

A degree-$l\,$ spherical component
$\,U_{l}(\mbox{{\boldmath$r$}},\,\mbox{{\boldmath$r$}}^{\,*})\,$ in the point
$r$ is proportional to the degree-$l\,$ spherical component
$\,W_{l}(\mbox{{\boldmath${R}$}}\,,\,\mbox{{\boldmath$r$}}^{\;*})\,$ of the
perturbing potential in ${R}$ :

$\displaystyle U_{\it
l}(\mbox{{\boldmath$r$}},\,\mbox{{\boldmath$r$}}^{\,*})=\left(\frac{R}{r}\right)^{l+1}{k}_{l}\;W_{\it{l}}(\mbox{{\boldmath${R}$}}\,,\,\mbox{{\boldmath$r$}}^{\;*})~{}~{}.$
(28)

Then the additional exterior tidal potential of the primary can be written
down as:

$\displaystyle
U(\mbox{{\boldmath$r$}},\,\mbox{{\boldmath$r$}}^{\,*})~{}=~{}\sum_{{\it
l}=2}^{\infty}~{}U_{{l}}(\mbox{{\boldmath$r$}},\,\mbox{{\boldmath$r$}}^{\,*})~{}=~{}\sum_{{\it
l}=2}^{\infty}\,\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}k_{l}\;\,W_{\it{l}}(\mbox{{\boldmath${R}$}}\,,\;\mbox{{\boldmath$r$}}^{\;*})\quad.\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(29)

The quantities $\,k_{l}\,$ are the static Love numbers. Solution of a
boundary-value problem for potentials, deformation, and stress renders the
following expression:

$\displaystyle k^{\textstyle{{}^{(static)}}}_{\it l}\,=\;\frac{3}{2\,({\it
l}\,-\,1)}\;\,\frac{1}{1\;+\;{\cal{A}}^{\textstyle{{}^{(static)}}}_{l}}~{}=~{}\frac{3}{2\,({\it
l}\,-\,1)}\;\,\frac{1}{1\;+\;{\cal{B}}_{l}\,\mu}~{}=~{}\frac{3}{2\,({\it
l}\,-\,1)}\;\,\frac{1}{1\;+\;{\cal{B}}_{l}/J}~{}~{}~{},\qquad$ (30)

where

$\displaystyle{\cal{B}}_{l}\,\equiv~{}\frac{\textstyle{(2\,{\it{l}}^{\,2}\,+\,4\,{\it{l}}\,+\,3)}}{\textstyle{{\it{l}}\,\mbox{g}\,\rho\,R}}~{}=\;\frac{\textstyle{3\;(2\,{\it{l}}^{\,2}\,+\,4\,{\it{l}}\,+\,3)}}{\textstyle{4\;{\it{l}}\,\pi\,G\,\rho^{2}\,R^{2}}}~{}\qquad,\quad$
(31)

and

$\displaystyle{\cal{A}}^{\textstyle{{}^{(static)}}}_{\it
l}\equiv~{}{\cal{B}}_{l}\,\mu~{}=~{}{{\cal{B}}_{l}}/{J}\quad.\qquad$ (32)

In the above formulae, $\,\rho\,$, g , and $\,R\,$ are the density, surface
gravity, and radius of the body, while $\,G\,$ signifies the gravitational
constant. Since static  implies: relaxed, then the rigidity modulus $\,\mu\,$
and the compliance $\,J\equiv 1/\mu\,$ assume their relaxed  values:

$\displaystyle\mu~{}=~{}\mu(\infty)\qquad\mbox{and}\qquad
J~{}=~{}J(\infty)\quad,\qquad\mbox{in the static case only}.$ (33)

This will not be the case for evolving tide to be considered below. There,
$\,\mu\,$ and $\,J\,$ will assume their unrelaxed values – like in Section
3.2.

To draw this section to a close, we would mention that in the denominator of
(30) the term $\,1\,$ comes from self-gravitation, while the term
$\,{\cal{A}}^{\textstyle{{}^{(static)}}}_{l}\,$ originates owing to rheology.
This way, $\,{\cal{A}}^{\textstyle{{}^{(static)}}}_{l}\,$ serves as a
dimensionless measure of strength-dominated versus gravity-dominated static
behaviour; see Section 5.2.1 below.

###  4.2 Evolving tide. Love operators and complex Love numbers

In the case of a perturber “feeling” the tide it itself creates, we set
$\,\mbox{{\boldmath$r$}}=\mbox{{\boldmath$r$}}^{\,*}\,$. This enables us to
use a shorter notation $\,U(\mbox{{\boldmath$r$}})\,$ instead of
$\,U(\mbox{{\boldmath$r$}},\,\mbox{{\boldmath$r$}}^{\,*})\,$. When the tide is
evolving, the time-dependence has to be introduced, both in the overall tidal
potential $\,U(\mbox{{\boldmath$r$}},\,t)\,$ and in its Legendre components
$\,U_{l}(\mbox{{\boldmath$r$}},\,t)\,$.

Under evolving tidal disturbance, the expression (29) becomes a linear
operator (called the Love operator):

$\displaystyle
U_{l}(\mbox{{\boldmath$r$}},\,t)\;=\;\left(\frac{R}{r}\right)^{l+1}\int_{-\infty}^{t}{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})~{}W_{l}(\mbox{{\boldmath${R}$}}\,,\;\mbox{{\boldmath$r$}}^{\;*},\;t\,^{\prime})\,dt\,^{\prime}~{},$
(34)

where the Love functions $\,k_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$ are
analogues to the compliance functions $\,J(t-t\,^{\prime})\,$ introduced in
the equation (9). Overdot stands for time derivative.

In the frequency domain, the convolution (34) becomes 666 The equation (35) is
simplified, in that a more accurate treatment would render the spectral
components as functions of the tidal mode
$\,\omega=\omega_{\textstyle{{}_{lmpq}}}\,$ and not of the physical frequency
$\,\chi=\chi_{\textstyle{{}_{lmpq}}}=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,$.
Fortunately, though, the simplification is affordable. As explained in
Appendix A.4 below, the employment of the positive definite physical
frequencies instead of the tidal modes is legitimate, insofar as we do not
forget to include the sign multiplier Sgn$\,\omega_{\textstyle{{}_{lmpq}}}\,$
into an $\,lmpq\,$ term of the expansion for the torque. With this stipulation
in mind, in this article we shall always expand over $\,\chi\,$.

$\displaystyle\bar{U}_{\textstyle{{}_{l}}}(\chi)\;=\;\left(\,\frac{R}{r}\,\right)^{l+1}\bar{k}_{\textstyle{{}_{l}}}(\chi)\;\,\bar{W}_{\textstyle{{}_{l}}}(\chi)\;\;\;,$
(35)

with $\,\chi\,$ being the frequency. Here
$\,\bar{U}_{\textstyle{{}_{l}}}(\chi)\,$ and
$\,\bar{W}_{\textstyle{{}_{l}}}(\chi)\,$ are the Fourier or Laplace components
of the potentials $\,{U}_{\textstyle{{}_{l}}}(t)\,$ and
$\,{W}_{\textstyle{{}_{l}}}(t)\,$, while the complex Love numbers
$\,\bar{k}_{\textstyle{{}_{l}}}(\chi)\,$ are the Fourier or Laplace components
of the kernels $\,{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$.

The frequency-dependencies $\,\bar{k}_{\textstyle{{}_{l}}}(\chi)\,$ should be
derived from the expression for the complex compliance $\,\bar{J}(\chi)\,$ or
from its inverse, the complex rigidity $\,\bar{\mu}(\chi)=1/\bar{J}(\chi)\,$.
Into that derivation, also the size and mass of the body will enter. So the
tidal response will be defined by both rheology and gravity.

Referring the reader to Efroimsky (2012a,b) and the literature cited therein,
we state that for viscoelastic bodies the complex Love number
$\,\bar{k}_{l}(\chi)\,$ is related to the complex rigidity
$\,\bar{\mu}(\chi)\,$ by the same algebraic expression through which the
static Love number $\,{k}_{l}\,$ is related to the relaxed rigidity
$\,{\mu}(\infty)\,$. Equivalently, one can say that the complex Love number
$\,\bar{k}_{l}(\chi)\,$ is related to the complex compliance
$\,\bar{J}(\chi)\equiv 1/\bar{\mu}(\chi)\,$ by the same algebraic expression
through which the static Love number $\,{k}_{l}\,$ is related to the relaxed
compliance $\,{J}(\infty)\equiv 1/{\mu}(\infty)\;$: 777 This algebraic
similarity is but one manifestation of a more general theorem known as the
correspondence principle  or the elastic-viscoelastic analogy . Although the
discovery of this principle is sometimes attributed to Biot (1954), an early
version thereof can be found in the cornerstone work by Sir George Darwin
(1879).

$\displaystyle\bar{k}_{\it l}(\chi)~{}=~{}\frac{3}{2\,({\it
l}\,-\,1)}\;\,\frac{\textstyle 1}{\textstyle
1\;+\;{\cal{B}}_{l}\;\bar{\mu}(\chi)}~{}=~{}\frac{3}{2\,({\it
l}\,-\,1)}\;\,\frac{\textstyle 1}{\textstyle
1\;+\;{\cal{B}}_{l}/\bar{J}(\chi)}~{}~{}~{}.~{}\quad~{}$ (36)

Writing the degree-$l\,$ complex Love number as

$\displaystyle\bar{k}_{\it{l}}(\chi)\;=\;{\cal{R}}{\it{e}}\left[\bar{k}_{\it{l}}(\chi)\right]\;+\;{\it
i}\;{\cal{I}}{\it{m}}\left[\bar{k}_{\it{l}}(\chi)\right]\;=\;|\bar{k}_{\it{l}}(\chi)|\;e^{\textstyle{{}^{-{\it
i}\epsilon_{\it l}(\chi)}}}$ (37)

we conventionally denote the phase as $\,-\,\epsilon_{l}\,$, with a “minus”
sign. This convention imparts $\,\epsilon_{l}\,$ with the meaning of phase
lag, as can be understood from the relation (35). Equating the absolute values
of the right- and left-hand sides of the relation (35), we make another useful
observation: the role of the dynamical Love number is played by the absolute
value of the complex Love number:

$\displaystyle{k}_{l}~{}=~{}|\bar{k}_{l}(\chi)|\quad.$ (38)

Finally, we notice that

$\displaystyle|\bar{k}_{\it{l}}(\chi)|\;\sin\epsilon_{l}(\chi)\;=\;-\;{\cal{I}}{\it{m}}\left[\,\bar{k}_{\it{l}}(\chi)\,\right]\;\;\;,$
(39)

whence, using the expression (36), we arrive at

$\displaystyle\bar{k}_{\it{l}}(\chi)\;\sin\epsilon_{l}(\chi)~{}=~{}-~{}\frac{3}{2(l-1)}~{}\frac{{\cal{B}}_{\textstyle{{}_{l}}}\;{\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]}{\left({\cal{R}}{\it{e}}\left[\bar{J}(\chi)\right]+{\cal{B}}_{\textstyle{{}_{l}}}\right)^{2}+\left({\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]\right)^{2}}$
(40a) where $\,k_{l}(\chi)\equiv|\bar{k}_{\it{l}}(\chi)|\,$ and, as ever,
$\,\chi\,$ is a shortened notation for $\,\chi_{\textstyle{{}_{lmpq}}}\,$.

We see that the product
$~{}k_{l}(\chi_{\textstyle{{}_{lmpq}}})/Q_{l}(\chi_{\textstyle{{}_{lmpq}}})\,\equiv\,k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})\;$
standing in an $\,lmpq\,$ term of the expansion for the tidal torque can be
calculated consistently through the formulae (36 \- 40a), with the
coefficients $\,{\cal{B}}_{l}\,$ rendered by (31). For a homogeneous
incompressible sphere, the only information needed to calculate this product
is: the radius, the density, and the rheological law $\,\bar{J}(\chi)\,$.

The above expression can be cast into the form of

$\displaystyle\bar{k}_{\it{l}}(\chi)\;\sin\epsilon_{l}(\chi)=~{}-~{}\frac{3}{2(l-1)}~{}\;\frac{{\cal{A}}_{\textstyle{{}_{l}}}\;\Im(\chi)}{\left(\,\Re(\chi)\,+\,{\cal{A}}_{\textstyle{{}_{l}}}\,\right)^{2}\,+\,\Im(\chi)^{\,2}}\quad,\qquad$
(40b)

where

$\displaystyle\Re(\chi)\equiv{\cal{R}}{\it{e}}\left[\bar{J}(\chi)\right]/J\quad,\qquad\,\Im(\chi)\equiv{\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]/J\quad,$
(41)

are the real and imaginary components of the dimensionless  complex
compliance, while

$\displaystyle{\cal{A}}_{l}\,\equiv\,{\cal{B}}_{l}\,\mu\,=\,{\cal{B}}_{l}/J$
(42)

is the dimensionless rigidity. The dimensionless rigidity $\,{\cal{A}}_{l}\,$
relates to $\,{\cal{B}}_{l}\,$ in the same way as it used to do in the static
case, equation (32). Mind, though, that now $\,\mu\,$ and $\,J\,$ denote the
unrelaxed  values of the rigidity and compliance, not the relaxed ones as in
the static problem. So the values of $\,{\cal{A}}_{l}\,$ are different from
$\,{\cal{A}}^{\textstyle{{}^{(static)}}}_{l}\,$.

While the product
$~{}k_{l}(\chi_{\textstyle{{}_{lmpq}}})/Q_{l}(\chi_{\textstyle{{}_{lmpq}}})\,\equiv\,k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})\;$
shows up in an $\,lmpq\,$ term of the tidal torque, a product

$\displaystyle
k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\cos\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})\;=\;{\cal{R}}{\it{e}}\left[\bar{k}_{l}(\chi_{\textstyle{{}_{lmpq}}})\right]$
(43)

emerges in an $\,lmpq\,$ term of the expansion for the tidal potential. For
reference purposes, we provide in Appendix A.6 a general expression for this
product, as well as its versions for the Maxwell and other rheologies.

Do draw this section to a close, it would not hurt to mention that the
expression (36), as well as the ensuing formulae (4.2), are inapplicable to
fluid celestial bodies. 888 Tidal response of fluid objects (like Jupiters or
stars) cannot  be treated as a viscous limit of the Maxwell rheology.
Dissipation in these bodies is ruled by the equations of hydrodynamics, with
turbulence taken into account (Ogilvie & Lin 2004; Remus et al. 2012; Auclair-
Desrotour et al. 2015).

###  4.3 The tidal lag versus the “seismic” lag

From the expression (36), we obtain the actual dynamical Love numbers:

$\displaystyle{k}_{l}(\chi)\,\equiv\;|\,\bar{k}_{l}(\chi)\,|$ $\displaystyle=$
$\displaystyle\frac{3}{2\,({\it
l}\,-\,1)}\;\sqrt{\frac{\Re(\chi)^{2}+\Im(\chi)^{2}}{\left(\,\Re(\chi)+{\cal{A}}_{\textstyle{{}_{l}}}\right)^{2}+\Im(\chi)^{2}}}~{}~{}~{}.~{}\quad~{}$
(44)

The ratio of the formulae (40b) and (44) gives us the inverse tidal quality
factors:

$\displaystyle
Q_{l}(\chi)^{-1}\;\equiv\;\sin\epsilon_{l}(\chi)\;=\;-\;\frac{{\cal{A}}_{\textstyle{{}_{l}}}\,\Im(\chi)}{\sqrt{\Re(\chi)^{2}+\Im(\chi)^{2}\,}~{}\sqrt{\left(\,\Re(\chi)+{\cal{A}}_{\textstyle{{}_{l}}}\right)^{2}+\Im(\chi)^{2}}\,}$
(45)

The tidal quality factors should not be confused with the “seismic” quality
factor whose inverse is the sine of the phase lag $\,\delta\,$ showing up in
the rheological law:

$\displaystyle
Q(\chi)^{-1}\;\equiv\;\sin\delta(\chi)~{}=~{}-\;\frac{\Im(\chi)}{\sqrt{\Re(\chi)^{2}+\Im(\chi)^{2}\,}}\quad.$
(46)

The former and latter formulae interrelate the tidal and “seismic” phase lags:

$\displaystyle\sin\epsilon_{l}(\chi)\;=\;\frac{{\cal{A}}_{\textstyle{{}_{l}}}}{\sqrt{\left(\,\Re(\chi)+{\cal{A}}_{\textstyle{{}_{l}}}\right)^{2}+\Im(\chi)^{2}}\,}~{}\sin\delta(\chi)\quad.$
(47)

To the best of our knowledge, this relation has never appeared in the
literature hitherto.

As one would expect, the lags coincide when rheology “beats” gravity – i.e.,
when
$\,{\cal{A}}_{\textstyle{{}_{l}}}\,\gg\,|\,\Re(\chi)\,|\,,~{}\,|\,\Im(\chi)\,|\,$.
Otherwise, the tidal phase lags deviate considerably from the nominal
“seismic” lag $\,\delta\,$. This happens on approach to a spin-orbit
resonance, i.e., to a zero frequency. While the “seismic” lag stays constant
(and approaches $\,\pi/2\,$, in the case of Maxwell body), the tidal lags
continuously go through zero – and so do the products
$\,k_{\textstyle{{}_{l}}}\,\sin\epsilon_{\textstyle{{}_{l}}}\,$, as can be
seen from Figure 1. In this figure, we depicted
$\,k_{\textstyle{{}_{l}}}\,\sin\epsilon_{\textstyle{{}_{l}}}\,$ not as a
function of the positive definite physical frequency
$\,\chi=\chi_{\textstyle{{}_{lmpq}}}\,$, but as a function of the Fourier mode
$\,\omega=\omega_{\textstyle{{}_{lmpq}}}\,$ that can have either sign. 999
Recall that
$\,\chi_{\textstyle{{}_{lmpq}}}=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,$. The
shape of this function is about the same for all linear rheologies. For a
Maxwell body, the extrema are located at

$\displaystyle\omega_{peak}\,=\;\pm\;\frac{1}{\tau_{{}_{M}}\,{\cal{B}}_{l}\;\mu}\;=\;\pm\;\frac{\textstyle{8\,\pi\,G\,\rho^{2}\,R^{2}}}{\textstyle{57\;\eta}}\quad.$
(48)

Figure 1:  A typical shape of the quality function
$\,k_{l}(\omega)\,\sin\epsilon_{l}(\omega)\,$, where $\,\omega\,$ is a
shortened notation for the tidal Fourier mode
$\,\omega_{\textstyle{{}_{lmpq}}}\,$. (From Noyelles et al. 2014.)

The peaks’ amplitude is virtually insensitive to our choice of the viscosity
$\,\eta\,$. At the same time, the spread between the extrema depends on
$\,\eta\,$. As we see from the formula (48), for a higher viscosity the peaks
are located close to the point of resonance. As the viscosity assumes smaller
values, the peak frequency spreads out, eventually reaching and bypassing the
orbital frequency $\,n\,$. In realistic settings, however, this would require
extremely low viscosities.

Outside the inter-peak interval, the function
$\,k_{l}(\omega)\,\sin\epsilon_{l}(\omega)\,$ behaves as the inverse $\,Q\,$
in seismology, and changes very slowly with frequency.

Between the peaks, the products $\,k_{l}(\omega)\,\sin\epsilon_{l}(\omega)\,$
are almost linear in $\,\omega=\omega_{\textstyle{{}_{lmpq}}}\,$, and go
continuously through zero. This enables the tidal torque to transcend spin-
orbit resonances continuously (Makarov & Efroimsky 2013, Makarov et al. 2012,
Noyelles et al. 2014).

It can be demonstrated that the linearity of $\,k_{l}\,\sin\epsilon_{l}\,$ in
$\,\omega\,$ is equivalent to the time lag being frequency independent:
$\,\Delta t_{l}(\omega_{\textstyle{{}_{lmpq}}})\,=\,\Delta t\,$, see Efroimsky
& Makarov (2013). 101010 Some authors call the constant-$\Delta t$ tidal model
viscous . This is not a good term, since in the viscous limit of the Maxwell
model we still arrive at a kink shape of
$\,k_{l}(\omega)\,\sin\epsilon_{l}(\omega)\,$, like in Figure 1. The CTL
(constant time lag) tidal model is a better term for this approach. This means
that the tidal response of an actual terrestrial body can be approximated with
the constant-$\Delta t\,$ model only when all considered tidal frequencies are
lower than $\,|\,\omega_{peak}\,|\;$ – or, roughly speaking, when all mean
motions and spin rates are lower than $\,|\,\omega_{peak}\,|\,$. To achieve
this, we would have to endow the viscosity with very low values. This
observation absolutely prohibits the application of the CTL (constant time
lag) tidal model to solid or semi-molten silicate planets (while for liquified
planets this entire formalism is not intended anyway).

##  5 Complex Love numbers for different rheologies

For homogeneous near-spherical objects, the formulae (4.2) provide a turn-of-
the-crank method. Inserting therein the rheology $\,\bar{J}(\chi)\,$, along
with the values of the radius and density, we obtain the functions
$~{}k_{l}(\chi)/Q_{l}(\chi)\,\equiv\,k_{\textstyle{{}_{l}}}(\chi)~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi)\;$,
with $\,\chi=\chi_{\textstyle{{}_{lmpq}}}\,$. The complex compliance
$\,\bar{J}(\chi)\,$ is a function of the tidal frequency and rheological
parameters. This is why the tidal torque is sensitive both to the frequency
and the rheological parameters (and, thereby, also to the temperature). A
similar conclusion is valid for the tidal heating rate (Henning et al. 2009,
Makarov & Efroimsky 2014).

###  5.1 Self-gravitating Maxwell sphere

Insertion of the complex compliance (17) into the formulae (36) and (4.2)
furnishes:

$\displaystyle\bar{k}_{\textstyle{{}_{l}}}(\chi)$ $\displaystyle=$
$\displaystyle\frac{3}{2(l-1)}\;\frac{1}{\;\textstyle
1\;+~{}\frac{\textstyle\tau_{\textstyle{{}_{M}}}\;{\it
i}\;\chi}{\textstyle\;1\;+\;\tau_{\textstyle{{}_{M}}}\;{\it
i}\;\chi\;}\;\,{\cal{B}}_{\textstyle{{}_{l}}}\,\mu\;}$ (49a) $\displaystyle=$
$\displaystyle\frac{3}{2(l-1)}\;\frac{1}{\;\textstyle
1\;+\;\frac{\textstyle\tau_{\textstyle{{}_{M}}}\;{\it i}\;\chi}{\textstyle
1\;+\;\tau_{\textstyle{{}_{M}}}\;{\it
i}\;\chi\;}~{}\,{\cal{A}}_{\textstyle{{}_{l}}}\;\;}$ (49b)

and

$\displaystyle\frac{k_{l}(\chi)}{Q_{l}(\chi)}\,\equiv\,k_{\textstyle{{}_{l}}}(\chi)~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi)$
$\displaystyle=$
$\displaystyle\frac{3}{2(l-1)}\;\frac{{\cal{B}}_{l}~{}\frac{\textstyle
1}{\textstyle\eta\,\chi}}{~{}(J\,+\,{\cal{B}}_{l})^{2}\,+~{}\frac{\textstyle
1}{\textstyle\eta^{2}\,\chi^{2}}~{}}$ (50a) $\displaystyle=$
$\displaystyle\frac{3}{2(l-1)}\;\frac{{\cal{A}}_{l}~{}\frac{\textstyle
1}{\textstyle\tau_{{}_{M}}\,\chi}}{~{}(1\,+\,{\cal{A}}_{l})^{2}\,+~{}\frac{\textstyle
1}{\textstyle\tau_{{}_{M}}^{2}\,\chi^{2}}~{}}\quad.\qquad~{}$ (50b)

Given by (31), the factors $\,{\cal{B}}_{l}\,$ are dimensional, while
$\,{\cal{A}}_{l}=\,{\cal{B}}_{l}\,\mu\,=\,{\cal{B}}_{l}/J\,$ are
dimensionless. Recall that, in distinction from the static case, here
$\,\mu\,$ and $\,J\equiv 1/\mu\,$ denote the unrelaxed  (not static) values of
the rigidity and compliance, wherefore our $\,{\cal{A}}_{l}\,$ should not be
misidentified with their counterparts
$\,{\cal{A}}^{\textstyle{{}^{(static)}}}_{l}$ from Section 4.1.

Generalisation of the formula (50) to the Andrade model can be found in
Efroimsky (2012 a, eqn. 170).

###  5.2 Three extreme cases

####  5.2.1 Self-gravitating drop of an ideal fluid. No rheology at all

In the “no-rheology” limit, the material becomes an ideal fluid wherein shear
deformation entails no stress. This implies:

$\displaystyle\eta\rightarrow 0\quad,\qquad\mu\rightarrow 0\quad,\qquad
J\equiv\frac{\,1\,}{\,\mu\,}\rightarrow\,\infty\quad,\qquad\qquad$ (51a)
$\displaystyle\mbox{wherefrom}\qquad\bar{J}(\chi)\rightarrow\infty\qquad\mbox{and}\qquad{\cal{A}}_{\textstyle{{}_{l}}}\rightarrow
0\quad,\qquad\qquad\qquad$ (51b)

though the real part of the dimensionless  compliance stays finite:
$\,\Re(\chi)~{}\rightarrow~{}1\,$. Thence,

$\displaystyle\bar{k}_{l}(\chi)~{}\rightarrow~{}\frac{3}{2(l-1)}\quad,$ (52)
$\displaystyle
k_{\textstyle{{}_{l}}}(\chi)~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi)~{}\rightarrow~{}0\quad.$
(53)

From (52) we see that in this limit the complex Love number becomes real. Due
to the absence of rheology as such, this value of the Love number originates
solely from self-gravitation. From (53) we understand that all terms in the
expression for torque become zero. No heat is produced either.

####  5.2.2 Self-gravitating elastic body

This limit is implemented through

$\displaystyle\tau_{{}_{M}}\rightarrow\infty\quad,\qquad\eta\rightarrow\infty\quad,\qquad\mbox{with}~{}\,J\,~{}\mbox{staying}~{}\,\mbox{finite}~{}~{}.\quad$
(54a) Thence the complex compliance becomes real:
$\displaystyle\bar{J}(\chi)~{}\rightarrow~{}J\quad.$ (54b)

This again renders the complex Love numbers real:

$\displaystyle\bar{k}_{\textstyle{{}_{l}}}(\chi)\;\rightarrow\;\frac{\textstyle
3}{\textstyle 2\,(l-1)}~{}\frac{\textstyle 1}{\textstyle
1\,+\,{\cal{B}}_{{}_{l}}\,\mu}~{}=~{}\frac{\textstyle 3}{\textstyle
2\,(l-1)}~{}\frac{\textstyle 1}{\textstyle 1\,+\,{\cal{A}}_{{}_{l}}}\quad,$
(55) $\displaystyle
k_{\textstyle{{}_{l}}}(\chi)~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi)~{}\rightarrow~{}0\quad.\qquad\qquad\qquad\qquad\qquad\qquad\quad$
(56)

Phase lags vanish, no tidal torque emerges, and no dissipation is taking
place.

####  5.2.3 Self-gravitating perfectly viscous body

In this situation, we have to get rid of rigidity by setting
$\,\mu\rightarrow\infty\,$ (yes, $\,\infty\,$) :

$\displaystyle\tau_{{}_{M}}\rightarrow
0\quad,\qquad\mu\rightarrow\infty\quad,\qquad
J\equiv\frac{\,1\,}{\,\mu\,}\rightarrow\,0\quad,\qquad\qquad$ (57)
$\displaystyle\mbox{wherefrom}\qquad\bar{J}(\chi)\rightarrow~{}-~{}\frac{i}{\eta\chi}\qquad\mbox{and}\qquad{\cal{A}}_{\textstyle{{}_{l}}}\rightarrow\infty\quad.\qquad\qquad\qquad$
(58)

In this limit, the expressions (49a) and (50a) become:

$\displaystyle\bar{k}_{\textstyle{{}_{l}}}(\chi)~{}\rightarrow~{}\frac{3}{2(l-1)}\;\frac{1}{\;\textstyle
1\;+~{}\textstyle
i~{}\eta~{}\chi\;\,{\cal{B}}_{\textstyle{{}_{l}}}\;}\quad,~{}\qquad~{}\qquad\qquad~{}\qquad~{}$
(59) $\displaystyle
k_{\textstyle{{}_{l}}}(\chi)~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi)~{}\rightarrow~{}\frac{3}{2(l-1)}~{}\frac{{\cal{B}}_{\textstyle{{}_{l}}}\;\frac{\textstyle
1}{\textstyle\eta\,\chi}}{{\cal{B}}_{\textstyle{{}_{l}}}^{\,2}~{}+~{}\frac{\textstyle
1}{\textstyle\eta^{2}\,\chi^{2}}}~{}=~{}~{}\frac{3}{2(l-1)}~{}\frac{{\cal{B}}_{\textstyle{{}_{l}}}\;{\textstyle\eta\,\chi}}{{\cal{B}}_{\textstyle{{}_{l}}}^{\,2}\,\eta^{2}\,\chi^{2}~{}+~{}1}\quad.$
(60)

It should however be noted that the viscous limit in these formulae cannot be
interpreted too literally. Recall that all our algebra rests on the
cornerstone fact that the complex $\,\bar{k}_{\textstyle{{}_{l}}}(\chi)\,$
relates to the complex $\,\bar{J}(\chi)\,$ in the same manner as their static
counterparts do. This is an approximation that stays valid insofar as the
accelerations and inertial forces can be neglected in the equation of motion
(see Efroimsky 2012 a and references therein). In the absence of rigidity, the
Coriolis force comes into the picture and entails complicated hydrodynamics
(Ogilvie & Lin 2004, Auclair-Desrotour et al. 2015). So the compliance $\,J\,$
cannot be set exactly zero. It should sooner be understood that the formulae
(59) and (60) pertain to a finite $\,J\ll\eta\chi\,$. If such highly viscous
objects exist in the world of small bodies, then the tidal torque (and
dissipation) for these objects is proportional not to the rigidity $\,\mu\,$
but to $\,\eta\chi\,$, as can be seen from the above expression.

##  6 Numbers

Given our sparse knowledge of the actual parameters and structure of asteroids
and comets, we are not sure how good approximation the Maxwell model is. We
shall employ it for the time being, due to the lack of anything better.

As for the parameters’ values, we only know that the mean density of most
asteroids is about $\,\rho=2\times 10^{3}\,$ kg/m${}^{3}\,$, up to a factor of
1.5 (Pravec & Harris 2007), while the density of comets is around $\,0.6\times
10^{3}\,$ kg/m${}^{3}\,$ (Britt et al. 2006).

###  6.1 Rigidity beats gravity – and goes out

Assuming the nominal values $\,\mu_{\textstyle{{}_{(rock)}}}(\infty)=5\times
10^{10}\,$ Pa and $\,\rho=3.5\times 10^{3}\,$ kg/m${}^{3}\,$, Goldreich & Sari
(2009, Section 3.1) obtain, for rubble piles:

$\displaystyle{\cal{A}}_{2}^{\textstyle{{}^{(static)}}}\propto~{}10^{4}\quad\mbox{for}\quad
R~{}\approx~{}10^{4}~{}\mbox{m}\quad,$ (61a)
$\displaystyle{\cal{A}}_{2}^{\textstyle{{}^{(static)}}}\propto~{}10^{2}\quad\mbox{for}\quad
R~{}\approx~{}10^{6}~{}\mbox{m}\quad.$ (61b)

Although the unrelaxed rigidity differs from its relaxed (static) counterpart,
for regular solids the difference is not dramatic. We presume that for rubble
piles, too, the said difference does not exceed two orders of magnitude. If
this presumption is right, we can accept, for the unrelaxed dimensionless
rigidity, that

$\displaystyle{\cal{A}}_{l}\,\gg\,1\quad,\quad\mbox{for}\quad
R\,\lesssim\,10^{6}~{}\mbox{m}~{}~{}.$ (62)

This should not come as a surprise, because $\,{\cal{A}}_{l}\,$ being larger
than unity implies that in small bodies “rigidity beats gravity”.

Now we can safely assume in the formulae (49 \- 50) that, for all asteroids
and comets (and for not too large satellites),
$\,1+{\cal{A}}_{l}\approx{\cal{A}}_{l}\,$ and, equivalently,
$\,J+{\cal{B}}_{l}\approx{\cal{B}}_{l}\,$. Hence the following approximation
for the squared Love number:

$\displaystyle
k_{\textstyle{{}_{l}}}^{2}(\chi)~{}=~{}\left[\,\frac{3}{2(l-1)}\,\right]^{2}\frac{1~{}+~{}\tau_{{}_{M}}^{2}\,\chi^{2}}{1~{}+~{}\tau_{{}_{M}}^{2}\,\chi^{2}~{}(1\,+\,{\cal{A}}_{l})^{2}}~{}\approx~{}\left[\,\frac{3}{2(l-1)}\,\right]^{2}\frac{1~{}+~{}\eta^{2}\,\chi^{2}\,J^{2}}{1~{}+~{}\eta^{2}\,\chi^{2}~{}{\cal{B}}_{l}^{2}}\quad,\quad$
(63)

and for $\,k_{l}/Q_{l}\;$:

$\displaystyle
k_{\textstyle{{}_{l}}}(\chi)~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi)~{}=~{}\frac{3}{2(l-1)}\;\frac{{\cal{B}}_{l}~{}\frac{\textstyle
1}{\textstyle\eta\,\chi}}{~{}(J\,+\,{\cal{B}}_{l})^{2}\,+~{}\frac{\textstyle
1}{\textstyle\eta^{2}\,\chi^{2}}~{}}~{}\approx~{}\frac{3}{2(l-1)}\;\frac{{\cal{B}}_{l}~{}\frac{\textstyle
1}{\textstyle\eta\,\chi}}{~{}{\cal{B}}_{l}^{2}\,+~{}\frac{\textstyle
1}{\textstyle\eta^{2}\,\chi^{2}}~{}}\quad.\qquad$ (64)

We see that the Love number becomes rigidity-independent for
$\,\eta\,\chi\,\ll\,\mu\,$. Moreover, our second entity, $\,k_{l}/Q_{l}\,$,
does not depend upon rigidity at all. 111111 We can use the equation (64) to
express $\,k_{l}/Q_{l}\,$ as a function of the dimensionless rigidity
$\,{\cal{A}}_{l}\;$:  $\displaystyle
k_{\textstyle{{}_{l}}}(\chi)~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi)~{}=~{}\frac{3}{2(l-1)}\;\frac{{\cal{A}}_{l}~{}\,\frac{\textstyle
1}{\textstyle\tau_{{}_{M}}\,\chi}}{~{}{\cal{A}}_{l}^{2}\,+~{}\frac{\textstyle
1}{\textstyle\tau_{{}_{M}}^{2}\,\chi^{2}}~{}}\quad.\qquad~{}$ However, since
$\,{\cal{A}}_{l}={\cal{B}}_{l}\,\mu\,$ and $\,\tau_{{}_{M}}=\eta/\mu\,$,
casting our expression into the above form will only camouflage the absence of
an actual dependence upon $\,\mu\,$ or $\,J\,$.

The latter fact warrants a seemingly paradoxical conclusion that the rigidity
of small bodies is too large to be relevant  for description of tidal lagging
in these bodies; too large – even with a reduction derived by Goldreich & Sari
(2009). In reality, this conclusion works also for monoliths, because in the
expression for $\,{\cal{A}}_{l}\,$ a higher rigidity of a solid rock is
partially compensated by its higher density squared. So the conclusion works
for all small bodies, including satellites, – provided they are (a) not too
large, so the inequalities (62) hold; and (b) may be regarded as Maxwell
bodies.

###  6.2 Viscosity versus self-gravitation. A practical criterion

Our next goal is to examine the denominators of the right-hand sides of the
expressions (63) and (64). In both denominators, it is important whether
$\,{\cal{B}}_{l}\,\chi\,\eta\,$ is much larger or much smaller than unity.
With the expression (31) for $\,{\cal{B}}_{l}\,$ kept in mind, this furnishes
us with the following criterion.

For tidal frequencies and effective viscosities obeying

$\displaystyle\frac{57}{8\,\pi}~{}\frac{\chi~{}\eta}{G\,\rho^{2}\,R^{2}}~{}\gg~{}1\quad,$
(65)

the expressions for $\,k_{l}\,$ and $\,k_{l}/Q_{l}\,$ read as: 121212
Evidently, for even higher viscosities, the Love numbers become simply
$\,k_{l}\approx\,\frac{\textstyle 3}{\textstyle 2(l-1)}~{}\frac{\textstyle
1}{\textstyle{\cal{A}}_{\textstyle{{}_{l}}}}~{}$.

$\displaystyle
k_{\textstyle{{}_{l}}}(\chi)~{}\approx~{}\frac{3}{2(l-1)}\;\frac{~{}\sqrt{1~{}+~{}\eta^{2}\,\chi^{2}\,J^{2}}~{}}{~{}{\cal{B}}_{l}~{}\eta\;\chi}\quad,\quad$
(66) $\displaystyle
k_{\textstyle{{}_{l}}}(\chi)~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi)~{}\approx~{}\frac{3}{2(l-1)}~{}\frac{\textstyle
1}{\textstyle{\cal{B}}_{l}\,\eta\,\chi}\quad.\qquad$ (67)

They are good everywhere except in the low-frequency limit. There, the
inequality (65) is no longer satisfied, and we have to switch from (67) to the
full expression (64).

If the product $\,\chi\,\eta\,$ assumes low values, we arrive at a situation
with 131313 The numerical factor on the left-hand side of (68) corresponds to
$\,l=2\,$. If higher degrees are included, the factor should be chosen as
$~{}3\,(2l^{2}+4l+3)/(4l\pi)~{}$ for the maximal $\,l\,$ taken into account.

$\displaystyle\frac{57}{8\,\pi}~{}\frac{\chi~{}\eta}{G\,\rho^{2}\,R^{2}}~{}\ll~{}1\quad,$
(68)

so the expressions of our concern become:

$\displaystyle
k_{\textstyle{{}_{l}}}(\chi)~{}\approx~{}\frac{3}{2(l-1)}\quad,\qquad$ (69)
$\displaystyle
k_{\textstyle{{}_{l}}}(\chi)~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi)~{}\approx~{}\frac{3}{2(l-1)}\;\,{\cal{B}}_{l}\;\eta\;\chi\quad.\qquad$
(70)

The latter expression can be used to explore tidal entrapment into spin-orbit
states.

##  7 Conclusions

In the article, we have shown that in asteroidal binaries the intensity of
tidal interactions (i.e., the tidal torques and the ensuing spin and orbit
evolution) depends primarily on the product of the tidal frequency $\,\chi\,$
by the viscosity $\,\eta\,$. Contrary to a long-entrenched opinion, the
rigidity $\,\mu\,$ exerts virtually no influence on tidal evolution.

We have developed a quantitative criterion that enabled us to separate two
types of the functional dependence of $\,k_{l}/Q_{l}\,$ upon the said product
$\,\chi\eta\,$. For higher values of this parameter, we obtained:
$\,k_{l}/Q\propto 1/(\eta\chi)\;$, see the equations (65 \- 67). For lower
values of $\,\chi\eta\,$, we got: $\,k_{l}/Q\propto\eta\chi\,$, see the
formulae (68 \- 70). The dependencies (67) and (70) determine the spin
histories, including tidal captures into spin-orbit resonances. 141414 Be
mindful that the approximation (67) cannot be used in the vicinity of a
resonance where $\,\chi\,$ vanishes. In that vicinity, one has to employ the
full expression (64) or its approximation (70).

Our study rests on an assumption that small bodies can be described with the
Maxwell model. The model is well applicable to rigid rocks at low frequencies,
so our conclusions are safely valid for monoliths. We apply this approach also
to rubble, for the lack of a better model so far. As we learn more about
mechanics of granular mixtures in a weak gravity field, we should be prepared
to amend the tidal theory with other rheological parameters, ones that do not
show up in viscoelastic mechanics.

It should be finally added that the old misbelief about the allegedly key role
of the rigidity has pervaded also the literature on inelastic relaxation of
precessing (wobbling) asteroids and comets (Prendergast 1958; Burns & Safronov
1973; Efroimsky 2001, 2002; Molina et al. 2003; Ryabova 2004; Sharma et al.
2005; Breiter et al. 2012; Breiter & Murawiecka 2015). In an upcoming work
(Frouard & Efroimsky 2015), we shall demonstrate that in the said setting,
too, the leading role belongs to viscosity.

## Acknowledgments

I wish to thank Konstantin Batygin, Julien Frouard, Valéry Lainey and Valeri
Makarov for our useful discussions on topics related to this project.

##  Appendix A  Appendix.

In this Appendix, we briefly summarise the basic facts from the theory of
bodily tides. For more detailed explanation, see Efroimsky & Makarov (2013)
and references therein.

###  A.1 Fourier modes and physical frequencies

As we mentioned in Section 2.1, both the potential from the perturber and the
additional tidal potential of the perturbed body can be expanded over the
Fourier modes

$\displaystyle\omega_{\textstyle{{}_{lmpq}}}~{}\approx~{}(l-2p+q)\;n\,-\,m\;\dot{\theta}\quad,\qquad$
(71)

with $\,n\,\equiv\;\stackrel{{\scriptstyle\bf\centerdot}}{{\cal{M\,}}}$ being
the “anomalistic” mean motion and $\,{\bf{\dot{\theta\,}}}\,$ being the spin
rate of the tidally perturbed body. These modes can assume either sign, while
the physical frequencies of stresses and strains in the perturbed body are
positive definite:

$\displaystyle\chi_{\textstyle{{}_{lmpq}}}\,=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|~{}\approx~{}|\,(l-2p+q)\;n\,-\,m\;\dot{\theta}\,|\,~{}.$
(72)

###  A.2 Phase lags and dynamical Love numbers

The tidal reaction always lags in time, compared to action. Each Fourier mode
$\,\omega_{\textstyle{{}_{lmpq}}}\,$ has a time lag of its own, $\,\Delta
t_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$. Owing to causality, time lags are
positive definite:

$\displaystyle\Delta t_{l}(\omega_{\textstyle{{}_{lmpq}}})\;>\;0\quad.$ (73)

For physical reasons, the time lags are defined by the forcing frequencies
$\,\chi_{\textstyle{{}_{lmpq}}}\equiv\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,$,
and are ignorant of the sign of the modes
$\,\omega_{\textstyle{{}_{lmpq}}}\;$:

$\displaystyle\Delta t_{l}(\,-\,\omega_{\textstyle{{}_{lmpq}}})\;=\;\Delta
t_{l}(\omega_{\textstyle{{}_{lmpq}}})\;=\;\Delta
t_{l}(\chi_{\textstyle{{}_{lmpq}}})\quad.$ (74)

In the Darwin-Kaula theory of tides, the phase lags are products of the modes
$\,\omega_{lmpq}\,$ by the corresponding time lags:

$\displaystyle\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})$ $\displaystyle=$
$\displaystyle\omega_{\textstyle{{}_{lmpq}}}~{}\,\Delta
t_{l}(\omega_{\textstyle{{}_{lmpq}}})$ (75a) $\displaystyle=$
$\displaystyle\chi_{\textstyle{{}_{lmpq}}}~{}\,\Delta
t_{l}(\chi_{\textstyle{{}_{lmpq}}})~{}\,\mbox{Sgn}\,(\,\omega_{\textstyle{{}_{lmpq}}}\,)\,~{}.$
(75b)

The functional forms of the lags and Love numbers (as functions of the Fourier
mode $\,\omega_{lmpq}\,$) are defined only by the degree $\,l\,$. Hence the
notation: $\,k_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$,
$\,\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$, $\,\Delta
t_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$. This, however, is
true only for a homogeneous spherical body. 151515 For oblate homogeneous
bodies, the functional dependencies of the lags and Love numbers on modes are
defined by both the degree $\,l\,$ and order $\,m\,$ (Dehant 1987, Ogilvie
2013). So one should write:
$\,k_{\textstyle{{}_{lm}}}(\omega_{\textstyle{{}_{lmpq}}})\,$,
$\,\epsilon_{\textstyle{{}_{lm}}}(\omega_{\textstyle{{}_{lmpq}}})\,$, and
$\,\Delta t_{\textstyle{{}_{lm}}}(\omega_{\textstyle{{}_{lmpq}}})\,$. These
Love numbers and lags differ from the Love numbers and lags of the spherical
reference body by terms of the order of the flattening, so a small non-
sphericity can be neglected.

Keeping in mind that the time lag $\,\Delta
t_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,$ is positive
definite, we see from (75a) that the signs of $\,\epsilon_{l}\,$ and
$\,\omega_{\textstyle{{}_{lmpq}}}\,$ always coincide. So the phase lags are
odd functions of $\,\omega_{lmpq}\,$ and thus can be written as

$\displaystyle\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}=~{}|\,\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,|\,~{}\mbox{Sgn}\,(\omega_{\textstyle{{}_{lmpq}}})$
(76a) or, equivalently, as
$\displaystyle\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}=~{}\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\,\mbox{Sgn}\,(\omega_{\textstyle{{}_{lmpq}}})~{}~{},$
(76b)

with
$~{}\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})\,=\,|\,\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,|~{}$
being positive-definite.

Just as the time lags, the dynamical Love numbers depend on the actual
physical frequencies $\,\chi_{\textstyle{{}_{lmpq}}}\,$. This is the same as
to say that they are even functions of $\,\omega_{\textstyle{{}_{lmpq}}}\;$:

$\displaystyle
k_{\textstyle{{}_{l}}}(\,-\,\omega_{\textstyle{{}_{lmpq}}})~{}=~{}k_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}=~{}k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}~{}.$
(77)

From the latter two equations, it ensues that

$\displaystyle
k_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}=~{}k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\mbox{Sgn}\,(\omega_{\textstyle{{}_{lmpq}}})\,~{}.$
(78)

###  A.3 Observable geometric lags

Geometrically, tides comprise a system of superimposed bulges moving around
the body in different directions, at different angular, and with different
lags. The principal, semidiurnal bulge has an elliptic shape. As seen from the
surface, it moves at the same angular rate as the perturber, with a positive
or negative lag. Lesser bulges have more involved shapes and different
velocities. By geodetic measurements, it is possible to single out those
bulges, and to measure their magnitudes and geometric lags (Benjamin et al.
2006, Eanes & Bettadpur 1996).

The geometric lag angle of an $\,lmpq\,$ bulge is (Efroimsky & Makarov 2013,
Eqn 26):

$\displaystyle\delta_{lmpq}\,=~{}\frac{\omega_{lmpq}}{m}~{}\Delta
t_{l}(\omega_{lmpq})\,~{}.$ (79)

For example, the geometric lag of the principal bulge is
$\,\delta_{2200}=\,\frac{\textstyle\omega_{2200}}{\textstyle 2}~{}\Delta
t_{2}\,=\,(n-\dot{\theta})\,\Delta t_{2}\,$, where the time lag is taken at
the appropriate, semidiurnal mode: $\,\Delta t_{2}=\Delta
t_{2}(\omega_{2200})\,$. Also mind that the tide components with $\,m=0\,$ are
longitude-independent and the notion of geometric lag becomes meaningless –
see the equation (25) in Ibid.

###  A.4 Why positive frequencies are insufficient

While the general formula for a Fourier expansion of a field includes
integration or summation over both positive and negative modes, in the case of
real fields it is sufficient to expand over positive frequencies only. 161616
The condition of a field being real requires that the real part of a Fourier
term at a negative mode is equal to the real part of the term at an opposite,
positive mode. Thus one can get rid of the terms with negative modes, at the
cost of doubling the appropriate terms with positive modes. (The convention is
that the actual field is the real part of a complex expression.) Surprisingly,
the tidal theory makes a rare exception from this rule. This theory cannot be
expressed via the physical frequencies
$\,\chi_{\textstyle{{}_{lmpq}}}\,=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,$
solely, with no mentioning of the modes $\,\omega_{\textstyle{{}_{lmpq}}}\,$.
It turns out that a contribution from a Fourier mode into the additional tidal
potential is not  physically equivalent to the contribution from the mode of
an opposite sign.

To understand the reason for this, recall that the expansion for the tidal
potential is needed to obtain expressions for tidal force and torque. These
are infinite series too; and in each of these two series an $\,lmpq\,$ term
contains explicitly the sign of the appropriate tidal mode
$\,\omega_{\textstyle{{}_{lmpq}}}\,$. Specifically, an $\,lmpq\,$ term in the
expressions for the tidal force or torque contains a factor

$\displaystyle
k_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}=~{}k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\mbox{Sgn}\,(\omega_{\textstyle{{}_{lmpq}}})\,~{},$
(80)

where we made use of the formulae (76b) and (77).

This way, if we choose to expand the tidal torque or the tidal force over the
positive definite frequencies $\,\chi_{\textstyle{{}_{lmpq}}}\,$ only, we
shall have to insert “by hand” the multipliers
Sgn$\,\omega_{\textstyle{{}_{lmpq}}}\,$ into each $\,lmpq\,$ term of these
expansions (Efroimsky 2012a,b).

###  A.5 A typical shape of the function
$~{}k_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}$

The function
$~{}k_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})\,=\,k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})/Q_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\mbox{Sgn}\,(\omega_{\textstyle{{}_{lmpq}}})~{}$
enters the $\,lmpq\,$ term in the expression (5) for the tidal torque. The
kink shape of this function is amazingly robust: it is about the same for all
linear rheological models, including the most extreme ones (say, purely
viscous mantle with no rigidity). It will exhibit a sharp kink, as in Figure
1, with two peaks having opposite signs. The fact that the function goes
continuously through zero enables the torque to transcend spin-orbit
resonances continuously.

Close to $\,\omega_{\textstyle{{}_{lmpq}}}=\,0\,$, the frequency dependence
makes two extrema, and demonstrates a slow frequency-dependence at higher
frequencies.

###  A.6 Expression for the function
$~{}k_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}\cos\epsilon_{\textstyle{{}_{l}}}(\omega_{\textstyle{{}_{lmpq}}})~{}$

As was mentioned in Section 4.2 above, a product

$\displaystyle
k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\cos\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})\;=\;{\cal{R}}{\it{e}}\left[\bar{k}_{l}(\chi_{\textstyle{{}_{lmpq}}})\right]$
(81)

stands in an $\,lmpq\,$ term of the expansion for the tidal potential $\,U\,$.
It is equal to

$\displaystyle
k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\cos\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})$
$\displaystyle=$
$\displaystyle\frac{3}{2(l-1)}~{}\;\frac{\left(\,{\cal{R}}{\it{e}}\left[\bar{J}(\chi)\right]\,+\,{\cal{B}}_{\textstyle{{}_{l}}}\,\right)\;{\cal{R}}{\it{e}}\left[\bar{J}(\chi)\right]\,+\,\left({\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]\,\right)^{2}}{\left(\,{\cal{R}}{\it{e}}\left[\bar{J}(\chi)\right]\,+\,{\cal{B}}_{\textstyle{{}_{l}}}\,\right)^{2}\,+\,\left({\cal{I}}{\it{m}}\left[\bar{J}(\chi)\right]\,\right)^{2}}\qquad\qquad$
(82a) $\displaystyle=$
$\displaystyle\frac{3}{2(l-1)}~{}\;\frac{\left(\,\Re(\chi)\,+\,{\cal{A}}_{\textstyle{{}_{l}}}\,\right)\;\Re(\chi)\,+\,\Im(\chi)^{\,2}}{\left(\,\Re(\chi)\,+\,{\cal{A}}_{\textstyle{{}_{l}}}\,\right)^{2}\,+\,\Im(\chi)^{\,2}}\quad.$
(82b)

For a self-gravitating Maxwell sphere, this expression becomes:

$\displaystyle
k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\cos\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}=~{}\frac{3}{2(l-1)}~{}\left(\,1\;-\;\frac{{\cal{A}}_{l}}{{\cal{A}}_{l}^{2}\,+\,\frac{\textstyle
1}{\textstyle\chi^{2}\,\tau^{2}_{{}_{M}}}}\,\right)$ (83)

Of a special interest are the three extreme cases considered in Section 5.2:

$\displaystyle\mbox{self-gravitating ideal-fluid drop}:\qquad
k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\cos\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})$
$\displaystyle\rightarrow$ $\displaystyle\frac{3}{2(l-1)}\quad,\qquad\qquad$
(84) $\displaystyle\mbox{self-gravitating elastic sphere}:\qquad
k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\cos\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})$
$\displaystyle\rightarrow$
$\displaystyle\frac{3}{2(l-1)}~{}\left(\,1\;-\;{\cal{A}}_{l}^{-1}\,\right)\quad,\qquad\qquad$
(85) $\displaystyle\mbox{self-gravitating viscous sphere}:\qquad
k_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})~{}\cos\epsilon_{\textstyle{{}_{l}}}(\chi_{\textstyle{{}_{lmpq}}})$
$\displaystyle\rightarrow$
$\displaystyle\frac{3}{2(l-1)}~{}\frac{1}{1\,+\,{\cal{B}}_{l}^{\,2}\,\chi^{2}\,\tau^{2}_{{}_{M}}}\quad.\qquad\qquad$
(86)

Recall, however, that the Kaula theory of bodily tides becomes inapplicable in
the limit of a perfectly viscous body, as was explained in Section 5.2.3.

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