$\displaystyle\epsilon\,^{\prime}_{l}(\omega)\,\equiv\,-~{}\left[\,\varphi_{\textstyle{{}_{U_{l}}}}(\omega)~{}+~{}\varphi_{\textstyle{{}_{W_{l}}}}(-\omega)\,\right]\,=~{}\varphi_{\textstyle{{}_{W_{l}}}}(\omega)~{}-~{}\varphi_{\textstyle{{}_{U_{l}}}}(\omega)~{}-~{}\varphi_{\textstyle{{}_{W_{l}}}}(\omega)~{}-~{}\varphi_{\textstyle{{}_{W_{l}}}}(-\omega)~{}$
$\displaystyle=~{}\epsilon_{l}(\omega)~{}-~{}\left[\,\varphi_{\textstyle{{}_{W_{l}}}}(\omega)~{}+~{}\varphi_{\textstyle{{}_{W_{l}}}}(-\omega)\,\right]\,~{}.$
(G3)

At first glance, this result is most unphysical. Usually, to calculate
dissipation rate, we have to sum, over physical frequencies or over Fourier
modes, terms proportional to the sines of phase lags at those modes. The
addition of a finite phase to those lags looks bizarre. However, an accurate
calculation carried out in the Appendix H shows that the phase consists of two
parts. One is equal to
$\,\left[\,(\,-\,1)^{\,l}\,-\,1\,\right]~{}{\pi}/{2}\,$, so its presence
renders an overall factor of $\,(-1)^{\,l}\,$. The other part of the phase is
$\,(m\,^{\prime}\,+\,m)\,\lambda\,$, so after integration over the surface, it
results in a $\,\delta(m\,^{\prime}\,+\,m)\,$ factor, 272727 The finite phase
assumes the value of
$\,\left[(\,-\,1)^{\,l}-1\right]\,{\pi}/{2}\,+\,(m\,^{\prime}+m)\lambda\,$,
with the integer $\,m\,^{\prime}\,$ being the order of
$\,\omega\,^{\prime}=\omega_{\textstyle{{}_{lm^{\prime}p^{\prime}q^{\prime}}}}\,$,
and $\,m\,$ being that of $\,\omega=\omega_{\textstyle{{}_{lmpq}}}\,$. The
presence of $\,\left[\,(\,-\,1)^{\,l}\,-\,1\,\right]\,{\pi}/{2}\,$ in the
phase is equivalent to multiplying the sum by $\,(-1)^{\,l}\,$. So, for even
$\,l\,$, this part of the phase can be ignored. The presence of the term
$\,(m\,^{\prime}\,+\,m)\,\lambda\,$ in the phase tells us that, after
integration over the surface, only the terms with
$\,m\,^{\prime}\,=\,m\,=\,0\,$ stay. Hence, after integration, we are
effectively left with
$\,\epsilon^{\prime}_{l}(\omega)\,=\,(-1)^{l}\,\epsilon_{l}(\omega)\,\delta(m\,^{\prime}\,+\,m)\,$.
where $\,m\,$ is the second index of
$\,\omega_{\textstyle{{}_{lmpq}}}\,=\,\omega\,$, while $\,m\,^{\prime}\,$ is
the second index of
$\,\omega_{\textstyle{{}_{lm^{\prime}p^{\prime}q^{\prime}}}}\,=\,-\,\omega~{}$:

$\displaystyle\langle
P\rangle~{}=~{}\frac{1}{4\,\pi\,G\,R}~{}\sum_{l=2}^{\infty}(2\,l\,+\,1)\,\int
dS~{}\langle\,W(t)~{}\dot{U}(t)\,\rangle~{}=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\frac{1}{4\,\pi\,G\,R}\,\sum_{\omega}\,(2\,l\,+\,1)~{}\frac{\omega}{2}\,\int
dS~{}W_{l}(\omega)\,\left[\,W_{l}(\omega)\,+\,(-1)^{l}\,\delta(m\,^{\prime}\,+\,m)\,W_{l}(-\omega)\right]~{}k_{l}(\omega)~{}\sin\epsilon_{l}(\omega)\,~{}.~{}\qquad$
(G4)

The indices $\,m\,$ and $\,m\,^{\prime}\,$ being nonnegative (see the equation
4), the emergence of $\,\delta(m\,^{\prime}\,+\,m)\,$ indicates that the
summation in the second part must be reduced to
$\,m\,=\,m\,^{\prime}\,=\,0~{}$:

$\displaystyle\langle P\rangle~{}=~{}\frac{1}{4\,\pi\,G\,R}~{}\int
dS~{}\left[\,{\sum_{\omega\,=\,\omega_{\textstyle{{}_{{}_{lmpq}}}}}}~{}\frac{\omega}{2}~{}(2l+1)~{}k_{l}(\omega)~{}\sin\epsilon_{l}(\omega)~{}W^{2}_{l}(\omega)\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.+~{}{\sum_{\omega\,=\,\omega_{\textstyle{{}_{{}_{l0pq}}}}}}~{}\frac{\omega}{2}~{}(2l+1)~{}k_{l}(\omega)~{}\sin\epsilon_{l}(\omega)~{}~{}(-1)^{l}\,W_{l}(\omega)\,W_{l}(-\omega)\,\right]\,~{}.\quad$
(G5)

We then see what the superscript $\,\sharp\,$ introduced in (67 \- 68)
actually implies:

$\displaystyle{\sum_{\omega}}^{\textstyle{{}^{\textstyle\,\sharp~{}}}}.\,.\,.~{}W_{l}^{2}(\omega)~{}\equiv~{}{\sum_{\omega\,=\,\omega_{\textstyle{{}_{{}_{lmpq}}}}}}~{}.\,.\,.~{}W^{2}_{l}(\omega)~{}+~{}{\sum_{\omega\,=\,\omega_{\textstyle{{}_{{}_{l0pq}}}}}}~{}.\,.\,.~{}(-1)^{l}~{}W_{l}(\omega)~{}W_{l}(-\omega)\,~{},$
(G6)

where the first sum on the right-hand side is complete (i.e., goes over all
modes), while the second sum is only over the modes with a vanishing second
index.

Now, what is $\,W_{l}(\omega)\,$? Naïvely,
$\,W_{l}(\omega)\,\equiv\,W_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$ should be
the real magnitude of the term $\,W_{\textstyle{{}_{lmpq}}}\,$ of Kaula’s
series (4). In reality, we have degeneracy of modes, so in each of the two
Fourier series (for $\,W\,$ and for $\,U\,$) we first must group together the
terms corresponding to each actual value of mode, and only afterward should we
multiply the series by one another and perform time averaging. This
calculation is presented in Appendix H.

##  Appendix H Heat production over tidal modes. An accurate derivation of the
formula (69)

As was explained above, the actual $\,W_{l}(\omega)\,$ will be a sum of
several $\,W_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$, over all the sets
$\,lmpq\,$ furnishing the same value of $\,\omega\,$. Thence, in (G6) and
elsewhere, summation over $\,\omega\,$ will be not a sum over $\,lmpq\,$, but
a sum over all distinct  values of $\,\omega_{\textstyle{{}_{lmpq}}}\,$.

These details would be irrelevant, were we summing terms linear in
$\,W_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$. In that case, to group terms and
then to sum the groups would be the same as to sum all the terms at once. We
however are dealing with the expression (G6) quadratic in $\,W_{l}\,$,
wherefore the said details matter a lot. Incorporation of those complicates
the calculation technically, though the main idea remains the same as in (G5).

###  H.1 Prefatory algebra

First, it would be convenient to rewrite the formula (4) for the perturbing
potential as

$\displaystyle
W(\mbox{{\boldmath${R}$}}\,,\;\mbox{{\boldmath$r$}}^{\;*})\,=\sum_{l=2}^{\infty}~{}W_{l}~{}~{},~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}$
(H1)

where

$\displaystyle
W_{l}\,=\sum_{m^{\prime}p^{\prime}q^{\prime}}\,W_{lm^{\prime}p^{\prime}q^{\prime}}\,=~{}\sum_{m^{\prime}=0}^{l}\;\sum_{p^{\prime}=0}^{l}\;\sum_{j=\,-\,\infty}^{\infty}A_{lm^{\prime}p^{\prime}q^{\prime}}~{}\cos\left(v_{lm^{\prime}p^{\prime}q^{\prime}}\,-~{}m^{\prime}\,(\lambda\,+\,\theta)~{}+~{}\psi_{lm^{\prime}}\right)\,~{},$
(H2) $\displaystyle
A_{lm^{\prime}p^{\prime}q^{\prime}}~{}=~{}-~{}\frac{G\,M^{*}}{a}\;\left(\,\frac{R}{a}\,\right)^{\textstyle{{}^{l}}}\;\frac{(l-m^{\prime})!}{(l+m^{\prime})!}\;\left(\,2\,-\,\delta_{0m^{\prime}}\,\right)\;P_{lm^{\prime}}(\sin\phi)\;F_{lm^{\prime}p^{\prime}}(i)\;G_{lp^{\prime}q^{\prime}}(e)~{}~{},$
(H3)
$\displaystyle\psi_{lm^{\prime}}~{}=~{}\left[\,(\,-\,1)^{\,l-m^{\prime}}\,-\,1\,\right]~{}\frac{\pi}{4}\;~{}.~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}$
(H4)

Similarly, the formula (16) for the additional tidal potential at
$\,\mbox{{\boldmath$r$}}=\mbox{{\boldmath${R}$}}\,$ should be cast as

$\displaystyle
U(\mbox{{\boldmath${R}$}}\,,\;\mbox{{\boldmath$r$}}^{\;*})\,=\sum_{l=2}^{\infty}\;U_{l}~{}~{},~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}$
(H5)

where

$\displaystyle
U_{l}\,=\sum_{mpq}\,U_{lmpq}\,=\sum_{m=0}^{l}\;\sum_{p=0}^{l}\;\sum_{q=\,-\,\infty}^{\infty}B_{lmpq}~{}\cos\left(v_{lmpq}\,-~{}m\,(\lambda\,+\,\theta)~{}+~{}\psi_{lm}~{}-~{}\epsilon_{lmpq}\right)~{},~{}\quad$
(H6) $\displaystyle B_{lmpq}~{}=~{}k_{l}(\omega_{lmpq})~{}A_{lmpq}\;~{},$ (H7)

$\psi_{lm}\,$ is given by (H4), while $\,\epsilon_{lmpq}\,$ is the phase lag
(17b). In (H7) we omitted the multiplier
$\,\left(\,{R}/{r}\,\right)^{\textstyle{{}^{l+1}}}\,$, as we are interested in
the values of $\,U\,$ over the surface where $\,r=R\,$.

The product of the quantities $\,W_{l}\,$ and $\,\dot{U}_{l}\,$ reads as

$\displaystyle
W_{l}\,\,\dot{U}_{l}\,=~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}~{}\qquad~{}\,~{}\qquad~{}$
(H8)
$\displaystyle-\sum_{m^{\prime}p^{\prime}q^{\prime}}\sum_{mpq}\,A_{lm^{\prime}p^{\prime}q^{\prime}}\,\cos\left(v_{lm^{\prime}p^{\prime}q^{\prime}}-\,m^{\prime}(\lambda+\theta)+\psi_{lm^{\prime}}\right)\omega_{lmpq}B_{lmpq}~{}\sin\left(v_{lmpq}-m(\lambda+\theta)+\psi_{lm}-\epsilon_{lmpq}\right)$
$\displaystyle=~{}-~{}\frac{1}{2}\,\sum_{m^{\prime}p^{\prime}q^{\prime}}\sum_{mpq}\,A_{lm^{\prime}p^{\prime}q^{\prime}}\,\omega_{lmpq}\,B_{lmpq}~{}\sin\left(\,v_{lm^{\prime}p^{\prime}q^{\prime}}+v_{lmpq}-(m^{\prime}+m)\,(\lambda+\theta)+\,\psi_{lm^{\prime}}+\,\psi_{lm}\,-\,\epsilon_{lmpq}\,\right)$
$\displaystyle+~{}\frac{1}{2}~{}\sum_{m^{\prime}p^{\prime}q^{\prime}}\sum_{mpq}\,A_{lm^{\prime}p^{\prime}q^{\prime}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin\left(~{}v_{lm^{\prime}p^{\prime}q^{\prime}}-v_{lmpq}-\,(m^{\prime}-m)\,(\lambda+\theta)\,+\,\psi_{lm^{\prime}}\,-\,\psi_{lm}\,+\,\epsilon_{lmpq}~{}\right)~{}~{}.$

The formula (65b) for the heating rate contains a time-average of the product
$\,W_{l}\,\dot{U}_{l}\,$, integrated over the surface. In the first sum of
(H8), integration over the longitude $\,\lambda\,$ will leave only the terms
with $\,m^{\prime}\,=\,m\,=\,0\,$. (Recall that $\,m^{\prime}\,$ and $\,m\,$
are nonnegative.) In the second sum, integration over $\,\lambda\,$ will
eliminate all terms except the ones with $\,m^{\prime}\,=\,m\,$. Therefore,

$\displaystyle W_{l}\,\,\dot{U}_{l}\,=$ $\displaystyle-$
$\displaystyle\frac{1}{2}~{}\sum_{p^{\prime}q^{\prime}}\sum_{pq}\,A_{l0p^{\prime}q^{\prime}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\sin\left(~{}v_{l0p^{\prime}q^{\prime}}\,+\,v_{l0pq}\,+\,2\,\psi_{l0}\,-\,\epsilon_{l0pq}~{}\right)$
(H9a) $\displaystyle+$
$\displaystyle\frac{1}{2}~{}\sum_{m}\sum_{p^{\prime}q^{\prime}}\sum_{pq}\,A_{lmp^{\prime}q^{\prime}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin\left(~{}v_{lmp^{\prime}q^{\prime}}\,-\,v_{lmpq}\,+\,\epsilon_{lmpq}~{}\right)~{}+~{}~{}.~{}.~{}.~{}~{}~{},~{}\qquad\qquad~{}$
the ellipsis denoting the terms which are to vanish after integration over the
longitude $\,\lambda\,$. The presence of the phase
$\,2\psi_{l0}=\left[(\,-\,1)^{\,l}-1\right]\,{\pi}/{2}\,$ in the first sum is
equivalent to multiplying the sum by $\,(-1)^{\,l}\,$. Combined with the
formula (5) for $\,v_{lmpq}\,$, this observation gives us: $\displaystyle
W_{l}\,\dot{U}_{l}\,=~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}~{}$
$\displaystyle-~{}\frac{(-1)^{\,l}}{2}~{}\sum_{p^{\prime}q^{\prime}}\sum_{pq}\,A_{l0p^{\prime}q^{\prime}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\sin\left(\,2\,(l-p\,^{\prime}-p)~{}\omega+\,(2\,(l-p\,^{\prime}-p)\,+\,q\,^{\prime}\,+\,q)\,{\cal{M}}-\,\epsilon_{l0pq}\right)~{}\qquad~{}$
$\displaystyle+~{}\frac{1}{2}\,\sum_{m}\sum_{p^{\prime}q^{\prime}}\sum_{pq}\,A_{lmp^{\prime}q^{\prime}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin\left(\,2\,(p-p\,^{\prime})\,\omega+\,(2p-q-2p\,^{\prime}+q\,^{\prime})\,{\cal{M}}+\,\epsilon_{lmpq}\,\right)\,+~{}.~{}.~{}.~{}~{},~{}\qquad$
(H9b) ${\cal{M}}\,$ and $\,\omega\,$ being the mean anomaly and the argument
of the pericentre of the perturber. In Section 2.3, we defined the mean motion
$\,n\,$ as the mean anomaly rate
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{\cal{M}\,}}\,$. Thence 282828 The
epoch $\,t_{0}=0\,$ is set at the instant of periapse crossing whence
$\,{\cal{M}}\,$ is reckoned. So we write the mean anomaly simply as
$\,{\cal{M}}\,=\,n\,t\,$. Calculating the present rate of dissipation, we
average over one or several cycles of tidal flexure, and not over the entire
time span since the epoch (which may be distant). For a Keplerian orbit,
$\,{\cal{M}}\,$ changes uniformly with time and always assumes a value of
$\,2\,\pi\,N\,$ at a periapse crossing, with $\,N\,$ being integer. So we can
always set $\,{\cal{M}}=0\,$ at a recent crossing and can reckon time from
there (i.e, set $\,t_{0}=0\,$ at that moment).
In realistic situations,
$\,{\cal{M}}\,=\,\int_{t_{0}}^{t}n(t\,^{\prime})\,dt\,^{\prime}~{}$. (No
$\,{\cal{M}}_{0}\,$ term here, as we agreed to define $\,n\,$ as
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{\cal{M}\,}}$.) Despite the apsidal
precession, $\,{\cal{M}}\,$ will still be changing by $\,2\pi\,$ between two
subsequent periapse crossings. It however will no longer be right to
substitute the integral with a product of $\,n\,$ by a time interval. Despite
this, we still can set $\,t_{0}=0\,$ and $\,{\cal{M}}=0\,$ at a recent
crossing, and can approximate the integral with $\,n\,t\,$ over several
rotations sufficient for our averaging. $\displaystyle
W_{l}\,\,\dot{U}_{l}\,=~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}$
$\displaystyle\,\frac{(-1)^{\,l}}{2}\,\sum_{p^{\prime}q^{\prime}}\sum_{pq}A_{l0p^{\prime}q^{\prime}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\sin\left(\,-\,2\,(l-p\,^{\prime}-p)~{}\omega-\,(2\,(l-p\,^{\prime}-p)\,+\,q\,^{\prime}\,+\,q)\,n\,t\,+\,\epsilon_{l0pq}\,\right)\qquad\quad$
$\displaystyle\left.\,\right.+\,\frac{1}{2}~{}\sum_{m}\sum_{p^{\prime}q^{\prime}}\sum_{pq}\,A_{lmp^{\prime}q^{\prime}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin\left(~{}2\,(p-p\,^{\prime})\,\omega+\,(2p-q-2p\,^{\prime}+q\,^{\prime})\,n\,t\,+\,\epsilon_{lmpq}~{}\right)\,+~{}.~{}.~{}.~{}~{},\qquad$
(H9c) where we have moved the “minus” sign inside the sine function in the
first sum.

Selecting the secular terms in the above sum, we notice that some of those
contain not only the phase lag but also finite phases proportional to the
initial value of the pericentre. Naturally, such ugly terms come in pairs
containing equal but opposite initial phase. Each such pair renders a mutual
input proportional to the sine of the phase lag $\,\epsilon_{lmpq}\,$.

To demonstrate this, single out the initial phases in each term of (H9c). The
second sum in (H9c) will be split into two subsums, one corresponding to
$\,p\,^{\prime}\neq p\,$, another to $\,p\,^{\prime}=p~{}$:

$\displaystyle W_{l}\,\,\dot{U}_{l}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\,l}}{2}\,\sum_{p^{\prime}q^{\prime}}\sum_{pq}A_{l0p^{\prime}q^{\prime}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\sin(\,-\,2\,(l-p\,^{\prime}-p)~{}\omega_{0}\,-\,(\omega_{\textstyle{{}_{l0p^{\prime}q^{\prime}}}}\,+\,\omega_{\textstyle{{}_{l0pq}}})\,t\,+\,\epsilon_{l0pq}\,)$
(H9d)
$\displaystyle+~{}\frac{1}{2}~{}\sum_{m}\sum_{p\,q}\sum_{\stackrel{{\scriptstyle\textstyle{{}^{~{}p\,^{\prime}\,q\,^{\prime}}}}}{{p\,^{\prime}\neq
p~{}}}}\,A_{lmp^{\prime}q^{\prime}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin(~{}2\,(p-p\,^{\prime})\,\omega_{0}\,+\,(\omega_{\textstyle{{}_{lmp^{\prime}q^{\prime}}}}\,-\,\omega_{\textstyle{{}_{lmpq}}})\,t\,+\,\epsilon_{lmpq}~{})$
$\displaystyle\left.\,\right.+\,\frac{1}{2}~{}\sum_{m}\sum_{pqq^{\prime}}\,A_{lmpq^{\prime}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin\left(\,(q\,^{\prime}\,-\,q)\,n\,t\,+\,\epsilon_{lmpq}~{}\right)\,+~{}.~{}.~{}.\,~{}\qquad$

###  H.2 Selection of secular terms in (H9d). The role of rheology

The purpose of the below-presented algebraic development of (H9d) is to single
out secular terms, i.e., the ones whose time average vanishes. Involvement of
time averages immediately brings in rheological properties of the tidally
perturbed body. Indeed, the averaging remains legitimate over so long a time
scale as keeps the rheological response linear, not plastic. Very roughly, we
may link the borderline to the Maxwell time of the mantle,
$\,\tau_{{}_{M}}\,$. If the period of the apsidal motion is shorter than
$\,\tau_{{}_{M}}\,$ or, at least, does not exceed it by more than an order or
two, we have to keep the apsidal rate $\,\dot{\omega}\,$ in the expression (7)
for the Fourier mode and, accordingly, have to average the product
$\,W_{l}\,\,\dot{U}_{l}\,$ over the apsidal period. If however the apsidal
timescale exceeds $\,\tau_{{}_{M}}\,$ considerably, then averaging of the
product $\,W_{l}\,\,\dot{U}_{l}\,$ over such a time scale will lack physical
meaning, and the apsidal rate in the expression (7) should be set zero. As we
shall see below, this will make difference in our further selection of the
secular parts of the first two subsums in (H9d).

####  H.2.1 Symmetrisation

Processing the third subsum in (H9d) is easy: we retain the secular terms
(those with $\,q=q\,^{\prime}\,$) and, in anticipation of time averaging,
ignore the rest of the subsum.

The second subsum in (H9d) can be split into two equal halves, with $\,pq\,$
and $\,p\,^{\prime}q\,^{\prime}\,$ swapped in the second half:

$\displaystyle\frac{1}{2}~{}\sum_{m}\,\sum_{\stackrel{{\scriptstyle\textstyle{{}^{\,p\,,\,p\,^{\prime}}}}}{{p\,^{\prime}\neq
p~{}}}}\,\sum_{q\,,\,q\,^{\prime}}\,~{}A_{lmp\,^{\prime}q\,^{\prime}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin(~{}2\,(p-p\,^{\prime})\,\omega_{0}\,+\,(\omega_{\textstyle{{}_{lmp\,^{\prime}q\,^{\prime}}}}\,-\,\omega_{\textstyle{{}_{lmpq}}})\,t\,+\,\epsilon_{lmpq}~{})~{}=$
$\displaystyle\frac{1}{4}~{}\sum~{}\left[\,A_{lmp\,^{\prime}q\,^{\prime}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin(~{}2\,(p-p\,^{\prime})\,\omega_{0}\,+\,(\omega_{\textstyle{{}_{lmp\,^{\prime}q\,^{\prime}}}}\,-\,\omega_{\textstyle{{}_{lmpq}}})\,t\,+\,\epsilon_{lmpq}~{})~{}+\right.~{}\quad~{}\qquad~{}\qquad$
$\displaystyle\left.A_{lmpq}~{}\omega_{lmp\,^{\prime}q\,^{\prime}}~{}B_{lmp\,^{\prime}q\,^{\prime}}~{}\sin(~{}2\,(p\,^{\prime}-p)\,\omega_{0}\,+\,(\omega_{\textstyle{{}_{lmpq}}}\,-\,\omega_{\textstyle{{}_{lmp\,^{\prime}q\,^{\prime}}}})\,t\,+\,\epsilon_{lmp\,^{\prime}q\,^{\prime}}~{})\,\right]~{}=~{}\qquad\qquad$
(H10a)
$\displaystyle\frac{1}{4}~{}\sum~{}\left[\,A_{lmp\,^{\prime}q\,^{\prime}}~{}\omega_{lmpq}~{}A_{lmpq}~{}k_{l}(\omega_{lmpq})~{}\sin(~{}2\,(p-p\,^{\prime})\,\omega_{0}\,+\,(\omega_{\textstyle{{}_{lmp\,^{\prime}q\,^{\prime}}}}\,-\,\omega_{\textstyle{{}_{lmpq}}})\,t\,+\,\epsilon_{lmpq}~{})~{}+\right.\quad\qquad$
$\displaystyle\left.A_{lmpq}~{}\omega_{lmp\,^{\prime}q\,^{\prime}}~{}A_{lmp\,^{\prime}q\,^{\prime}}~{}k_{l}(\omega_{lmp\,^{\prime}q\,^{\prime}})~{}\sin(~{}2\,(p\,^{\prime}-p)\,\omega_{0}\,+\,(\omega_{\textstyle{{}_{lmpq}}}\,-\,\omega_{\textstyle{{}_{lmp\,^{\prime}q\,^{\prime}}}})\,t\,+\,\epsilon_{lmp\,^{\prime}q\,^{\prime}}~{})\,\right]~{}\,.\qquad$
(H10b)

Prior to time averaging of (H10), we, generally, have
$~{}\omega_{lmpq}~{}k_{l}(\omega_{lmpq})\,\neq\,\omega_{lmp\,^{\prime}q\,^{\prime}}~{}k_{l}(\omega_{lmp\,^{\prime}q\,^{\prime}})~{}$
and $~{}\epsilon_{lmpq}\,\neq\,\epsilon_{lmp\,^{\prime}q\,^{\prime}}~{}$. The
averaging, however, leaves us only the terms with
$\,\omega_{lmpq}\,=\,\omega_{lmp\,^{\prime}q\,^{\prime}}\,$. Hence the secular
part of the above expression is:

$\displaystyle\frac{1}{4}\sum_{m}\sum_{\stackrel{{\scriptstyle\textstyle{{}^{\,p\,,\,p\,^{\prime}}}}}{{p\,^{\prime}\neq
p}}}\sum_{q}\,\left[\,A_{lmp\,^{\prime}q\,^{\prime}}\,\right]_{\textstyle{{}_{\omega_{\textstyle{{}_{{}_{lmpq}}}}=\,\omega_{\textstyle{{}_{{}_{lmp\,^{\prime}q\,^{\prime}}}}}}}}\omega_{lmpq}\,B_{lmpq}\,\left[\,\sin(2\,(p-p\,^{\prime})\,\omega_{0}\,+\,\epsilon_{lmpq})\,+\,\sin(\,-2\,(p-p\,^{\prime})\,\omega_{0}+\epsilon_{lmpq})\,\right]~{}~{}\,~{}\qquad$
$\displaystyle=~{}\frac{1}{2}~{}\sum_{m}\,\sum_{\stackrel{{\scriptstyle\textstyle{{}^{\,p\,,\,p\,^{\prime}}}}}{{p\,^{\prime}\neq
p~{}}}}\,\sum_{q}\,\left[\,A_{lmp\,^{\prime}q\,^{\prime}}\,\right]_{\textstyle{{}_{\omega_{lmpq}\,=\,\omega_{lmp\,^{\prime}q\,^{\prime}}}}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\cos(\,2\,(p\,^{\prime}\,-\,p)\,\omega_{0}\,)~{}\sin\epsilon_{lmpq}~{}~{}.~{}\qquad~{}\,~{}\qquad\qquad\qquad$
(H11)

where the value of $\,q\,^{\prime}\,$ is defined by the fact that the secular
terms obey

$\displaystyle
0\,=\,\omega_{lmpq}\,-\,\omega_{lmp\,^{\prime}q\,^{\prime}}\,=\,-\,2\,(p-p\,^{\prime})\,\dot{\omega}\,+~{}\left[\,-\,2\,(p\,-\,p\,^{\prime})\,+\,q\,-\,q\,^{\prime}\,\right]\,n~{}~{}.$
(H12)

Similarly to (H10), the first subsum in (H9d) can be rewritten as

$\displaystyle\frac{(-1)^{\,l}}{2}\,\sum_{\,p\,^{\prime},\,q\,^{\prime}\,}\sum_{\,p,\,q}A_{l0p\,^{\prime}q\,^{\prime}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\sin(\,-\,2\,(l-p\,^{\prime}-p)~{}\omega_{0}\,-\,(\omega_{\textstyle{{}_{l0p\,^{\prime}q\,^{\prime}}}}\,+\,\omega_{\textstyle{{}_{l0pq}}})\,t\,+\,\epsilon_{l0pq}\,)~{}=~{}\qquad$
$\displaystyle\frac{(-1)^{\,l}}{4}\sum_{\stackrel{{\scriptstyle\textstyle{{}^{\,p\,,\,p\,^{\prime}\,}}}}{{}}}\sum_{\,q\,,\,q\,^{\prime}}~{}\left[\,A_{l0p\,^{\prime}q\,^{\prime}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\sin(\,-\,2\,(l-p\,^{\prime}-p)\,\omega_{0}\,-\,(\omega_{\textstyle{{}_{l0p^{\prime}q^{\prime}}}}+\omega_{\textstyle{{}_{l0pq}}})\,t\,+\,\epsilon_{l0pq}\,)\right.~{}\qquad~{}\qquad~{}$
$\displaystyle\left.~{}+~{}\,A_{l0pq}~{}\omega_{l0p\,^{\prime}q\,^{\prime}}~{}B_{l0p\,^{\prime}q\,^{\prime}}~{}\sin(\,-\,2\,(l-p-p\,^{\prime})\,\omega_{0}\,-\,(\omega_{\textstyle{{}_{l0pq}}}\,+\,\omega_{\textstyle{{}_{l0p\,^{\prime}q\,^{\prime}}}})~{}t\,+\,\epsilon_{l0p\,^{\prime}q\,^{\prime}})\,\right]~{}=~{}\qquad~{}$
(H13a)
$\displaystyle\frac{(-1)^{\,l}}{4}\sum_{\stackrel{{\scriptstyle\textstyle{{}^{\,p\,,\,p\,^{\prime}\,}}}}{{}}}\sum_{\,q\,,\,q\,^{\prime}}~{}\left[\,A_{l0p\,^{\prime}q\,^{\prime}}~{}\omega_{l0pq}~{}A_{l0pq}~{}k_{l}(\omega_{l0pq})\;\sin(\,-\,2\,(l-p\,^{\prime}-p)\,\omega_{0}\,-\,(\omega_{\textstyle{{}_{l0p^{\prime}q^{\prime}}}}+\omega_{\textstyle{{}_{l0pq}}})\,t\,+\,\epsilon_{l0pq}\,)\right.\quad\quad$
$\displaystyle\left.~{}+~{}\,A_{l0pq}~{}\omega_{l0p\,^{\prime}q\,^{\prime}}~{}A_{l0p\,^{\prime}q\,^{\prime}}~{}k_{l}(\omega_{l0p\,^{\prime}q\,^{\prime}})~{}\sin(\,-\,2\,(l-p-p\,^{\prime})\,\omega_{0}\,-\,(\omega_{\textstyle{{}_{l0pq}}}\,+\,\omega_{\textstyle{{}_{l0p\,^{\prime}q\,^{\prime}}}})~{}t\,+\,\epsilon_{l0p\,^{\prime}q\,^{\prime}})\,\right]~{}\,.\qquad$
(H13b)

After time-averaging of (H13b), the factor
$\,\omega_{l0p\,^{\prime}q\,^{\prime}}\,$ accompanying the second sine becomes
$~{}-\,\omega_{l0pq}\,$, while $\,\epsilon_{l0p\,^{\prime}q\,^{\prime}}\,$
becomes $~{}-\,\epsilon_{l0pq}\,$. Hence the secular part of the above
expression is:

$\displaystyle\frac{(-1)^{\,l}}{2}\,\sum_{\stackrel{{\scriptstyle\textstyle{{}^{~{}p\,,\,p\,^{\prime}}}}}{{p\,^{\prime}\neq
p~{}}}}\,\sum_{q}\,\left[A_{l0p\,^{\prime}q\,^{\prime}}\right]_{\textstyle{{}_{\omega_{l0pq}\,=\,-\,\omega_{l0p\,^{\prime}q\,^{\prime}}}}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\cos(\,2\,(l\,-\,p\,^{\prime}\,-\,p)\,\omega_{0}\,)~{}\sin\epsilon_{l0pq}~{}\,~{},~{}\qquad~{}\,~{}$
(H14)

where the value of $\,q\,^{\prime}\,$ is determined from the condition of the
terms being secular:

$\displaystyle
0\,=\,\omega_{lmpq}\,+\,\omega_{lmp\,^{\prime}q\,^{\prime}}\,=\,2\,(l\,-\,p\,-\,p\,^{\prime})\,\dot{\omega}\,+~{}\left[\,2\,(l\,-\,p\,-\,p\,^{\prime})\,+\,q\,+\,q\,^{\prime}\,\right]\,n~{}~{}.$
(H15)

####  H.2.2 The case of a quiescent pericentre

When the period of the apsidal motion is much (by orders of magnitude) longer
than the Maxwell time $\,\tau_{{}_{M}}\,$, we cannot count on a linear tidal
response at such long time scales. Instead, we should treat the pericentre as
fixed and should nullify $\,\dot{\omega}\,$ in the expression (7) for the
Fourier mode, as well as in (H12) and (H15). From (H11) and (H12), we see that
the time average of the second subsum in (H9d) acquires the form of

$\displaystyle\frac{1}{2}~{}\sum_{m}\,\sum_{\stackrel{{\scriptstyle\textstyle{{}^{\,p\,,\,p\,^{\prime}}}}}{{p\,^{\prime}\neq
p~{}}}}\,\sum_{q}\,\left[\,A_{lmp\,^{\prime}q\,^{\prime}}\,\right]_{\textstyle{{}_{q\,^{\prime}\,=\,q-2(p-p\,^{\prime})}}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\cos(\,2\,(p\,^{\prime}\,-\,p)\,\omega_{0}\,)~{}\sin\epsilon_{lmpq}~{}~{}.$
(H16)

From (H14) and (H15), we infer that the average of the first subsum in (H9d)
is

$\displaystyle\frac{(-1)^{\,l}}{2}\,\sum_{\stackrel{{\scriptstyle\textstyle{{}^{~{}p\,,\,p\,^{\prime}}}}}{{}}}\,\sum_{q}\,\left[A_{l0p\,^{\prime}q\,^{\prime}}\right]_{\textstyle{{}_{q\,^{\prime}\,=\,-q-2(l-p-p\,^{\prime})}}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\cos(\,2\,(l\,-\,p\,^{\prime}\,-\,p)\,\omega_{0}\,)~{}\sin\epsilon_{l0pq}~{}\,~{}.~{}\qquad~{}\,~{}$
(H17)

Altogether, the secular part of the expression (H9d) acquires the form of

${}^{\textstyle{{}^{(\dot{\omega}=0)}}}\langle\,W_{l}\,\,\dot{U}_{l}\,\rangle$
$\displaystyle=$
$\displaystyle\frac{(-1)^{\,l}}{2}~{}\sum_{p\,q}\sum_{\stackrel{{\scriptstyle\textstyle{{}^{~{}p\,^{\prime}}}}}{{}}}\,\left[A_{l0p\,^{\prime}q\,^{\prime}}\right]_{\textstyle{{}_{q\,^{\prime}\,=\,-q-2(l-p-p\,^{\prime})}}}~{}~{}\omega_{l0pq}~{}B_{l0pq}~{}\cos(\,2\,(l\,-\,p\,^{\prime}\,-\,p)\,\omega_{0}\,)~{}\sin\epsilon_{l0pq}$
(H18a) $\displaystyle+$
$\displaystyle\frac{1}{2}~{}\sum_{m}\sum_{p\,q}\sum_{~{}p\,^{\prime}\neq
p}\,\left[\,A_{lmp\,^{\prime}q\,^{\prime}}\,\right]_{\textstyle{{}_{q\,^{\prime}\,=q-2(p-p\,^{\prime})}}}~{}\omega_{lmpq}~{}B_{lmpq}~{}\cos(\,2\,(p\,^{\prime}\,-\,p)\,\omega_{0}\,)~{}\sin\epsilon_{lmpq}~{}$
$\displaystyle+$
$\displaystyle\frac{1}{2}~{}\sum_{m}\sum_{pq}\,A_{lmpq}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin\epsilon_{lmpq}~{}+\,~{}.~{}.~{}.~{}\,~{},\qquad$
where the last two sums may now be reunited into one:
${}^{\textstyle{{}^{(\dot{\omega}=0)}}}\langle\,W_{l}\,\,\dot{U}_{l}\,\rangle$
$\displaystyle=$
$\displaystyle\frac{(-1)^{\,l}}{2}\sum_{p\,p\,^{\prime}\,q}\left[A_{l0p\,^{\prime}q\,^{\prime}}\right]_{\textstyle{{}_{q\,^{\prime}=-q-2(l-p-p\,^{\prime})}}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\cos(\,2\,(l-p\,^{\prime}-p)\,\omega_{0}\,)~{}\sin\epsilon_{l0pq}$
(H18b) $\displaystyle+$
$\displaystyle\frac{1}{2}\,\sum_{m}\sum_{p\,p\,^{\prime}\,q}\,\left[A_{lmp\,^{\prime}q\,^{\prime}}\right]_{\textstyle{{}_{q\,^{\prime}\,=q-2(p-p\,^{\prime})}}}\,\omega_{lmpq}~{}B_{lmpq}\,\cos(\,2\,(p\,^{\prime}-p)\,\omega_{0}\,)\,\sin\epsilon_{lmpq}\,+\,.\,.\,.\,~{},~{}\qquad~{}~{}~{}$

the ellipsis denoting the terms which will vanish after integration over the
longitude $\,\lambda\,$.

####  H.2.3 The case of a moving pericentre

Consider a situation where the evolution rate of the pericentre is relatively
fast, i.e., not much slower than the Maxwell timescale of the mantle material.
In this situation, the time derivative $\,\dot{\omega}\,$ should be taken into
account in the expression for the Fourier tidal modes.

Just as in the case of an idle pericentre, here we begin with the second
subsum in (H9d). As we saw above, its secular part can be transformed into the
expression (H11), where the index $\,q\,^{\prime}\,$ is no longer independent
but is constrained by the condition (H12). For a vanishing $\,\dot{\omega}\,$,
that condition became $\,q\,^{\prime}\,=\,q\,-\,2\,(p\,-\,p\,^{\prime})\,$,
thus making the secular part look as (H16).

Now the situation is different, and the equality
$\,q\,^{\prime}\,=\,q\,-\,2\,(p\,-\,p\,^{\prime})\,$ no longer ensues from
(H12). There, though, is an alternative method of satisfying (H12). For a
chosen set of integers $\,\\{p,\,p\,^{\prime},\,q,\,q\,^{\prime}\\}\,$, it is
possible to pick up commensurate values of $\,\dot{\omega}\,$ and $\,n\,$,
obeying

$\displaystyle\frac{\,\dot{\omega}\,}{n\,}\,=~{}-~{}\,\frac{\,2\,(p\,-\,p\,^{\prime})\,-\,(q\,-\,q\,^{\prime})\,}{2\,(p\,-\,p\,^{\prime})}~{}=~{}-~{}1~{}+~{}\frac{q\,-\,q\,^{\prime}}{\,2\,(p\,-\,p\,^{\prime})\,}\,~{}.$
(H19)

This equality will then be true also for any set
$\,\\{Kp,\,Kp\,^{\prime},\,Kq,\,Kq\,^{\prime}\\}\,$ with a nonzero integer
$\,K\,$. We should not, however, accept the formulae at their face value, but
should always keep in mind the physical meaning of the orbital elements of our
concern. While formally  solution of a perturbed Keplerian problem does not
prohibit the pericentre rate from being comparable to $\,n\,$, such a
proximity is unlikely to be attained in real life – except in the situations
where an orbit is undegoing a rapid and considerable change. Usually,
$\,\dot{\omega}\,$ is much  smaller in absolute value than $\,n\,$. Therefore,
$\,(q-q\,^{\prime})/(2(p-p\,^{\prime}))\,$ should be very close to, but not
exactly equal to $\,1\,$. There are no combinations of integers to make this
happen, because $\,2(p-p\,^{\prime})\,$ takes values between $\,-2\,l\,$ and
$\,2\,l\,$, so the smallest absolute value of the ratio
$\,(q-q\,^{\prime})/(2(p-p\,^{\prime}))\,$ is $\,1/(2\,l)\,$. For $\,l=2\,$,
this would make (H19) look $\,{\,\dot{\omega}\,}/{n\,}\,=~{}-~{}\,3/4\,$,
while for a larger $\,l\,$ the ratio $\,{\,\dot{\omega}\,}/{n\,}\,$ would be
even closer to unity, a situation irrelevant to actual settings. Therefore,
the entire second subsum in (H9d) must drop out.

Consider the first subsum in (H9d). As we saw, its secular part can be
transformed into the expression (H14), where the index $\,q\,^{\prime}\,$ is
constrained by the condition (H15). For a vanishing $\,\dot{\omega}\,$, the
condition used to entail $\,q\,^{\prime}=\,-\,q-2(l-p-p\,^{\prime})\,$, so the
secular part used to assume the form (H17). This will no longer be so in the
presence of $\,\dot{\omega}\,$. We then have two ways of satisfying the
condition (H15). One is to impose, simultaneously, the conditions

$\displaystyle l\,=\,p\,+\,p\,^{\prime}\,~{}~{},\qquad
q\,=\,-\,q\,^{\prime}\,~{},$ (H20)

in which case the secular part of the discussed subsum will be simplified to

$\displaystyle\frac{(-1)^{\,l}}{2}\,\sum_{p\,q}\,\left[A_{l0p\,^{\prime}q\,^{\prime}}\right]_{\textstyle{{}_{\,q\,^{\prime}\,=\,-q}}}^{\textstyle{{}^{\,l=p\,^{\prime}+p}}}~{}\,\omega_{l0pq}~{}B_{l0pq}~{}\sin\epsilon_{l0pq}~{}\,~{}.~{}\qquad~{}\,~{}$
(H21)

Another way is to pick up, for some set
$\,\\{l,\,p,\,p\,^{\prime},\,q,\,q\,^{\prime}\\}\,$, commensurate
$\,\dot{\omega}\,$ and $\,n\,$ obeying

$\displaystyle\frac{\,\dot{\omega}\,}{n\,}~{}=~{}-~{}1~{}-~{}\,\frac{q\,+\,q\,^{\prime}}{2\,(l\,-\,p\,-\,p\,^{\prime})}\,~{}.$
(H22)

This will work also for any set
$\\{Kl,\,Kp,\,Kp\,^{\prime},\,Kq,\,Kq\,^{\prime}\\}$ with a nonzero integer
$K$. Just as in the formula (H19), this option is not viable, as no set
$\\{l,\,p,\,p\,^{\prime},\,q,\,q\,^{\prime}\\}$ makes the pericentre rate much
lower than $\,n\,$. So the only realistic possibility is implemented by (H20
\- H21).

All in all, the secular part of the expression (H9d) acquires the form

${}^{\textstyle{{}^{(\dot{\omega}\neq 0)}}}\langle
W_{l}\,\,\dot{U}_{l}\rangle\,=~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad$
$\displaystyle\frac{(-1)^{\,l}}{2}\,\sum_{\,p\,q\,}\,\left[A_{l0p\,^{\prime}q\,^{\prime}}\right]_{\textstyle{{}_{q\,^{\prime}\,=\,-q}}}^{\textstyle{{}^{\,l=p\,^{\prime}+p}}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\sin\epsilon_{l0pq}~{}+~{}\frac{1}{2}~{}\sum_{m}\sum_{\,p\,q\,}\,A_{lmpq}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin\epsilon_{lmpq}\,+~{}.~{}.~{}.~{}~{},\quad$
(H23)

the ellipsis signifying the part vanishing after integration over the
longitude $\,\lambda\,$.

####  H.2.4 Comparison of the two cases

Comparing formulae (H18b) and (H23), we see that a whole group of terms,
absent in the moving-pericentre case, shows up when the pericentre stays idle.
These are the terms obtained by summation over $\,p\,^{\prime}\,$. When
averaging over the apsidal period is legitimate, most of them vanish and the
value of $\,p\,^{\prime}\,$ gets fixed. Otherwise these terms stay.

As we explained above, averaging is physically meaningful over timescales
which do not exceed the Maxwell time by too many orders of magnitude. Over
longer periods, linear viscoelastic reaction is not guaranteed. Specifically,
we cannot be sure that full rebound is possible. At such long times, it would
be physical to regard the periapse idle – which will change the averaging
procedure and will leave unaveraged the terms numbered with all
$\,p\,^{\prime}\,$.

To understand whether the periapse should be regarded quiescent or not, its
period must be compared with the Maxwell time. The latter is shorter (and may
thus fall short of the apsidal period) at higher temperatures. So for heated-
up planets the periapse may have to be regarded as idle, thus giving birth to
the $\,p\,^{\prime}\,\neq\,p\,$ terms and making the heat production more
intensive.

###  H.3 The low-inclination approximation

A considerable simplification becomes available if we agree to omit the
$\,O(i^{2})\,$ terms. To appreciate this, examine the expressions for the
inclination functions $\,F_{lmp}(i)\,$.

For $\,l=2\,$, we have:

$F_{201}(i)~{}=~{}-~{}\frac{1}{2}~{}+~{}O(i^{2})\quad,\qquad
F_{220}(i)~{}=~{}3~{}+~{}O(i^{2})~{}~{},$

the other $\,F_{2mp}(i)\,$ being of the order of $\,i\,$ or higher. Moreover,
as the functions $\,F_{200}(i)\,$ and $\,F_{202}(i)\,$ are of the order of
$\,i^{2}\,$, the only relevant product with $\,m=0\,$ is the diagonal one:
$\,F_{201}^{\,2}(i)\,=\,{1}/{4}\,+\,O(i^{2})~{}$, the cross products
$\,F_{20p\,^{\prime}}(i)\,F_{20p}(i)\,$ being $\,O(i^{2})\,$. Similarly, since
$\,F_{221}(i)\,$ and $\,F_{222}(i)\,$ are of the order of $\,i^{2}\,$ or
higher, the only relevant product with $\,m=2\,$ is diagonal:
$\,F_{220}^{\,2}(i)~{}=~{}9~{}+~{}O(i^{2})~{}$,

Performing the same check for $\,l=3\,$, we write down:

$F_{311}(i)~{}=~{}-~{}\frac{3}{2}~{}+~{}O(i^{2})\quad,\qquad
F_{330}(i)~{}=~{}15~{}+~{}O(i^{2})~{}~{},$

and notice that the other $\,F_{3mp}(i)\,$ are of the order of $\,i\,$ or
higher. Above that, as the functions $\,F_{310}(i)\,$, $\,F_{312}(i)\,$, and
$\,F_{313}(i)\,$ are of the order of $\,i^{2}\,$ or higher, the only surviving
product with $\,m=1\,$ is the diagonal one,
$\,F^{\,2}_{311}(i)\,=\,{9}/{4}\,+\,O(i^{2})\,$, the cross products being
$\,O(i^{2})\,$. By the same token, since $\,F_{331}(i)\,$, $\,F_{332}(i)\,$,
and $\,F_{333}(i)\,$ are of the order of $\,i^{2}\,$ or higher, the only
remaining term with $\,m=3\,$ is diagonal:
$\,F^{\,2}_{330}(i)\,=\,225\,+\,O(i^{2})\,$.

A similar examination of the functions with $\,l=4\,$ demonstrates that only
the diagonal terms $\,F_{402}^{2}(i)\,$, $\,F_{421}^{2}(i)\,$, and
$\,F_{440}^{2}(i)\,$ survive, the cross terms being $\,O(i^{2})\,$ or higher-
order.

We have performed this check for all $\,l\leq 7\,$, and made sure that only
the $\,p\,^{\prime}=p\,$ terms exceed $\,O(i^{2})\,$. As the terms with
$\,l>7\,$ are unlikely to play any role in practical calculations ever, 292929
Each product $\,A_{lmp\,^{\prime}q\,^{\prime}}\,B_{lmpq}\,$ is proportional to
$\,(R/a)^{2l}\,$, and an extra factor of $\,R/a\,$ will show up later, when we
plug everything into the expression for heat production rate. Indeed,
integration over the surface will give us a factor of $\,R^{2}\,$ in (H27).
Then the overall multiplier $\,1/(4\pi GR)\,$ standing in (65b) must be taken
into account. All together, this leaves us with an extra $\,R\,$, thus making
the final expression for dissipation rate proportional to $\,(R/a)^{2l+1}\,$.
we stop our check here and reduce our sum to the diagonal
($\,p\,^{\prime}=p\,$) terms only.

Then, for a quiescent pericentre, the secular part of (H9d) will be:

${}^{\textstyle{{}^{(\dot{\omega}=0)}}}\langle\,W_{l}\,\,\dot{U}_{l}\,\rangle$
$\displaystyle=$
$\displaystyle\frac{(-1)^{\,l}}{2}~{}\sum_{p\,q}\,\left[A_{l0pq\,^{\prime}}\right]_{\textstyle{{}_{q\,^{\prime}\,=\,-q-2(l-2p)}}}~{}~{}\omega_{l0pq}~{}B_{l0pq}~{}\cos(\,2\,(l\,-\,2\,p)\,\omega_{0}\,)~{}\sin\epsilon_{l0pq}~{}$
(H24) $\displaystyle+$
$\displaystyle\frac{1}{2}~{}\sum_{m}\sum_{pq}\,A_{lmpq}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin\epsilon_{lmpq}~{}+\,~{}.~{}.~{}.~{}\,~{},\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

For a moving pericentre, the secular part of (H9d) will assume the form of

${}^{\textstyle{{}^{(\dot{\omega}\neq 0)}}}\langle
W_{l}\,\,\dot{U}_{l}\rangle\,=~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad~{}\qquad$
$\displaystyle\frac{(-1)^{\,l}}{2}\,\sum_{\,p\,q\,}\,\left[A_{l0pq\,^{\prime}}\right]_{\textstyle{{}_{q\,^{\prime}\,=\,-q}}}^{\textstyle{{}^{\,p=l/2}}}~{}\omega_{l0pq}~{}B_{l0pq}~{}\sin\epsilon_{l0pq}~{}+~{}\frac{1}{2}~{}\sum_{m}\sum_{\,p\,q\,}\,A_{lmpq}~{}\omega_{lmpq}~{}B_{lmpq}~{}\sin\epsilon_{lmpq}\,+~{}.~{}.~{}.~{}~{}.\quad$
(H25)

Recall that the ellipsis denotes the part vanishing after integration over the
longitude. In the first sum, the condition $\,p=l/2\,$ means that the first
sum is nil for odd values of $l$.

The formulae (H24) and (H25) have been derived here for reference purposes
solely. While they enable one to write down much shorter expressions for the
tidal dissipation rate, below we shall write the expressions which are general
and can be used for an arbitrary $\,i\,$.

###  H.4 The dissipation rate

####  H.4.1 The case of a quiescent pericentre

Insertion of the formulae (H3), (H7) and (H18b) into the equation (65b) gives
us the damping rate in the case when the pericentre is fixed and no averaging
over the apsidal period is required:

$\displaystyle\langle\,P\,\rangle~{}=~{}\frac{1}{4\pi
GR}~{}\sum_{l}(2l+1)~{}\int dS~{}\left\langle
W_{l}\,\dot{U}_{l}\right\rangle~{}=~{}$ $\displaystyle\frac{1}{4\pi
GR}\,\left(\frac{G\,M^{*}}{a}\right)^{2}\,\sum_{l=2}^{\infty}\,(2l+1)\,\left(\frac{R}{a}\right)^{\textstyle{{}^{2\,l}}}\left\\{~{}\frac{(-1)^{\,l}}{2}~{}\sum_{p=0}^{l}\sum_{p\,^{\prime}=0}^{l}F_{l0p}(i)~{}F_{l0p\,^{\prime}}(i)\,\sum_{q\,=-\infty}^{\infty}\,G_{lpq}(e)~{}\right.$
$\displaystyle\left[\,G_{lp\,^{\prime}q\,^{\prime}}(e)\,\right]_{\textstyle{{}_{q\,^{\prime}\,=\,-\,q\,-\,2\,(l-p-p\,^{\prime})}}}\,~{}\omega_{\textstyle{{}_{l0pq}}}\,\,k_{l}(\omega_{\textstyle{{}_{l0pq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{l0pq}}})~{}\cos\left(\,2\,(l\,-\,p\,^{\prime}\,-\,p)\,\omega_{0}\,\right)\int\,dS\;\left[\,P_{l0}(\sin\phi)\,\right]^{2}$
$\displaystyle+~{}\frac{1}{2}~{}\sum_{m=0}^{l}\,\left[~{}\frac{(l-m)!}{({\it
l}+m)!}\;\left(\,2~{}-~{}\delta_{0m}\,\right)~{}\right]^{\textstyle{{}^{2}}}~{}\sum_{p=0}^{l}F_{lmp}(i)\;\sum_{p\,^{\prime}=0}^{l}F_{lmp\,^{\prime}}(i)~{}\sum_{q\,=-\infty}^{\infty}G_{lpq}(e)~{}\,\qquad\qquad\qquad~{}$
$\displaystyle\left.\left[\,G_{lp\,^{\prime}q\,^{\prime}}(e)\,\right]_{\textstyle{{}_{q\,^{\prime}\,=\,q\,-\,2\,(p-p\,^{\prime})}}}~{}\cos\left(\,2\,(p\,^{\prime}\,-\,p)\,\omega_{0}\,\right)\,~{}\omega_{\textstyle{{}_{lmpq}}}\,\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,\int\,dS\,\left[\,P_{{\it{l}}m}(\sin\phi)\,\right]^{2}\right\\}~{}~{}.~{}~{}~{}~{}$
(H26)

Using the normalisation (A3b) and recalling that
$dS=R^{2}\,\cos\phi\,d\phi\,d\lambda\,$, write the integral as

$\displaystyle\int\,dS\;\left[\,P_{{\it{l}}m}(\sin\phi)\,\right]^{2}\,=~{}R^{2}\,2~{}\pi~{}\frac{2}{2l\,+\,1}\,~{}\frac{(l+m)!}{(l-m)!}~{}\,,$
(H27)

whence

$\displaystyle\langle\,P\,\rangle~{}=~{}$
$\displaystyle\frac{G\,{M^{*}}^{\,2}}{2~{}a}\,\sum_{l=2}^{\infty}\,\left(\frac{R}{a}\right)^{\textstyle{{}^{2\,l\,+\,1}}}\left\\{~{}(-1)^{\,l}~{}~{}\delta_{0m}~{}\sum_{p=0}^{l}\sum_{p\,^{\prime}=0}^{l}F_{lmp}(i)~{}F_{lmp\,^{\prime}}(i)~{}\quad\right.$
$\displaystyle\,\sum_{q\,=-\infty}^{\infty}\,G_{lpq}(e)~{}\left[\,G_{lp\,^{\prime}q\,^{\prime}}(e)\,\right]_{\textstyle{{}_{q\,^{\prime}\,=\,-\,q\,-\,2\,(l-p-p\,^{\prime})}}}\,~{}\cos\left(\,2\,(l\,-\,p\,^{\prime}\,-\,p)\,\omega_{0}\,\right)$
$\displaystyle+~{}\sum_{m=0}^{l}~{}\frac{(l-m)!}{({\it
l}+m)!}\;\left(\,2~{}-~{}\delta_{0m}\,\right)^{2}~{}\sum_{p=0}^{l}F_{lmp}(i)\;\sum_{p\,^{\prime}=0}^{l}F_{lmp\,^{\prime}}(i)~{}\,\qquad\qquad~{}\,\qquad\qquad~{}$
$\displaystyle\left.~{}\sum_{q\,=-\infty}^{\infty}G_{lpq}(e)~{}\left[\,G_{lp\,^{\prime}q\,^{\prime}}(e)\,\right]_{\textstyle{{}_{q\,^{\prime}\,=\,q\,-\,2\,(p-p\,^{\prime})}}}~{}\cos\left(\,2\,(p\,^{\prime}\,-\,p)\,\omega_{0}\,\right)~{}\right\\}\,~{}\omega_{\textstyle{{}_{lmpq}}}\,\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}~{}~{}.~{}\qquad$
(H28)

The expression will be simplified considerably, if we notice that the first
sum in the curly brackets can be absorbed by the second sum. To appreciate
this, introduce auxiliary integers

$\displaystyle\tilde{p}~{}\equiv~{}l~{}-~{}p\,^{\prime}\,~{}\quad\mbox{and}~{}\qquad\tilde{q}~{}\equiv~{}-~{}q\,^{\prime}\,~{}.$
(H29)

Just as $\,p\,^{\prime}\,$, the integer $\,\tilde{p}\,$ assumes the values
from $\,0\,$ through $\,l\,$. Like $\,q\,^{\prime}\,$, the integer
$\,\tilde{q}\,$ runs from $\,-\,\infty\,$ to $\,\infty\,$. In terms of
$\,\tilde{p}\,$ and $\,\tilde{q}\,$, the first sum becomes

$\displaystyle(-1)^{\,l}\,\delta_{0m}\sum_{p=0}^{l}\sum_{\tilde{p}=0}^{l}F_{lmp}(i)\,F_{lm(l-\tilde{p})}(i)\sum_{q\,=-\infty}^{\infty}G_{lpq}(e)\,\left[\,G_{l(l-\tilde{p})(-\tilde{q})}(e)\,\right]_{\textstyle{{}_{\tilde{q}\,=\,q\,-\,2\,(p\,-\,\tilde{p})}}}~{}\cos\left(2(\tilde{p}-p)\,\omega_{0}\,\right)~{}\qquad$
(H30a)
$\displaystyle=~{}\delta_{0m}\sum_{p=0}^{l}\sum_{\tilde{p}=0}^{l}F_{lmp}(i)\,F_{lm\tilde{p}}(i)\sum_{q\,=-\infty}^{\infty}G_{lpq}(e)\,\left[\,G_{l\tilde{p}\tilde{q}}(e)\,\right]_{\textstyle{{}_{\tilde{q}\,=\,q\,-\,2\,(p\,-\,\tilde{p})}}}~{}\cos\left(2(\tilde{p}-p)\,\omega_{0}\,\right)\,~{},~{}~{}\qquad$
(H30b)

where we made use of the relations 303030 The equality (H32) is borrowed from
Giacaglia (1976a, eqn. 23), while (H33) comes from Gooding & Wagner (2008, eqn
10). The formula (H34) stems from the integral expression (e.g., Giacaglia
1976b, eqn 4)  $\displaystyle
F_{lmp}(i)~{}=~{}\frac{1}{2\,\pi}~{}\int_{-\pi}^{\pi}P^{m}_{l}(\sin\phi)~{}e^{\textstyle{{}^{i\,[m\lambda\,-\,(l-2p)u]}}}\,du~{}~{}.$
(H31)

$\displaystyle G_{lpq}(e)~{}=~{}G_{l(l-p)(-q)}(e)~{}~{},$ (H32) $\displaystyle
F_{lmp}(i)~{}=~{}(-1)^{l-m}~{}F_{lm(l-p)}(\pi-i)$ (H33)

and

$\displaystyle F_{l0p}(i)~{}=~{}F_{l0p}(\pi-i)~{}~{}.$ (H34)

Evidently, (H30b) is equal to the $\,m=0\,$ part of the second sum in the
curly brackets in (H28). Denoting $\,\tilde{p}\,$ with $\,p\,^{\prime}\,$
(which is legitimate, as these are just dummy indices), we can absorb the
expression (H30b) into the second sum. The outcome is:

$\displaystyle\langle\,P\,\rangle~{}=~{}\frac{G\,{M^{*}}^{\,2}}{a}\,\sum_{l=2}^{\infty}\,\left(\frac{R}{a}\right)^{\textstyle{{}^{2\,l\,+\,1}}}\sum_{m=0}^{l}~{}\frac{(l-m)!}{({\it
l}+m)!}\;\left(\,2-\delta_{0m}\,\right)\,\sum_{p=0}^{l}F_{lmp}(i)\;\sum_{p\,^{\prime}=0}^{l}F_{lmp\,^{\prime}}(i)$
$\displaystyle\sum_{q\,=-\infty}^{\infty}G_{lpq}(e)~{}\left[\,G_{lp\,^{\prime}q\,^{\prime}}(e)\,\right]_{\textstyle{{}_{q\,^{\prime}\,=\,q\,-\,2\,(p-p\,^{\prime})}}}~{}\cos\left(\,2\,(p\,^{\prime}\,-\,p)\,\omega_{0}\,\right)\,~{}\omega_{\textstyle{{}_{lmpq}}}\,\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\,~{}.~{}~{}\qquad$
(H35)

####  H.4.2 The case of a uniformly moving pericentre

Similarly to the above, insertion of the equations (H23), (H3) and (H7) into
the formula (65b) will give us the heating rate in the case when the
pericentre is moving and our time averaging includes that over the apsidal
period:

$\displaystyle\langle\,P\,\rangle~{}=~{}\frac{1}{4\pi
GR}~{}\sum_{l}(2l+1)~{}\int dS~{}\left\langle
W_{l}\,\dot{U}_{l}\right\rangle~{}=~{}$ $\displaystyle\frac{1}{4\pi
GR}\,\left(\frac{G\,M^{*}}{a}\right)^{2}\,\sum_{l=2}^{\infty}\,(2l+1)\,\left(\frac{R}{a}\right)^{\textstyle{{}^{2\,l}}}\left\\{~{}\frac{(-1)^{\,l}}{2}~{}\sum_{p=0}^{l}F^{\,2}_{l0p}(i)~{}(-1)^{l}\right.$
$\displaystyle\,\sum_{q\,=-\infty}^{\infty}\,G^{\,2}_{lpq}(e)~{}\omega_{\textstyle{{}_{l0pq}}}\,\,k_{l}(\omega_{\textstyle{{}_{l0pq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{l0pq}}})~{}\int\,dS\;\left[\,P_{l0}(\sin\phi)\,\right]^{2}\,+$
$\displaystyle\frac{1}{2}~{}\sum_{m=0}^{l}\,\left[~{}\frac{(l-m)!}{({\it
l}+m)!}\;\left(\,2~{}-~{}\delta_{0m}\,\right)~{}\right]^{\textstyle{{}^{2}}}~{}\sum_{p=0}^{l}F^{\,2}_{lmp}(i)\,\qquad\qquad\qquad~{}$
$\displaystyle\left.\sum_{q\,=-\infty}^{\infty}G^{\,2}_{lpq}(e)~{}\omega_{\textstyle{{}_{lmpq}}}\,\,k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\int\,dS\;\left[\,P_{{\it{l}}m}(\sin\phi)\,\right]^{2}~{}\right\\}\,~{}~{},~{}~{}$
(H36)

where we also employed the equalities (H32 \- H34). Further insertion of (H27)
leads us to:

$\displaystyle\langle\,P\,\rangle~{}=~{}\frac{G\,{M^{*}}^{\,2}}{2~{}a}\sum_{l=2}^{\infty}\left(\,\frac{R}{a}\,\right)^{\textstyle{{}^{2\,l\,+\,1}}}\,\sum_{m=0}^{l}\sum_{p=0}^{l}F^{\,2}_{lmp}(i)$
$\displaystyle\sum_{q\,=-\infty}^{\infty}G^{\,2}_{lpq}(e)\left[\delta_{0m}~{}+~{}\frac{(l-m)!}{({\it
l}+m)!}\left(2-\delta_{0m}\right)^{2}\right]\,\omega_{\textstyle{{}_{lmpq}}}k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}=~{}\quad$
(H37a)
$\displaystyle\frac{G\,{M^{*}}^{\,2}}{a}~{}\sum_{l=2}^{\infty}\left(\frac{R\,}{a}\right)^{\textstyle{{}^{2l+1}}}\,\sum_{m=0}^{l}\,\sum_{p=0}^{l}F^{\,2}_{lmp}(i)\sum_{q\,=-\infty}^{\infty}G^{\,2}_{lpq}(e)\frac{(l-m)!}{({\it
l}+m)!}\left(2-\delta_{0m}\right)\,\omega_{\textstyle{{}_{lmpq}}}k_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,~{}.\qquad$
(H37b)

It would be instructive to compare this expression with the corresponding
result from the classical paper by Peale and Cassen (1978). To this end, three
items must be kept in mind.

  * 1\.

The authors of Ibid. tacitly assumed that averaging should be carried out not
only over the tidal period but also over the apsidal period — this can be
understood from how their formulae (21) transformed into (22). This is why
their resulting formula (31) is appropriate to compare with our expression
(H37b).

  * 2\.

As the derivation in Ibid. was intended for the incompressible case and for
$\,l=2\,$ solely, we should use, for the purpose of comparison, the equality
$\,k_{2}(\omega_{\textstyle{{}_{2mpq}}})=3h_{2}(\omega_{\textstyle{{}_{2mpq}}})/5\,$
derived in Appendix F.

  * 3\.

In Ibid., only the case of synchronous rotation was addressed, with
$\,\omega_{2mpq}=\,(2-2p)\dot{\omega}\,+\,(2-2p+q)\,n\,+m\,(\dot{\Omega}-\dot{\theta})\,\approx\,(2-2p+q-m)\,n\,$,
where $\,n\,$ is the apparent mean motion of the perturber. Librations were
ignored.

Taking all this into account, we write, for the purpose of comparison, an
appropriately simplified version of our expression (H37b):

${}^{\textstyle{{}^{(synchronous)}}}\langle\,P\,\rangle^{\textstyle{{}^{(incompress)}}}_{\textstyle{{}_{~{}l=2}}}~{}=~{}$
$\displaystyle\frac{G\,{M^{*}}^{\,2}}{a}\left(\frac{R\,}{a}\right)^{\textstyle{{}^{2l+1}}}\sum_{m=0}^{2}\sum_{p=0}^{2}F^{\,2}_{2mp}(i)\sum_{q\,=-\infty}^{\infty}G^{\,2}_{2pq}(e)\frac{(2-m)!}{(2+m)!}\left(2-\delta_{0m}\right)\,\omega_{\textstyle{{}_{2mpq}}}~{}\frac{3}{5}~{}h_{2}(\omega_{\textstyle{{}_{2mpq}}})~{}\sin\epsilon_{2}(\omega_{\textstyle{{}_{2mpq}}})\,~{}.~{}~{}~{}$
(H38)

As the time lag in our formula (17a) is always positive-definite, the sign of
the phase lag $\,\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$ coincides
with that of the tidal mode $\,\omega_{\textstyle{{}_{lmpq}}}\,$, wherefore
the product
$\,\omega_{\textstyle{{}_{lmpq}}}~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$
can always be written down as a product of absolute values:

$\displaystyle\omega_{\textstyle{{}_{lmpq}}}~{}\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})~{}=~{}|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,\cdot\,|\,\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,|~{}=~{}\frac{\chi_{\textstyle{{}_{lmpq}}}}{Q_{\textstyle{{}_{lmpq}}}}~{}~{},$
(H39)

where $\,\frac{\textstyle 1~{}}{\textstyle
Q_{\textstyle{{}_{lmpq}}}}\,\equiv\,|\,\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})\,|\,$
is the inverse quality factor, while
$\,\chi_{\textstyle{{}_{lmpq}}}\,\equiv\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,$
is the positive-definite physical forcing frequency. For synchronous spin and
$\,l=2\,$, the forcing frequency is
$\,\chi_{\textstyle{{}_{2mpq}}}\approx\,|\,2\,-\,2\,p\,+q\,-m\,|\,n\,$, whence
the quadrupole input into the power is:

${}^{\textstyle{{}^{(synchronous)}}}\langle\,P\,\rangle^{\textstyle{{}^{(incompress)}}}_{\textstyle{{}_{~{}l=2}}}~{}=~{}$
$\displaystyle\frac{G\,{M^{*}}^{\,2}}{a}\left(\frac{R\,}{a}\right)^{\textstyle{{}^{2l+1}}}\sum_{m=0}^{2}\sum_{p=0}^{2}F^{\,2}_{2mp}(i)\sum_{q\,=-\infty}^{\infty}G^{\,2}_{2pq}(e)\frac{(2-m)!}{(2+m)!}\left(2-\delta_{0m}\right)~{}\frac{3}{5}~{}h_{2}~{}\frac{|\,2-2p+q-m\,|}{Q_{\textstyle{{}_{2mpq}}}}~{}n~{}~{},\qquad$
(H40)

with an absolute value in the numerator.

Peale & Cassen (1978) had in their formula (31) simply
$\,(\,2\,-\,2\,p\,+q\,-m\,)\,$ instead of $\,|\,2\,-\,2\,p\,+q\,-m\,|\,$. As a
result, their expression for dissipation rate contained negative inputs from
some Fourier modes (i.e., for some sets of $\,m\,p\,q\,$). Being of the order
of $\,e^{4}\,$, such inputs lead to underestimation of the heat production
rate in the situations where the eccentricity is high. The presence of such
inputs in the formula (31) from Ibid. was pointed out by Makarov (2013) who
explored tidal heat production in the Moon in the cause of its orbital
evolution. Presumably, the Moon was formed much closer to the Earth than it is
today, and could be captured into a three-body resonance with the Sun, driving
the orbital eccentricity to high values for a limited timespan (Touma & Wisdom
1994).

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Table 1: Symbol key  Notation | Description
---|---
$\mbox{{\boldmath$r$}}^{\,*}$ | the position of the star relative to the centre of the planet
$r^{\,*}$ | the star-planet distance
$\phi^{\,*}$ | the declination of the star relative to the equator of the planet
$\lambda^{\,*}$ | the right ascension of the star relative to a fixed meridian on the planet
${R}$ | a point on the surface of the planet
$r$ | a point outside the planet, located above the surface point ${R}$
$R$ | the radius of the planet
$r$ | distance from the centre of the planet to an exterior point $r$
$\phi$ | the latitude of the point ${R}$ on the surface of the planet
| (also the declination of an exterior point $r$ located above the surface
point ${R}$)
$\lambda$ | the longitude of a point ${R}$ on the surface of the planet
| (also the right ascension of an exterior point $r$ located above the surface
point ${R}$)
$W(\mbox{{\boldmath${R}$}},\,\mbox{{\boldmath$r$}}^{\,*})$ | tide-raising potential at a surface point ${R}$ of the planet
$W_{l}(\mbox{{\boldmath${R}$}},\,\mbox{{\boldmath$r$}}^{\,*})$ |  the $l$-degree part of the tide-raising potential at a surface point ${R}$ of the planet
$U(\mbox{{\boldmath$r$}},\,\mbox{{\boldmath$r$}}^{\,*})$ | additional tidal potential in a point $r$ outside the planet
$U_{l}(\mbox{{\boldmath$r$}},\,\mbox{{\boldmath$r$}}^{\,*})$ |  $.\,.\,.$ the $l$-degree part of the additional tidal potential in a point $r$ outside the planet
$V^{\,0}$ | the constant-in-time spherically symmetrical potential of an undeformed planet
$V\,^{\prime}$ | the total perturbation of the potential of the planet ($\,V\,^{\prime}\,=\,W\,+\,U\,$)
$V$ | the overall potential of the planet ($\,V\,=\,V^{\,0}\,+\,V\,^{\prime}\,=\,V^{\,0}\,+\,W\,+\,U\,$)
$\omega_{\textstyle{{}_{lmpq}}}$ | the Fourier modes of the tide
$\chi_{\textstyle{{}_{lmpq}}}$ | the physical frequencies of the tidal stresses and strains ($\,\chi_{\textstyle{{}_{lmpq}}}\,=\,|\,\omega_{\textstyle{{}_{lmpq}}}\,|\,$)
$\chi$ | a shorter notation for $\,\chi_{\textstyle{{}_{lmpq}}}\,$
$M$ | the mass of the planet
$M^{\,*}$ | the mass of the star
$a$ | the semimajor axis
$e$ | the eccentricity
$i$ | the obliquity (the inclination of the star as seen from the planet)
$\omega$ | the argument of the pericentre of the star as seen from the planet
$\omega$ | when there is no risk of confusion, $\omega$ is also used as a short notation for $\omega_{\textstyle{{}_{lmpq}}}$
$\Omega$ | the longitude of the node of the star as seen from the planet
${\cal{M}}$ | the mean anomaly of the star as seen from the planet
$n$ | the mean motion
$G$ | Newton’s gravitational constant
$\theta$ | the rotation angle of the planet
$\stackrel{{\scriptstyle\bf\centerdot}}{{\theta\,}}$ | the spin rate of the planet
$k_{l},\,h_{l}\,$ | the degree-$l\,$ static Love numbers of the planet
$k_{l}(\omega_{\textstyle{{}_{lmpq}}}),\,h_{l}(\omega_{\textstyle{{}_{lmpq}}})\,$ | the degree-$l\,$ dynamical Love numbers of the planet
$\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}})$ | the degree-$l\,$ tidal phase lag
$\Delta t_{l}(\omega_{\textstyle{{}_{lmpq}}})$ | the degree-$l\,$ tidal time lag
$Q_{l}(\omega_{\textstyle{{}_{lmpq}}})$ | the degree-$l\,$ tidal quality factor defined as $\,Q_{l}(\omega_{\textstyle{{}_{lmpq}}})\,\equiv\,1/\sin\epsilon_{l}(\omega_{\textstyle{{}_{lmpq}}}\,$)
$Q_{\textstyle{{}_{lmpq}}}$ | the $\,Q\,$ factor, in the notation of Kaula (1964) and Peale & Cassen (1978)
$F_{lmp}(i)$ | the inclination functions
$G_{lpq}(e)$ | the eccentricity polynomials
$\mu$ | the mean rigidity of the mantle
$J$ | the mean compliance of the mantle ($\,J\,=\,1/\mu\,$)
$J$ | when there is no risk of confusion, $J$ also denotes the Jacobian
$\rho$ | the mean density of the planet
g | the surface gravity on the planet
${\bf{u}}$ | tidal displacement in the planet
${\bf{v}}$ | the velocity of tidal displacement in the planet
$P$ | the power exerted on the planet by the tidal stresses
$\langle P\rangle$ | the time-averaged power (averaged over one or several cycles of flexure)