text:
\section{Introduction}
\label{sec:intro}

The classic equations for tidal evolution in two-body systems derived or utilized in seminal papers [e.g.,
\citet{macd64,gold66moon,gold66,mign79,mign81}], reviews [e.g., \citet{burn77,weid89,peal99}], and
textbooks [e.g.,~\citet{murr99,danb92}] are based upon the underlying assumption that the two spherical
components in the system are separated by several times the radius of the larger primary component.  While
this assumption is valid in planet-satellite systems\footnote{There are small natural satellites of the
outer planets that orbit very close to their primaries, but we must keep in mind that these satellites are
part of much more complex dynamical systems than simple two-component binaries in addition to having
negligible masses compared to their primaries.} such as Earth-Moon, Jupiter-Galilean satellites, and
Saturn-Titan, as well as for Pluto-Charon and the majority of binary main-belt asteroids (with 100-km-scale
primaries), it is not completely accurate for all binary asteroids, especially those in the near-Earth
region.  Based upon the compilation by~\citet{wals06} of measured and estimated binary asteroid component
size and semimajor axis parameters, nearly 75\% of near-Earth and Mars-crossing binaries have inter-component
separations between 3 and 5 primary radii.  An updated compilation of parameters by~\citet{prav07} including
small main-belt binaries, those with primaries less than 10~km in diameter, confirms that 75\% of binary
systems among these three populations have close mutual orbits.  In addition, double asteroids, those
systems with equal-size components that were not counted in the above tallies, such as (69230)
Hermes~\citep{marg03,prav03,marg06iau}, (90) Antiope~\citep{merl00iauc,mich04,desc07}, (854) Frostia, (1089)
Tama, (1313) Berna, and (4492) Debussy~\citep{behr06}, have separations within 5 primary radii.  The favored
formation mechanism for near-Earth, Mars-crossing, and small main-belt binaries is rotational fission or
mass shedding~\citep{marg02s,rich06,desc08} most likely due to YORP spin-up~\citep{prav07}, a torque on the
asteroid spin state due to re-emission of absorbed sunlight~\citep{rubi00,vokr02}, where the typical
binaries produced have equatorial mutual orbits with semimajor axes between 2 and 4.5 primary radii and
eccentricities below 0.15~\citep{wals08yorp}.  Though all binary systems in these three populations may
not have separations of less than 5 primary radii at present, if formed via spin-up, these systems likely
have tidally evolved outward from a closer orbit.

Complex generalized formulae for tidal evolution are presented by~\citet{kaul64} and \citet{mign80} as
extensions of the work of~\citet{darw79a,darw79b,darw80} that account for higher-order terms in the
expansion of the tidal potential, though, nearly universally, even by Darwin, Kaula, and Mignard
themselves, only the leading order is applied in practice under the assumption of a distant secondary
and the negligibility of higher-order terms.  To date, the most common application of higher-order
expansions of the tidal potential is in the Mars-Phobos system where tides on Mars raised by Phobos
orbiting at 2.76 Mars radii are causing the gradual infall of Phobos's orbit.  As the separation between
Mars and Phobos decreases, higher-order terms in the potential expansion must gain importance.  With
this in mind, attempts to understand the observed secular acceleration of Phobos and the past history
of its orbit date back to~\citet{redm64} and have continued with~\citet{smit76},~\citet{lamb79},
and~\citet{szet83}, among others, with~\citet{bill05} presenting the most recent treatment of the subject.

Because many binaries exist in a regime where traditional assumptions break down, and because tidal
evolution is most important at small separations, we are motivated to examine tidal interactions in
close orbits.  Here, we expand the gravitational potential between two spherical bodies to arbitrary
order as well as allow for a secondary of non-negligible mass.  We then present the resulting equations
for the evolution of the component spin rates and the semimajor axis due to the tidal bulges raised on
both components when restricted to systems with mutual orbits that are both circular and equatorial as
suggested for small binaries formed via spin-up.  The effect of accounting for close orbits is examined
and compared to the effect of uncertainties in physical parameters of the binary system.

\section{Tidal Potential of Arbitrary Order}
\label{sec:tidesl}

The potential $V$ per unit mass at a point on the surface of the primary body of mass $M_{\rm p}$, radius
$R_{\rm p}$, and uniform density $\rho_{\rm p}$ due to a secondary of mass $M_{\rm s}$, radius $R_{\rm s}$,
and uniform density $\rho_{\rm s}$ orbiting on a prograde circular path in the equator plane of the
primary with semimajor axis $a$ measured from the center of mass of the primary is

\begin{equation}
V=-G\,\frac{M_{\rm s}}{\Delta},
\label{eq:Vfull}
\end{equation}

\noindent
where $G$ is the gravitational constant and $\Delta$ is the distance between the center of the secondary
and the point of interest given by

\begin{equation}
\Delta=a\left[1-2\left(\frac{R_{\rm p}}{a}\right)\cos\psi+\left(\frac{R_{\rm p}}{a}\right)^2\right]^{1/2},
\end{equation}

\noindent
with $\psi$ measured from the line joining the centers of the primary and secondary [e.g., \citet{murr99}].
In the spherical polar coordinate system ($r$, $\theta$, $\phi$) shown in Fig.~\ref{fig:sec2}, with the
polar angle $\theta$ measured from the rotation axis of the primary and the azimuthal angle $\phi$ measured
from an arbitrary reference direction fixed in space, the separation angle $\psi$ between the secondary and
the point of interest on the primary is

\begin{equation}
\cos\psi~=~\cos\theta_{\rm p}\cos\theta_{\rm s}+\sin\theta_{\rm p}\sin\theta_{\rm s}\cos\left(\phi_{\rm p}-\phi_{\rm s}\right).
\label{eq:psi}
\end{equation}

\noindent
For widely separated binary systems where the semimajor axis $a$ is much larger than the radius of the
primary $R_{\rm p}$, the potential is expanded in powers of the small term $R_{\rm p}/a$ such that

\begin{equation}
V=-G\,\frac{M_{\rm s}}{a}\left[1+\left(\frac{R_{\rm p}}{a}\right)\cos\psi+\left(\frac{R_{\rm p}}{a}\right)^2\,\frac{1}{2}\left(3\cos^2\psi-1\right)+\ldots\right].
\label{eq:Vexpansion}
\end{equation}

\noindent
The first term is independent of the position of the point of interest and thus produces no force on the
primary.  The second term provides the force that keeps the mass element at the point of interest in a
circular orbit about the center of mass of the system.  The third term is the tidal potential

\begin{equation}
U=-G\,\frac{M_{\rm s}R_{\rm p}^2}{a^3}\,\frac{1}{2}\left(3\cos^2\psi-1\right)
\label{eq:Ushort}
\end{equation}

\noindent
that is the focus of past studies of tidal evolution where the the bodies are widely separated such as in
the Earth-Moon system.  However, truncation of the expansion of $V$ in (\ref{eq:Vexpansion}) at three terms
accurately estimates the true potential in (\ref{eq:Vfull}) only for separations exceeding 5$R_{\rm p}$.
For smaller separations, as are often found among binary asteroids, higher orders in the expansion of $V$
are necessary.

The full expansion of the potential $V$ in~(\ref{eq:Vexpansion}) may be written concisely as the sum over
Legendre polynomials $P_{\ell}(\cos\psi)$, i.e., zonal harmonic or azimuthally independent
surface harmonic functions, as

\begin{equation}
V=-G\,\frac{M_{\rm s}}{a}\,\sum_{\ell=0}^{\infty}\left(\frac{R_{\rm p}}{a}\right)^{\ell} P_{\ell}\left(\cos\psi\right),
\label{eq:Vlegendre}
\end{equation}

\noindent
where the $\ell=2$ term of the expansion of $V$ is the dominant tidal term in (\ref{eq:Ushort}).  The full
tidal potential $U$ including all orders becomes

\begin{equation}
U=-G\,\frac{M_{\rm s}}{a}\,\sum_{\ell=2}^{\infty}\left(\frac{R_{\rm p}}{a}\right)^{\ell} P_{\ell}\left(\cos\psi\right).
\label{eq:Usum}
\end{equation}

\noindent
While we will derive the tidal evolution equations in terms of an arbitrary order $\ell$,
Table~\ref{tab:legendre} lists the order $\ell$ of the expansion necessary for accurate reproduction of the
potential $V$ at small separations.  At 2$R_{\rm p}$, the potential must be expanded to at least $\ell=6$,
requiring four additional, but manageable, terms in the expansion.  This separation is convenient in terms
of tidal evolution as it is the contact limit of a binary system with two equal-size components and a
reasonable initial separation for the onset of tidal evolution in a newly formed binary system, regardless
of component size, especially for systems formed through primary spin-up and mass shedding~\citep{wals08yorp}.
Proceeding inward of 2$R_{\rm p}$ rapidly requires an unwieldy number of terms in the expansion
(e.g., twice as many additional terms are needed at 1.5$R_{\rm p}$).

\section{Roche Limit}
\label{sec:roche}

The well-known classical fluid Roche limit is located at $a=2.46R_{\rm p}$~\citep{chan69} for equal
density components, so that if one considers a secondary just outside the fluid limit, one must include
the Legendre polynomials of orders $\ell\le4$ in the expansion for the potential felt by the primary.
For solid, cohesionless\footnote{A cohesionless material has zero shear strength in the absence of
confining pressure.  The interlocking of the constituent particles under pressure, however, can give the
material shear strength.} bodies (gravitational aggregates or so-called rubble piles) modeled as a dry
soil, the Roche limit falls approximately between $1.5R_{\rm p}$ and $2R_{\rm p}$~\citep{hols06,hols08,shar09}.
The cohesionless Roche limit is based upon a binary system that is not tidally evolving, but the secondary
remains stressed by its self-gravity, rotation (synchronized to the orbital period), and the difference in
gravity from its near, primary-facing side to its far side.  \citet{hols06,hols08} illustrate that the mass
ratio of the components has a negligible effect on the Roche limit, but one would expect that allowing the
secondary to have a more rapid spin or allowing for higher-order tidal terms due to its proximity to the
primary will increase the internal stresses and push the Roche limit farther from the primary, though, as
noted by~\citet{shar09}, these issues have not been studied in detail.

With a modest amount of cohesion, the secondary may exist within the stated Roche limit~\citep{hols08}.
For the rough properties of a near-Earth binary of $\rho_{\rm p, s}=2$~g/cm$^{3}$ and $R_{\rm s}=100$~m, a
cohesion value of $<100$~Pa is enough to hold the secondary together at the surface of the
primary\footnote{The cohesion needed to prevent disruption scales as the square of both the density and
size of the secondary.  Thus, for a main-belt binary with a $R_{\rm s}=10$~km, the necessary cohesion is of
order 10$^{6}$~Pa, similar to monolithic rock.}.  For comparison, the surface material of comet Tempel 1
excavated by the Deep Impact mission projectile is estimated to have a shear strength of
$<65$~Pa~\citep{aher05} and an effective strength of 10$^{3}$~Pa~\citep{rich07}; fine-grained terrestrial
sand is found to have cohesion values up to 250~Pa~\citep{schellart00}.  Therefore, it is not unreasonable
that in the tidal field of the primary, the secondary can stably exist at the very least within the fluid
Roche limit (even if cohesionless), if not also within the cohesionless Roche limit (with a cohesion
comparable to comet regolith or sand), justifying our later choice to work to order $\ell=6$ corresponding
to a separation of 2$R_{\rm p}$.

\section{External Potential of Arbitrary Order}
\label{sec:extpotl}

The tidal potential $U_{\ell}$ of arbitrary order $\ell\ge2$ felt by the primary, taken from (\ref{eq:Usum}),
may be written concisely as

\begin{equation}
U_\ell~=~-g_{\rm p}\,\zeta_{\ell, \rm p}\,P_{\ell}\left(\cos\psi\right),
\end{equation}

\noindent
where $g_{\rm p}=GM_{\rm p}/R_{\rm p}^2$ is the surface gravity of the primary and

\begin{equation}
\zeta_{\ell, \rm p}~=~\frac{M_{\rm s}}{M_{\rm p}}\left(\frac{R_{\rm p}}{a}\right)^{\ell+1}R_{\rm p}.
\label{eq:zeta}
\end{equation}

\noindent
The combination $\zeta_{\ell, \rm p}P_{\ell}\left(\cos\psi\right)$ is the equilibrium tide height, due
to the tidal potential of order $\ell$, that defines the equipotential surface about a primary that
is completely rigid (inflexible).  Because the mass ratio $M_{\rm s}/M_{\rm p}\le1$ and we assume
$a\ge2R_{\rm p}$, the quantity $\zeta_{\ell, \rm p}/R_{\rm p}\le 1/8$ for all binary systems, and typically
$\zeta_{\ell, \rm p}/R_{\rm p}\ll1$.

For a body with realistic rigidity, the tidal potential $U_{\ell}$ physically deforms the surface of the
primary by a small distance $\lambda_{\ell, \rm p}R_{\rm p}S_{\ell}$ as a function of position on the
primary, where $\lambda_{\ell, \rm p}\ll1$ and $S_{\ell}$ is a surface harmonic function.  \citet{darw79a}
and \citet{love27} lay the groundwork for showing that, in general, the deformation of a homogeneous
density, incompressible sphere

\begin{equation}
\lambda_{\ell, \rm p}\,R_{\rm p}\,S_{\ell}~=~-\,h_{\ell, \rm p}\,\frac{U_{\ell}}{g_{\rm p}}~=~h_{\ell,\,\rm p}\,\zeta_{\ell,\,\rm p}\,P_{\ell}\left(\cos\psi\right)
\label{eq:lambdadef}
\end{equation}

\noindent
is given in terms of the displacement Love number $h_{\ell, \rm p}$~\citep{munk60},

\begin{equation}
h_{\ell, \rm p}~=~\frac{2\ell+1}{2\left(\ell-1\right)}\frac{1}{1+\frac{\left(2\ell^2+4\ell+3\right)\mu_{\rm p}}{\ell g_{\rm p}\rho_{\rm p}R_{\rm p}}},
\label{eq:hlove}
\end{equation}

\noindent
introducing $\mu_{\rm p}$ as the rigidity or shear modulus of the primary\footnote{\citet{darw79a}
realized the correspondence between elastic and viscoelastic media and provides a generalized form for
the deformation of a viscous spheroid, a function equivalent to~(\ref{eq:lambdadef}) he calls $\sigma$,
that when applied to an elastic spheroid, in terms of rigidity rather than viscosity, is equivalent to
the expression found here.}.  For bodies less than 200 km in radius, as all components of binary asteroid
systems are, the rigidity $\mu$ dominates the stress due to self-gravity
$g\rho R\sim G\rho^2R^2$~\citep{weid89}, even for rubble-pile structures (i.e., the model proposed
by~\citet{gold09}), such that the Love number $h_{\ell}\ll1$ for small bodies.  With
$h_{\ell, \rm p}$ and $\zeta_{\ell, \rm p}/R_{\rm p}$ small, and noting from~(\ref{eq:lambdadef}) that
$\lambda_{\ell, \rm p}=h_{\ell, \rm p}\,\zeta_{\ell, \rm p}/R_{\rm p}$, the assumption of a small
deformation factor $\lambda_{\ell, \rm p}$ is justified.

Of particular interest is the external potential felt by the secondary now that the primary has
been deformed.  It is this external potential that produces the tidal torque that transfers
angular momentum through the system.  Here, we slightly alter our spherical coordinate system such
that $\theta$ now measures the angle from the axis of symmetry of the tidal bulge, as in~\citet{murr99},
such that the surface of the nearly spherical primary is now given by

\begin{equation}
R~=~R_{\rm p}\left(1+\sum_{\ell=2}^{\infty}\lambda_{\ell, \rm p} P_{\ell}\left(\cos\theta\right)\right).
\label{eq:shell}
\end{equation}

\noindent
The potential felt at a point external to the primary is the sum of the potential of a spherical
primary with radius $R_{\rm p}$ and that of the deformed shell.  However, only that due to the deformed
shell, called the non-central potential by~\citet{murr99}, will contribute to the torque.

In Fig.~\ref{fig:sec4}, the reciprocal of the distance $\Delta$ between the external point
($r$, $\theta$, $\phi$) and a point on the surface of the primary
($r^{\prime}$, $\theta^{\prime}$, $\phi^{\prime}$) separated by an angle $\psi$, where
$r^{\prime}=R$ from~(\ref{eq:shell}), is

\begin{equation}
\frac{1}{\Delta}~=~\frac{1}{r}\sum_{\ell=0}^{\infty}\left(\frac{R_{\rm p}}{r}\right)^{\ell}P_{\ell}\left(\cos\psi\right)~+~O\left(\lambda_{\ell^{\prime}, \rm p}\right).
\label{eq:delta}
\end{equation}

\noindent
The use of $\ell^{\prime}$ denotes terms based upon the surface deformation rather than the expansion
of the distance between the points of interest.  The non-central potential (per unit mass of the
object disturbed by the potential) due to the deformed shell with mass element
$\displaystyle{\rho_{\rm p}R_{\rm p}^3\sum_{\ell^{\prime}=2}^{\infty}\lambda_{\ell^{\prime}, \rm p}P_{\ell^{\prime}}\left(\cos\theta^{\prime}\right)d\left(\cos\theta^{\prime}\right)d\phi^{\prime}}$ is

\begin{equation}
U_{\rm nc}~=~-G\rho_{\rm p} R_{\rm p}^2\left(\frac{R_{\rm p}}{r}\right)\sum_{\ell^{\prime}=2}^{\infty}\sum_{\ell=0}^{\infty}\lambda_{\ell^{\prime}, \rm p}\left(\frac{R_{\rm p}}{r}\right)^{\ell}\int\int P_{\ell^{\prime}}\left(\cos\theta^{\prime}\right)P_{\ell}\left(\cos\psi\right)d\left(\cos\theta^{\prime}\right)d\phi^{\prime},
\end{equation}

\noindent
where the double integral goes over the surface of the primary.  The integral of the product of
two surface harmonics like the Legendre polynomials over a surface is zero unless $\ell=\ell^{\prime}$
such that for a specific order $\ell\ge2$~\citep{macr67},

\begin{eqnarray}
U_{\ell, \rm nc} & = & -G\rho_{\rm p} R_{\rm p}^2\left(\frac{R_{\rm p}}{r}\right)\lambda_{\ell, \rm p}\,\times\,\frac{4\pi}{2\ell+1}\left(\frac{R_{\rm p}}{r}\right)^{\ell}P_{\ell}\left(\cos\theta\right)\nonumber\\
 & = & -\frac{3}{2\ell+1}h_{\ell, \rm p}\zeta_{\ell, \rm p}g_{\rm p}\left(\frac{R_{\rm p}}{r}\right)^{\ell+1}P_{\ell}\left(\cos\theta\right).
\end{eqnarray}

\noindent
By defining the more familiar potential Love number

\begin{equation}
k_{\ell, \rm p}~=~\frac{3}{2\ell+1}h_{\ell, \rm p}~=~\frac{3}{2\left(\ell-1\right)}\frac{1}{1+\frac{\left(2\ell^2+4\ell+3\right)\mu_{\rm p}}{\ell g_{\rm p}\rho_{\rm p}R_{\rm p}}},
\label{eq:klove}
\end{equation}

\noindent
which is of a similar order as $h_{\ell, \rm p}$, the non-central potential is written in the form

\begin{equation}
U_{\ell, \rm nc}~=~-k_{\ell, \rm p}g_{\rm p}\zeta_{\ell, \rm p}\left(\frac{R_{\rm p}}{r}\right)^{\ell+1}P_{\ell}\left(\cos\theta\right)
\label{eq:Uexternal}
\end{equation}

\noindent
such that $U_{\ell, \rm nc}$ at the surface of the primary is simply $k_{\ell, \rm p}U_{\ell}$.
Because $\mu \gg g\rho R$ for small bodies, the Love number $k_{\ell, \rm p}$ may be approximated by

\begin{equation}
k_{\ell, \rm p}~\simeq~\frac{3}{2\left(\ell-1\right)}\,\frac{\ell}{2\ell^2+4\ell+3}\frac{g_{\rm p}\rho_{\rm p}R_{\rm p}}{\mu_{\rm p}}~=~\frac{2\pi}{\ell-1}\,\frac{\ell}{2\ell^2+4\ell+3}\frac{G\rho_{\rm p}^2 R_{\rm p}^2}{\mu_{\rm p}}.
\label{eq:kapprox}
\end{equation}

\noindent
Taking the external point to be the position of the secondary orbiting at a distance $a$ from the
primary, the complete\footnote{Here, by complete we mean accounting for all orders $\ell$.  We have,
however, limited the result to first order in the Love number $k_{\ell, \rm p}$ because terms of order
$\lambda_{\ell, \rm p}$ were ignored in (\ref{eq:delta}).  These would have produced higher-order terms
in the Love number in the final form of the potential in (\ref{eq:Ufull}), but because we have argued
$\lambda_{\ell, \rm p}$ and $k_{\ell, \rm p}$ are both small quantities, terms of second and higher
order in the Love number are negligible.} non-central potential per unit secondary mass due to tides
raised on the primary is

\begin{eqnarray}
U_{\rm nc} & = & -g_{\rm p}\sum_{\ell=2}^{\infty}k_{\ell, \rm p}\zeta_{\ell, \rm p}\left(\frac{a}{R_{\rm p}}\right)^{-\left(\ell+1\right)}P_{\ell}\left(\cos\theta\right) \nonumber \\
 & = & -\frac{GM_{\rm s}}{R_{\rm p}}\sum_{\ell=2}^{\infty}k_{\ell, \rm p}\left(\frac{a}{R_{\rm p}}\right)^{-2\left(\ell+1\right)}P_{\ell}\left(\cos\theta\right).
\label{eq:Ufull}
\end{eqnarray}

\noindent
The non-central potential drops off quickly with increasing separation as the separation to the sixth
power for $\ell=2$ and by an additional square of the separation for each successive order.  The
$\theta$ term in the Legendre polynomial accounts for the angular separation between the external
point of interest and the tidal bulge of the primary.  For the specific location of the secondary, we
define the angle $\delta$ as the geometric lag angle between the axis of symmetry of the tidal
bulge and the line connecting the centers of the two components.

\section{Tidal Dissipation Function $Q$}
\label{sec:Q}

In addition to the rigidity $\mu$, the response of a homogeneous, incompressible sphere to a disturbing
potential is characterized by the tidal dissipation function $Q$ defined by

\begin{equation}
Q^{-1}~=~\frac{1}{2\pi E^{*}}\oint \left(-\frac{dE}{dt}\right)dt,
\end{equation}

\noindent where $E^{*}$ is the maximum energy stored in the tidal distortion and the integral is the
energy dissipated over one cycle [see~\citet{gold63} or~\citet{efro09} for detailed discussions].  This
definition is akin to the quality factor in a damped, linear oscillator and does not depend on the
details of how the energy is dissipated.  Friction in the response of the body to a tide-raising
potential plus the rotation of the body itself (at a spin rate $\omega$ compared to the mean motion $n$
in the mutual orbit about the center of mass of the system) lead to misalignment by the geometric lag
angle $\delta$.

The geometric lag relates to a phase lag by $\epsilon_{\ell m p q}=-m\,\delta\,{\rm sign}\left(\omega-n\right)$,
where the $\ell m p q$ notation follows~\citet{kaul64}, and we have implicitly assumed a single tidal
bulge as done by~\citet{gers55} and~\citet{macd64} by using a single positive geometric lag $\delta$
independent of the tidal frequencies\footnote{The definition of the phase lag~\citep{kaul64,efro09},
when one ignores changes in the periapse and node, is
$\epsilon_{\ell m p q}= \left[\left(\ell-2p+q\right)n-m\omega\right]\Delta t_{\ell m p q}$, where the
bracketed term is the tidal frequency and $\Delta t_{\ell m p q}$ is the positive time lag in the response
of the material to the tidal potential.  In the potential expansion by~\citet{kaul64}, only terms satisfying
$\ell-2p=m$ and $q=0$ survive for mutual orbits that are circular and equatorial such that
$\epsilon_{\ell m p q}=-m\,|\omega-n|\,\Delta t_{\ell m p q}\,{\rm sign}\left(\omega-n\right)=-m\,\delta\,{\rm sign}\left(\omega-n\right)$,
assuming a constant time lag and a single (positive) value for geometric lag for all viable combinations
of ${\ell m p q}$.}.  The tidal dissipation function $Q$, in turn, relates to the phase angle as
$Q^{-1}_{\ell m p q}=|{\rm cot}\,\,\epsilon_{\ell m p q}| \simeq |\epsilon_{\ell m p q}| + O(\epsilon_{\ell m p q}^{2})$~\citep{efro09}
provided energy dissipation is weak ($Q_{\ell m p q} \gg 1$).  The absolute value of
$\epsilon_{\rm \ell m p q}$ is required on physical grounds to ensure that $Q_{\ell m p q}$ is positive.
Since the tidal dissipation function is related to the phase lag, a different $Q_{\ell m p q}$
technically applies to each tidal frequency.  Compared to the dominant order $\ell=2$, where only the
${\ell m p q}=2200$ term survives in the setup of our problem,

\begin{equation}
Q_{\ell m p q}^{-1}=m\delta=\frac{m}{2}\,Q_{2200}^{-1}=\frac{m}{2}\,Q^{-1}
\label{eq:Q}
\end{equation}

\noindent
in general, where we define $Q \equiv Q_{2200}$ such that $Q_{\ell m p q}$ for any tidal frequency is
proportional to a single value of $Q$.  This simple relation between $Q_{\ell m p q}$ and the $Q$ of the
dominant tidal frequency is a direct result of our assumption of a single geometric lag independent of
tidal frequency.  Such a choice may not be the most realistic physical model\footnote{In our model, $Q$
varies inversely with the tidal frequency.  \citet{efro09} argue in favor of a rheological model where
$Q$ scales to a positive fractional power of the tidal frequency (at least for terrestrial planets).  It
is unclear what rheological model is proper for gravitational aggregates like binary asteroids.}, but
does allow for simpler mathematical manipulation.  Because $Q$ is necessarily positive regardless of the
sign of the phase lag, we append ${\rm sign}\left(\omega-n\right)$ to our forthcoming equations, where
the spin rate $\omega$ relates to the tidally distorted component.  If $\omega > n$, the bulge leads; if
$\omega < n$, the bulge lags behind.

\section{Tidal Torques on the Components}
\label{sec:torques}

The force on the secondary due to the distorted primary at order $\ell$ is $-M_{\rm s}\nabla U_{\ell, \rm nc}$,
and because we have restricted the problem to a circular, equatorial mutual orbit, the tidal bulge
remains in the orbit plane, and the sole component of the force is tangential to the mutual orbit.
Returning to the notation where $\psi$ measures the angle from the axis of symmetry of the tidal
bulge\footnote{In this notation, the tidal potential in~(\ref{eq:Usum}) deforms the shape of the component
according to~(\ref{eq:lambdadef}) and produces the external potential~(\ref{eq:Ufull}) all in terms of the
single angle $\psi$.}, the force at the location of the secondary is proportional to
$\displaystyle{-\left.\partial P_{\ell}/\partial\psi\right|_{\psi=\delta}}$ and pointed in the
$-\hat{\psi}$ direction.  The value of $\delta$ is taken to be positive as stated in the previous section
such that, for $\delta$ small, the quantity $\displaystyle{-\left.\partial P_{\ell}/\partial\psi\right|_{\psi=\delta}}$
is positive and the primary bulge attracts the secondary.  For a prograde mutual orbit with
$\omega_{\rm p}>n$, the primary bulge pulls the secondary ahead in the orbit; if $\omega_{\rm p}<n$, the
primary bulge retards the motion of the secondary (see Fig.~\ref{fig:sec6}).  The resulting torque vector
acting upon the orbit of the secondary, which is located at position $\textbf{r}$ with respect to the
center of mass of the primary, is given by
\textbf{$\Gamma_{\ell, \rm p}$}$=\textbf{r}\times\left(-M_{\rm s}~\nabla U_{\ell, \rm nc}\right)$.  Thus,
the torque vector, in general, is proportional to
$\displaystyle{-\left.\partial P_{\ell}/\partial\psi\right|_{\psi=\delta}\,\left(\hat{\psi}\times\hat{r}\right)}$.
As defined, the direction (sign) of $\hat{\psi}\times\hat{r}$ depends on whether the tidal bulge leads or
lags, and we indicate this in the magnitude of the torque via the term ${\rm sign}\left(\omega-n\right)$
such that the torque on the orbit of the secondary due to the $\ell^{\rm th}$-order deformation of the
primary is

\begin{eqnarray}
\Gamma_{\ell, \rm p} & = & -M_{\rm s}\frac{\partial U_{\ell, \rm nc}}{\partial \psi_{\rm p}}\nonumber\\
 & = & k_{\ell, \rm p}\frac{GM_{\rm s}^2}{R_{\rm p}}\left(\frac{a}{R_{\rm p}}\right)^{-2\left(\ell+1\right)}\left(\left.-\frac{\partial P_{\ell}\left(\cos\psi_{\rm p}\right)}{\partial \psi_{\rm p}}\right|_{\psi_{\rm p}=\delta_{\rm p}}\right){\rm sign}\left(\omega_{\rm p}-n\right).
\label{eq:torque}
\end{eqnarray}

\noindent
where $\delta_{\rm p}$ is the geometric lag angle between the primary's tidal bulge and the line
of centers.  A positive (negative) torque increases (decreases) the energy of the orbit at a rate
$\Gamma_{\rm p}\,n$.  An equal and opposite torque alters the rotational energy of the primary at a rate
$-\Gamma_{\rm p} \omega_{\rm p}$ such that the total energy $E$ of the system is dissipated over time at
a rate $\dot{E}=-\Gamma_{\rm p}\left(\omega_{\rm p}-n\right)<0$ as heat inside the primary.  Though energy
is dissipated, angular momentum is conserved due to the equal and opposite nature of the torques on the
orbit and the rotation of the primary.  Conservation of angular momentum results in the evolution of the
mutual orbit and is discussed in the following section.

A similar torque arises from tides raised on the secondary.  By the symmetry of motion about the center
of mass, the torque $\Gamma_{\ell, \rm s}$ is given by swapping the subscripts p and s in (\ref{eq:torque})
such that

\begin{eqnarray}
\Gamma_{\ell, \rm s} & = & k_{\ell, \rm s}\frac{GM_{\rm p}^2}{R_{\rm s}}\left(\frac{a}{R_{\rm s}}\right)^{-2\left(\ell+1\right)}\left(-\left.\frac{\partial P_{\ell}\left(\cos\psi_{\rm s}\right)}{\partial\psi_{\rm s}}\right|_{\psi_{\rm s}=\delta_{\rm s}}\right)\,{\rm sign}\left(\omega_{\rm s}-n\right)\label{eq:torques}\\
 & = & k_{\ell, \rm p}\frac{GM_{\rm s}^2}{R_{\rm p}}\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^{2\ell-3}\frac{\mu_{\rm p}}{\mu_{\rm s}}\left(\frac{a}{R_{\rm p}}\right)^{-2\left(\ell+1\right)}\left(-\left.\frac{\partial P_{\ell}\left(\cos\psi_{\rm s}\right)}{\partial\psi_{\rm s}}\right|_{\psi_{\rm s}=\delta_{\rm s}}\right)\,{\rm sign}\left(\omega_{\rm s}-n\right),\nonumber
\end{eqnarray}

\noindent
where $\delta_{\rm s}$ is the geometric lag angle between the tidal bulge of the secondary and the line
of centers.  This torque changes the orbital energy at a rate $\Gamma_{\rm s}\,n$, and the equal and
opposite torque alters the rotational energy of the secondary at a rate $-\Gamma_{\rm s}\omega_{\rm s}$,
dissipating energy as heat in the secondary at a rate $\dot{E}=-\Gamma_{\rm s}\left(\omega_{\rm s}-n\right)$.
Torques on the primary and secondary weaken for higher orders of $\ell$ and increasing separations, as
expected, and do so in the same manner as the non-central potential in (\ref{eq:Ufull}).  Once the
rotation rate of a component synchronizes with the mean motion of the mutual orbit, the associated torque
goes to zero due to the ${\rm sign}\left(\omega-n\right)$ term\footnote{If the mutual orbit were not
circular, a radial tide owing to the eccentricity would continue to act despite the synchronization of
the component spin rate to the mean motion.}.  Note that we have ignored interactions between the tidal
bulges of the components as these will depend on the square (or higher powers) of the Love numbers, which
we have argued are negligible (see Footnote 5).

\section{Spin Rate and Semimajor Axis Evolution for Close Orbits}
\label{sec:eqns}

During tidal evolution, angular momentum is transferred between the spins of the components and the
mutual orbit.  For simplicity, assume that the primary and secondary have spin axes parallel to the
normal of the mutual orbit plane and rotate in a prograde sense.  Then, the torque on the distorted
primary alters its spin with time at a rate $\dot{\omega_{\rm p}}=-\Gamma_{\rm p}/I_{\rm p}$, where
$I_{\rm p}=\alpha_{\rm p}M_{\rm p}R_{\rm p}^2$ is the moment of inertia of the primary.  The pre-factor
$\alpha$ is $2/5$ for a uniform density sphere, but can vary with the internal structure of the body,
and is left as a variable here such that the change in spin rate of the primary is

\begin{equation}
\dot{\omega}_{\ell, \rm p}~=~-\frac{k_{\ell, \rm p}}{\alpha_{\rm p}}\,\frac{\kappa^2}{1+\kappa}\left(\frac{a}{R_{\rm p}}\right)^{-2\ell+1}n^2\left(-\left.\frac{\partial P_{\ell}\left(\cos\psi_{\rm p}\right)}{\partial \psi_{\rm p}}\right|_{\psi_{\rm p}=\delta_{\rm p}}\right){\rm sign}\left(\omega_{\rm p}-n\right),
\label{eq:wp}
\end{equation}

\noindent
recalling that $-\partial P_{\ell}/\partial\psi\ge0$ for small angles and defining the mass ratio
$\kappa \equiv M_{\rm s}/M_{\rm p}=\left(\rho_{\rm s}/\rho_{\rm p}\right)\left(R_{\rm s}/R_{\rm p}\right)^3$.
Also note that $n^2$, which is proportional to $\left(a/R_{\rm p}\right)^{-3}$, was introduced via
Kepler's Third Law, $n^2a^3=G\left(M_{\rm p}+M_{\rm s}\right)$.  For rapidly spinning primaries with
$\omega_{\rm p}>n$, the torque will slow the rotation.

To conserve angular momentum in the system, the change in spin angular momentum, given by the torque
$-\Gamma_{\ell, \rm p}$, plus the change in orbital angular momentum must be zero.  The orbital
angular momentum for a circular mutual orbit $M_{\rm p}M_{\rm s}/\left(M_{\rm p}+M_{\rm s}\right)\,na^{2}$
changes with time as $\left(1/2\right) M_{\rm p}M_{\rm s}/\left(M_{\rm p}+M_{\rm s}\right)\,na\dot{a}$
such that conservation requires

\begin{equation}
\left(\frac{\dot{a}}{R_{\rm p}}\right)_{\ell, \rm p}~=~2k_{\ell, \rm p}\,\kappa\left(\frac{a}{R_{\rm p}}\right)^{-2\ell}n\left.\left(-\frac{\partial P_{\ell}\left(\cos\psi_{\rm p}\right)}{\partial\psi_{\rm p}}\right|_{\psi_{\rm p}=\delta_{\rm p}}\right) {\rm sign}\left(\omega_{\rm p}-n\right)
\label{eq:adotp}
\end{equation}

\noindent
for each order $\ell$.  For rapidly spinning primaries, the orbit will expand as angular momentum is
transferred from the spin of the primary to the mutual orbit and, so long as the geometric lag remains
small, higher orders will cause both more rapid despinning of the primary and faster expansion of the
mutual orbit than $\ell=2$ alone.  A large secondary with $\kappa\sim1$ clearly causes the most rapid
tidal evolution.  A small secondary with $\kappa\ll1$ will not cause the primary to despin appreciably
due to the $\kappa^2$-dependence of (\ref{eq:wp}), but the separation will evolve more readily as
(\ref{eq:adotp}) scales as $\kappa$.

One can derive the change in the semimajor axis in (\ref{eq:adotp}) by other methods including the work
done on the orbit and Gauss's formulation of Lagrange's planetary equations.  Setting the time derivative
of the total energy of the orbit $-GM_{\rm p}M_{\rm s}/2a$, which is $GM_{\rm p}M_{\rm s}\dot{a}/2a^{2}$,
equal to the work done on the orbit $\Gamma_{\ell, \rm p}n$ simplifies to (\ref{eq:adotp}).  Using Gauss's
formulation for spherical bodies (see \citet{burn76} for a lucid derivation) and a circular mutual orbit,

\begin{equation}
\dot{a}_{\ell}~=~\frac{2}{n}\left(1+\kappa\right)T_{\ell},
\label{eq:gauss}
\end{equation}

\noindent
where $T_{\ell}$ is the tangential component of the disturbing force (per unit mass) from the previous
section, which is $\left(1/a\right)\partial U_{\ell, \rm nc}/\partial \psi~{\rm sign}\left(\omega_{\rm p}-n\right)$
with $U_{\ell, \rm nc}$ given by (\ref{eq:Uexternal}).  The $\left(1+\kappa\right)$ term is not typically
present in the Gauss formulation, but is appended here to the disturbing function\footnote{Algebraically,
from the time rate of change of the orbital energy, $\displaystyle{\dot{a}=2a^{2}\dot{E}/GM_{\rm p}M_{\rm s}}$,
and the change in orbital energy is further related to the velocity of the secondary ${\bf \dot{r}}$ and the
disturbing force ${\bf F}=-M_{\rm s}\nabla U_{\rm nc}$ such that $\dot{E}={\bf \dot{r}}\cdot{\bf F}=naM_{\rm s}T$
for a circular mutual orbit.  Replacing $\dot{E}$ by $naM_{\rm s}T$ and using Kepler's Third Law,
$n^{2}a^{3}=G\left(M_{\rm p}+M_{\rm s}\right)=GM_{\rm p}\left(1+\kappa\right)$, in the expression for $\dot{a}$
gives (\ref{eq:gauss}) for a specific order $\ell$.  If $M_{\rm s}$ were ignored in Kepler's Third Law, the
more familiar form of Gauss's formulation would emerge:  $\dot{a}=2T/n$.} due to the non-inertial nature of
the coordinate system centered on the primary~\citep{rubi73} and is necessary, as stated in~\citet{darw80},
because the primary ``must be reduced to rest.''  One can also argue the term is necessary to account for
the reaction of one body to the tidal action of the other~\citep{ferr08} as the perturbation is internal to
the binary system rather than an external element (e.g., drag force, third body).  The $\left(1+\kappa\right)$
term is absent in the formulae of~\citet{kaul64}, which is reasonable if the secondary is of negligible mass,
but we wish to allow for an arbitrary mass ratio.  Substitution of the disturbing force into (\ref{eq:gauss})
produces (\ref{eq:adotp}).  By including the $\left(1+\kappa\right)$ term in Kaula's equation (38), evaluating
Kaula's $F$ and $G$ functions with zero inclination and eccentricity, and recalling that Kaula's
$\epsilon_{\ell m p q} = -m\,\delta\,{\rm sign}\left(\omega_{\rm p}-n\right)$ in our notation, we find our
evolution of the semimajor axis in (\ref{eq:adotp}) is a special case of Kaula's generalization\footnote{The
product of Kaula's $F_{\ell m p}$ and $G_{\ell p q}$ functions is non-zero for circular, equatorial orbits
only if $\ell-2p=m$ and $q=0$.  The prefactors on each $\psi$ in the Legendre polynomials listed in
Table~\ref{tab:legendre} are the values of $m$ for each order $\ell$ that satisfy $\ell-2p=m$.  Thus, the
cosine terms in the Legendre polynomials we list correspond to $\ell m p q$ of 2200, 3110, 3300, 4210, 4400,
5120, 5310, 5500, 6220, 6410, and 6600.  This correspondence allows us to link our equations written in terms
of Legendre polynomials and a geometric lag to Kaula's equations written in terms of the phase lag
$\epsilon_{\ell m p q}$.  While the combinations 2010, 4020, and 6030 satisfy $\ell-2p=m$, terms with $m=0$
cannot contribute to the tidal evolution of the system because, by definition, these terms do not produce a
phase lag.  These three terms are responsible for the $\psi$-independent components of the Legendre polynomials
with $\ell=2,4,6$ that vanish upon differentiation with respect to $\psi$.}, as one would expect.

The tidal evolution of the secondary follows similarly.  The torque on the distorted secondary alters its
spin with time at a rate $\dot{\omega_{\rm s}}=-\Gamma_{\rm s}/I_{\rm s}$,

\begin{eqnarray}
\dot{\omega}_{\ell, \rm s} & = & -\frac{k_{\ell, \rm s}}{\alpha_{\rm s}}\frac{1}{\kappa\left(1+\kappa\right)}\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^{2\ell-1}\left(\frac{a}{R_{\rm p}}\right)^{-2\ell+1}n^2\left(-\left.\frac{\partial P_{\ell}\left(\cos\psi_{\rm s}\right)}{\partial \psi_{\rm s}}\right|_{\psi_{\rm s}=\delta_{\rm s}}\right){\rm sign}\left(\omega_{\rm s}-n\right)\label{eq:ws}\\
 & = & -\frac{k_{\ell, \rm p}}{\alpha_{\rm p}}\frac{\kappa}{1+\kappa}\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^{2\ell-5}\frac{\alpha_{\rm p}}{\alpha_{\rm s}}\,\frac{\mu_{\rm p}}{\mu_{\rm s}}\,\left(\frac{a}{R_{\rm p}}\right)^{-2\ell+1}n^2\,\left(-\left.\frac{\partial P_{\ell}\left(\cos\psi_{\rm s}\right)}{\partial \psi_{\rm s}}\right|_{\psi_{\rm s}=\delta_{\rm s}}\right){\rm sign}\left(\omega_{\rm s}-n\right),\nonumber
\end{eqnarray}

\noindent
and alters the semimajor axis of the mutual orbit at a rate of

\begin{equation}
\left(\frac{\dot{a}}{R_{\rm p}}\right)_{\ell, \rm s}~=~2k_{\ell, \rm p}\,\kappa\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^{2\ell-3}\,\frac{\mu_{\rm p}}{\mu_{\rm s}}\,\left(\frac{a}{R_{\rm p}}\right)^{-2\ell}n\,\left.\left(-\frac{\partial P_{\ell}\left(\cos\psi_{\rm s}\right)}{\partial\psi_{\rm s}}\right|_{\psi_{\rm s}=\delta_{\rm s}}\right) {\rm sign}\left(\omega_{\rm s}-n\right).
\label{eq:adots}
\end{equation}

\noindent
The Legendre polynomials in Table~\ref{tab:legendre} are written as sums of terms of the form $\cos\,m\psi$
where $m$ is an integer (see Footnote 11).  Thus, the derivative
$\displaystyle{\left.\partial P_{\ell}/\partial\psi\right|_{\psi=\delta}}$ is a sum of terms of the form
$\sin\,m\delta$.  For small geometric lag angles $(Q \gg 1)$,
$\displaystyle{\left.-\partial P_{\ell}/\partial\psi\right|_{\psi=\delta} \ge 0}$ and
$\sin\,m\delta \simeq m\delta$ such that
$\displaystyle{\left.-\partial P_{\ell}/\partial\psi\right|_{\psi=\delta}}$
$\displaystyle{\propto Q^{-1}}$.  Because the derivative of a Legendre polynomial is proportional to $Q^{-1}$,
only the size ratio of the components and their material properties, in terms of their respective $\mu Q$
values, determine the relative strength of the torques and the relative contributions to the orbit expansion,

\begin{equation}
\left|\frac{\Gamma_{\ell, \rm s}}{\Gamma_{\ell, \rm p}}\right|~=~\left|\frac{\dot{a}_{\ell, \rm s}}{\dot{a}_{\ell, \rm p}}\right|~=~\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^{2\ell-3}\,\frac{\mu_{\rm p}Q_{\rm p}}{\mu_{\rm s}Q_{\rm s}},
\label{eq:aratio}
\end{equation}

\noindent
with the relative contribution of the secondary decreasing at higher orders of $\ell$ and for smaller
secondaries.  Note that the relative strength of the torques is independent of the mass and
density\footnote{However, the absolute strengths of the torques in (\ref{eq:torque}) and (\ref{eq:torques})
do depend on the masses and densities of the components.}.  For classical $\ell=2$ tides on components with
similar material properties, the torque due to the distorted secondary is a factor of the size ratio
weaker than the torque due to the distorted primary.  For each higher order in the expansion, the relative
strength of the torque due to the distorted secondary weakens by the square of the size ratio.  The
changes in the spin rates compare as

\begin{equation}
\left|\frac{\dot{\omega}_{\ell, \rm s}}{\dot{\omega}_{\ell, \rm p}}\right|~=~\frac{1}{\kappa}\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^{2\ell-5}\,\frac{\alpha_{\rm p}}{\alpha_{\rm s}}\,\frac{\mu_{\rm p}Q_{\rm p}}{\mu_{\rm s}Q_{\rm s}}~=~\frac{\rho_{\rm p}}{\rho_{\rm s}}\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^{2(\ell-4)}\,\frac{\alpha_{\rm p}}{\alpha_{\rm s}}\,\frac{\mu_{\rm p}Q_{\rm p}}{\mu_{\rm s}Q_{\rm s}}.
\label{eq:wratio}
\end{equation}

\noindent
This differs from a generalization of Darwin's result [c.f. \citet{darw79b}, p. 521] because we
have included the ratio of the Love numbers of the components.  At the dominant orders, $\ell=2$ and $3$,
with similar densities, shapes, and material properties, the spin rate of the secondary changes faster
than the primary.  However, interestingly, for $\ell=4$, the contributions to the changes in spin rates are
equal, and for orders $\ell>4$, the contribution to the change in spin rate of the primary is greater than
that of the secondary.  As with the torques, the relative strength of the changes in spin rates weakens by
the square of the size ratio for each successive order $\ell$.  For smaller secondaries, the changes in
spin rates are smaller than for similar mass components, and, for all cases, the process of changing the
spin of the primary is slower than for the secondary.

Evaluating the Love number $k_{\ell, \rm p}$ in (\ref{eq:kapprox}) and $\partial P_{\ell}/\partial\psi_{\rm p}$
from Table~\ref{tab:legendre} explicitly for orders $\ell\le6$, assuming a small geometric lag angle
$\delta_{\rm p}$, and applying (\ref{eq:Q}), the spin of the primary changes as

\begin{eqnarray}
\dot{\omega}_{\rm p} & = & -\frac{8}{19}\frac{1}{\alpha_{\rm p}}\frac{\pi^2 G^2\rho_{\rm p}^{3}R_{\rm p}^2}{\mu_{\rm p}Q_{\rm p}}\,\kappa^2\left(\frac{a}{R_{\rm p}}\right)^{-6}\,{\rm sign}\left(\omega_{\rm p}-n\right)\nonumber\\
 & & \quad \times\left[1+\frac{19}{22}\left(\frac{a}{R_{\rm p}}\right)^{-2}+\frac{380}{459}\left(\frac{a}{R_{\rm p}}\right)^{-4}+\frac{475}{584}\left(\frac{a}{R_{\rm p}}\right)^{-6}+\frac{133}{165}\left(\frac{a}{R_{\rm p}}\right)^{-8}\right],
\label{eq:wpall}
\end{eqnarray}

\noindent
where $n$ has been replaced with Kepler's Third Law to show the full dependence upon the separation
of the components $a/R_{\rm p}$.  Using either (\ref{eq:ws}) or (\ref{eq:wratio}), the spin of the
secondary changes as

\begin{eqnarray}
\dot{\omega}_{\rm s} & = & -\frac{8}{19}\frac{1}{\alpha_{\rm s}}\frac{\pi^2 G^2\rho_{\rm p}^{3}R_{\rm p}^2}{\mu_{\rm s}Q_{\rm s}}\,\kappa\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^{-1}\left(\frac{a}{R_{\rm p}}\right)^{-6}\,{\rm sign}\left(\omega_{\rm s}-n\right)\nonumber\\
 & & \quad \times\left[1+\frac{19}{22}\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^2\left(\frac{a}{R_{\rm p}}\right)^{-2}+\frac{380}{459}\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^4\left(\frac{a}{R_{\rm p}}\right)^{-4}\right.\nonumber\\
 & & \quad \quad \left.+\frac{475}{584}\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^6\left(\frac{a}{R_{\rm p}}\right)^{-6}+\frac{133}{165}\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^8\left(\frac{a}{R_{\rm p}}\right)^{-8}\right].
\label{eq:wsall}
\end{eqnarray}

\noindent
Assuming similar densities for the components, the change in the spin rate of the primary scales as the
size ratio of the components to the sixth power ($\propto \kappa^{2}$);  the spin rate of the secondary
scales only as the square of the size ratio at leading order, reinforcing from (\ref{eq:wratio}) how the
spin of the secondary evolves more rapidly than that of the primary, especially for small size ratios.

For close orbits, the separation of the components changes as angular momentum is transferred to
or from the spins of the components such that the overall change in the orbital separation for
$\ell \le 6$ is the sum of (\ref{eq:adotp}) and (\ref{eq:adots}),

\begin{eqnarray}
\frac{\dot{a}}{R_{\rm p}} & = & \frac{8\sqrt{3}}{19}\frac{\pi^{3/2}G^{3/2}\rho_{\rm p}^{5/2}R_{\rm p}^2}{\mu_{\rm p}Q_{\rm p}}\,\kappa\left(1+\kappa\right)^{1/2}\left(\frac{a}{R_{\rm p}}\right)^{-11/2}\nonumber\\
 & & \enspace \times\left[{\rm sign}\left(\omega_{\rm p}-n\right)+\left(\frac{R_{\rm s}}{R_{\rm p}}\right)\frac{\mu_{\rm p}Q_{\rm p}}{\mu_{\rm s}Q_{\rm s}}{\rm sign}\left(\omega_{\rm s}-n\right)\right.\nonumber\\
 & & \enspace +\frac{19}{22}\left(\frac{a}{R_{\rm p}}\right)^{-2}\left({\rm sign}\left(\omega_{\rm p}-n\right)+\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^3\frac{\mu_{\rm p}Q_{\rm p}}{\mu_{\rm s}Q_{\rm s}}{\rm sign}\left(\omega_{\rm s}-n\right)\right)\nonumber\\
 & & \enspace +\frac{380}{459}\left(\frac{a}{R_{\rm p}}\right)^{-4}\left({\rm sign}\left(\omega_{\rm p}-n\right)+\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^5\frac{\mu_{\rm p}Q_{\rm p}}{\mu_{\rm s}Q_{\rm s}}{\rm sign}\left(\omega_{\rm s}-n\right)\right)\nonumber\\
 & & \enspace +\frac{475}{584}\left(\frac{a}{R_{\rm p}}\right)^{-6}\left({\rm sign}\left(\omega_{\rm p}-n\right)+\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^7\frac{\mu_{\rm p}Q_{\rm p}}{\mu_{\rm s}Q_{\rm s}}{\rm sign}\left(\omega_{\rm s}-n\right)\right)\nonumber\\
 & & \enspace \left.+\frac{133}{165}\left(\frac{a}{R_{\rm p}}\right)^{-8}\left({\rm sign}\left(\omega_{\rm p}-n\right)+\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^9\frac{\mu_{\rm p}Q_{\rm p}}{\mu_{\rm s}Q_{\rm s}}{\rm sign}\left(\omega_{\rm s}-n\right)\right)\right].
\label{eq:adotboth}
\end{eqnarray}

\noindent
Inside the square brackets, having a secondary of negligible size ($R_{\rm s}/R_{\rm p}\rightarrow0$)
has the same effect as having a synchronous secondary ($\omega_{\rm s}=n$); both make the contribution
from the secondary vanish.  Of course, if one considers the factor outside the square brackets, having a
secondary of negligible size makes the mass ratio $\kappa$ negligible, while having a synchronous secondary
does not directly affect $\kappa$.  The change in the mean motion of the mutual orbit follows from Kepler's
Third Law and (\ref{eq:adotboth}) as

\begin{equation}
\frac{\dot{n}}{n}~=~-\frac{3}{2}\left(\frac{a}{R_{\rm p}}\right)^{-1}\left(\frac{\dot{a}}{R_{\rm p}}\right).
\label{eq:ndot}
\end{equation}

\noindent
Note that in the above equations (\ref{eq:wpall}--\ref{eq:ndot}), any difference in density between the
components is accounted for in the mass ratio $\kappa$; otherwise, only the size ratio of the components is
involved in the terms due to tides raised on the secondary.  Obviously, the contribution of the secondary
is most important when the components are of similar size.  Not only is the contribution of the secondary
weakened because of its smaller size, it should also be despun faster than the primary such that its
contribution turns off when $\omega_{\rm s}=n$ long before the primary does the same.  Furthermore, each
equation has a strong inverse dependence on the separation of the components even at $\ell=2$, and while
the inclusion of higher-order terms will be strongest at small separations, the orbit of a typical
outwardly evolving system will expand to a wider separation rapidly.

\section{Effect of Close Orbit Expansion on Tidal Evolution}

Inclusion of higher-order terms for the changes in spin rates and semimajor axis in
(\ref{eq:wpall}--\ref{eq:adotboth}) speeds up the evolution of the system and decreases the tidal
timescales.  Using up to order $\ell=6$ compared to $\ell=2$ results in the spin rates of the
components changing up to 28\% faster at 2$R_{\rm p}$, but falling off quickly with increasing
separation (Fig.~\ref{fig:wdot}) to less than 4\% at 5$R_{\rm p}$.  The size ratio of the components
only affects $\dot{\omega}_{\rm s}$, where the higher-order terms are weaker for smaller secondaries.
Similarly, for the change in semimajor axis with time, assuming both components are causing the
separation to change in the same sense (${\rm sign}\left(\omega_{\rm p}-n\right)$ and
${\rm sign}\left(\omega_{\rm s}-n\right)$ have the same value), using up to order $\ell=6$
(Fig.~\ref{fig:adot}) results in a faster evolution by 21\% to 28\% at 2$R_{\rm p}$ and decreases
quickly with increasing separation.  Unlike the changes in spin rates, the largest effect on the
evolution of the semimajor axis occurs when the size ratio is either unity (equal size) or negligible
or when the spin of the secondary has synchronized to the mean motion such that the tidal torque
on the secondary vanishes.  The change in semimajor axis with time is least affected by the
higher-order terms for a size ratio of 0.53 with all other size ratios falling within these bounds.
According to (\ref{eq:ndot}) for the change in the mean motion with time, the value of $\dot{n}/n$
using higher-order terms compared to $\ell=2$ has the same form as the change in semimajor axis in
Fig.~\ref{fig:adot}.

The strengths of the contributions of the extra terms in the close-orbit correction to the change in
semimajor axis are listed in Table~\ref{tab:astrength}.  At 2$R_{\rm p}$, higher-order terms with
$\ell \ge 3$ account for nearly 25\% of the change in semimajor axis with time.  Although the $\ell=6$
term is necessary for accurate reproduction of the potential between the bodies to within 1\% at
2$R_{\rm p}$, it does not alter the change in semimajor axis with time at the 1\% level because of the
stronger dependence of (\ref{eq:adotp}) on separation compared to (\ref{eq:Vlegendre}).  The net
contribution of the higher-order terms in Table~\ref{tab:astrength} decreases by roughly 5\% at each
value of the separation from Table~\ref{tab:legendre} with only the $\ell=3$ term having much
consequence beyond 3$R_{\rm p}$.

The total change in the component spin rates as a function of separation, shown in Fig.~\ref{fig:wa},
is given by integration of the ratio of~(\ref{eq:wpall}) and~(\ref{eq:adotboth}) for the primary
and the ratio of~(\ref{eq:wsall}) and~(\ref{eq:adotboth}) for the secondary.  Depending on the size
ratio of the components, the total change in the spin rate of the primary is enhanced by up to 6\% at
2$R_{\rm p}$ over using $\ell=2$ tides only, but not by more than a few percent at larger separations.
For the secondary, perhaps counter-intuitively, despite the spin of the secondary evolving more rapidly
with time by adding higher-order terms (Fig.~\ref{fig:wdot}), its evolution with respect to the
separation is less than when using $\ell=2$ only; the deficit is as large as 22\% at 2$R_{\rm p}$ when
the size of the secondary is negligible.  This is because for smaller secondaries, the effect of
higher-order terms on $\dot{\omega}_{\rm s}$ in~(\ref{eq:wsall}) is reduced, while the effect of
higher-order terms on $\dot{\omega}_{\rm p}$ is independent of the size ratio.  Thus, for a rapidly
rotating primary, the higher-order terms transfer more angular momentum from the spin of the primary
to the orbit, expanding the separation faster than by $\ell=2$ tides alone and faster than the spin
rate of the secondary changes such that the net effect on $\Delta\omega_{\rm s}(a)$ is smaller.

Integration of~(\ref{eq:adotboth}) provides the separation as a function of time.  For tidal
evolution from an initial separation of 2$R_{\rm p}$ to a final separation of 5$R_{\rm p}$
(Fig.~\ref{fig:at}), the close-orbit correction is strongest at the onset, expanding the
separation more rapidly than $\ell=2$ tides, but only by about 2\% over the same time interval.
The contributions from the higher-order terms lose strength over time as the separation increases
resulting in a net effect of expanding the separation by $\sim$1\% extra by using $\ell=6$
instead of $\ell=2$.  From Figs.~\ref{fig:wa} and~\ref{fig:at}, the integrated effects of the
close-orbit correction are small, typically of order a few percent; the effects are more
noticeable in the instantaneous rates of change of the spin rates, separation, or mean motion
due to the rapid fall-off in strength of the higher-order terms with increasing separation and
how rapidly the system tidally evolves from small separations.

Rearrangement and integration of (\ref{eq:adotboth}) allows one to calculate the combination of the
material properties of the components $\mu Q$ (assuming $\mu_{\rm p}Q_{\rm p}=\mu_{\rm s}Q_{\rm s}$)
and the age of the binary $\Delta t$ based on measurable system parameters.  For brevity, we retain
only terms due to tides raised on the primary giving

\begin{eqnarray}
\frac{\mu Q}{\Delta t} & = & \frac{8\sqrt{3}}{19}\pi^{3/2}G^{3/2}\rho_{\rm p}^{5/2}R_{\rm p}^{2}\,\kappa\left(1+\kappa\right)^{1/2}\nonumber\\
 & & \times \left[\int_{2}^{a_{\rm f}/R_{\rm p}}\frac{x^{11/2}}{1+\frac{19}{22}x^{-2}+\frac{380}{459}x^{-4}+\frac{475}{584}x^{-6}+\frac{133}{165}x^{-8}}\,{\rm d}x\right]^{-1}
\label{eq:muq}
\end{eqnarray}

\noindent
with $x=a/R_{\rm p}$.  Because both terms on the left-hand side of (\ref{eq:muq}) are unknown, one
may either estimate the material properties by assuming binary ages~\citep{marg02s,marg03,tayl07dpstides},
estimate binary ages by assuming material properties~\citep{wals06,gold09}, or consider both
avenues~\citep{marc08ecc,marc08circ,tayl10material}.  Furthermore, precisely because both terms are
unknown, assuming a value for one has an intimate effect on the calculation of the other as changing
one's value by an order of magnitude changes the result of the other by an order of magnitude.  Thus,
when one wishes to find $\mu Q$, for instance, choosing an age for the binary injects a great source
of uncertainty into the calculation.

The close-orbit correction enhances the rate at which the separation changes such that, to provide
the same tidal evolution over the same timescale $\Delta t$, the product $\mu Q$ must increase to
compensate for the inclusion of the higher-order terms.  For classical $\ell=2$ tides, the denominator
of the integrand in (\ref{eq:muq}) vanishes such that the effect of including terms up to $\ell=6$
alters $\mu Q$ according to

\begin{equation}
\frac{\mu Q_{\ell=6}}{\mu Q_{\ell=2}}~=~\frac{\int_{2}^{a_{\rm f}/R_{\rm p}}x^{11/2}\,{\rm d}x}{\int_{2}^{a_{\rm f}/R_{\rm p}}\frac{x^{11/2}}{1+\frac{19}{22}x^{-2}+\frac{380}{459}x^{-4}+\frac{475}{584}x^{-6}+\frac{133}{165}x^{-8}}\,{\rm d}x}
\label{eq:muqchange}
\end{equation}

\noindent
and is shown as a function of the final separation in Fig.~\ref{fig:muq}.  Note that in
Fig.~\ref{fig:muq}, the contribution of the secondary is included in the numerical integration of
(\ref{eq:adotboth}) although it is not explicitly given in (\ref{eq:muqchange}) above.  Evolution
from a close initial separation of 2$R_{\rm p}$ to a wide separation of 10$R_{\rm p}$ results in
only a $\sim$1\% increase in $\mu Q$ over the classical value for all size ratios.  Thus, the basic
$\ell=2$ tidal mechanism is sufficient for well-separated binaries.  On the other hand, if the final
separation is smaller, as is the case for most near-Earth binaries, the correction is larger,
increasing to 5\% for evolution from 2$R_{\rm p}$ to 5$R_{\rm p}$ and 15\% for evolution from
2$R_{\rm p}$ to 3$R_{\rm p}$.  When making a coarse estimate of the material properties of the
system, taking the close orbit into account is not of paramount importance; classical tides will
easily provide an order-of-magnitude estimate of $\mu Q$ for even the closest of binary asteroids,
though the result will be slightly underestimated.  Complementarily, if higher-order terms are
included and $\mu Q$ is held fixed, the age of the binary must decrease by the same factor as in
Fig.~\ref{fig:muq} meaning that $\ell=2$ tides provide an upper bound on ages for systems with a
given $\mu Q$ value.

The use of higher-order terms up to $\ell=6$ is sufficient for exploring the tidal evolution of
binary systems with separations greater than 2$R_{\rm p}$.  Additional terms with $\ell>6$ make
inconsequential changes to tidal evolution at these separations as illustrated by the rapid fall-off
of the contributions of the higher-order terms beyond 2$R_{\rm p}$ in Table~\ref{tab:astrength}.
Moreover, terms with $\ell>6$ leave Figs.~\ref{fig:wdot}--\ref{fig:muq} unchanged, only having an
effect within 2$R_{\rm p}$.  Thus, if one wishes to proceed inward of 2$R_{\rm p}$, simply using
orders of up to $\ell=6$ is insufficient as higher-order terms gain importance the closer one
proceeds to the primary.  Though we stated earlier that the number of terms required can rapidly
become unwieldy, one can approximate their strength.  For an arbitrary order $\ell > 2$, the
term within the square brackets of (\ref{eq:adotboth}) is approximately

\begin{equation}
0.8\left(\frac{a}{R_{\rm p}}\right)^{-2\left(\ell-2\right)}\left({\rm sign}\left(\omega_{\rm p}-n\right)+\left(\frac{R_{\rm s}}{R_{\rm p}}\right)^{2\ell-3}\frac{\mu_{\rm p} Q_{\rm p}}{\mu_{\rm s} Q_{\rm s}}{\rm sign}\left(\omega_{\rm s}-n\right)\right),
\label{eq:xterm}
\end{equation}

\noindent
allowing additional terms to be included without explicit calculation of the Love numbers
$k_{\ell, \rm p}$ or manipulation of the Legendre polynomials.  Similar terms follow for the
changes in spin rates.  One must keep in mind that the approximation in~(\ref{eq:xterm}) is only
valid so long as the small angle approximation holds
$\left(\sin\,m\delta\simeq m\delta\propto Q^{-1}\right)$ with $m\le\ell$, which requires $Q>10$ to
retain 1\% accuracy at $m=6$ and larger $Q$ as $m$ increases\footnote{We have applied (\ref{eq:Q}) to
estimate the value of $Q$ required.  In general, the small angle approximation holds to within 1\% for
$Q_{\ell m p q} \sim 4$ or greater.} (e.g., $Q>20$ for $m=10$).  Also, having separations of
less than 2$R_{\rm p}$ requires smaller secondaries, since contact occurs at a separation of
$(1+R_{\rm s}/R_{\rm p})\,R_{\rm p}$, which reduces the contribution of the secondary due to dependencies
upon the size ratio, in addition to demanding consideration of the Roche limit for the system (see
Section~\ref{sec:roche}).

\section{Comparison to Measurement Errors}

Take, for example, 66391 (1999 KW4), the best-studied of the near-Earth binary
systems~\citep{ostr06,sche06s}.  Even with exhaustive analysis of radar imagery, production of
three-dimensional shape models of both components, and investigation of the system dynamics, physical
parameters of the system are not known with extreme precision.  The densities of the primary and
secondary components are known to approximately 12\% and 25\%, respectively.  The uncertainty in the
density of the primary alone can cause error of more than 30\% in $\dot{\omega}_{\rm p}$,
$\dot{\omega}_{\rm s}$, and $\dot{a}/R_{\rm p}$ according to (\ref{eq:wpall}--\ref{eq:adotboth}), more
than the close-orbit correction causes in Figs.~\ref{fig:wdot} and~\ref{fig:adot}.  The higher
estimated density of the secondary in the 1999 KW4 system of 2.81 g/cm$^{3}$, compared to 1.97 g/cm$^{3}$
for the primary, directly affects the mass ratio $\kappa$ applied in the equations of tidal evolution as
one typically assumes similar densities for the components.  Ignoring the density uncertainties, this
difference in component densities alone causes a 43\% change in $\kappa$ that, in turn, affects
$\dot{\omega}_{\rm p}$ by a factor of two and $\dot{\omega}_{\rm s}$ and $\dot{a}/R_{\rm p}$ by
approximately 40\% as well, again, a larger effect than the close-orbit correction to tidal evolution.
The calculated value of $\mu Q$ in~(\ref{eq:muq}) is affected by density and mass ratio uncertainties
in the same way as $\dot{a}/R_{\rm p}$.  Furthermore, uncertainties in densities and the dependence
of the mass ratio $\kappa$ on density differences between the components apply at all separations
unlike the close-orbit correction, which falls off quickly with increasing separation.

One must also consider the effect of the initial separation of the components at the onset of tidal
evolution, a property that is not known for individual systems, but can be estimated from simulations
of binary formation mechanisms [e.g.,~\citet{wals06,wals08yorp}] and given a lower bound by
the contact limit at $\left(1+R_{\rm s}/R_{\rm p}\right)\,R_{\rm p}$.  Assuming evolution over the
same timescale, if the system had an initial separation $a_{\rm i}$ instead of 2$R_{\rm p}$, the
effect on $\mu Q$ calculated with classical $\ell=2$ tides raised only on the primary is

\begin{equation}
\frac{\mu Q_{\rm i}}{\mu Q_{2}}~=~\frac{1-\left(\frac{2}{a_{\rm f}/R_{\rm p}}\right)^{13/2}}{1-\left(a_{\rm i}/a_{\rm f}\right)^{13/2}}.
\label{eq:initsep}
\end{equation}

\noindent
For a final separation $a_{\rm f}$ from 3$R_{\rm p}$ to 10$R_{\rm p}$, unless the actual initial
separation $a_{\rm i}$ is within 10\% of the final separation ($>$0.9$a_{\rm f}$), the value of $\mu Q$
is affected by less than a factor of two by assuming an initial separation of 2$R_{\rm p}$.  Using up
to $\ell=6$ and allowing for tides raised on the secondary with any size ratio do not cause a significant
difference in this result.

A similar result is found for the dependence on the final separation of the components, which one
typically takes to be the current separation.  If $a_{\rm f}$ is the final (current) separation,
then changing the separation to $a_{\rm f^{\,\prime}}$ due to, say, a measurement error causes the
calculated $\mu Q$ value for tidal evolution from 2$R_{\rm p}$ to change as

\begin{equation}
\frac{\mu Q_{\rm f^{\,\prime}}}{\mu Q_{\rm f}}~=~\frac{1-\left(\frac{2}{a_{\rm f}/R_{\rm p}}\right)^{13/2}}{\left(a_{\rm f^{\,\prime}}/a_{\rm f}\right)^{13/2}-\left(\frac{2}{a_{\rm f}/R_{\rm p}}\right)^{13/2}}.
\label{eq:finalsep}
\end{equation}

\noindent
We find $\mu Q$ is affected by less than a factor of two if the final (current) separation is known
within 10\%.  From the dependence on the initial and final separations, it is clear that the tidal
evolution near the final separation dominates over the early evolution where the close-orbit correction
is necessary.  In fact, if instead of calculating $\mu Q$, one considers the time taken to tidally
evolve to a final separation $a_{\rm f} \ge 4R_{\rm p}$ (by assuming a value of $\mu Q$ instead of an age),
the evolution of the separation from 0.9$a_{\rm f}$ to $a_{\rm f}$ takes roughly the same amount of time
as the evolution from $a_{\rm i}\le 2R_{\rm p}$ to 0.9$a_{\rm f}$.  Thus, precisely when the close-orbit
correction is most prominent is also when the system requires the least amount of time to evolve, which
causes the mild effect of the close-orbit correction found in Figs.~\ref{fig:at} and~\ref{fig:muq}.

Returning to a concrete example, for the 1999 KW4 system, using the equivalent spherical radius of
the primary shape model, the separation of the components $a/R_{\rm p}$ is known to 3\% as
3.87$\pm$0.12~\citep{ostr06}.  By~(\ref{eq:finalsep}), this small uncertainty can result in a roughly
20\% error in the calculated $\mu Q$, more than twice the effect of the close-orbit correction in
Fig.~\ref{fig:muq} at 3.87$R_{\rm p}$.  Together with the dependence of the $\mu Q$ calculation on
the density values for the components, the accuracy of measurements of physical parameters in the
1999 KW4 system is more important than accounting for the proximity of the components to one another.

\section{Discussion}
\label{sec:disc}

We have derived the equations of tidal evolution to arbitrary order in the Legendre polynomial expansion
of the separation between two spherical bodies in a circular and equatorial mutual orbit allowing for
accurate representation of evolution within five primary radii.  Equations written in terms of the Love
number $k_{\ell}$ are applicable to any binary system, while equations where the Love number has been
evaluated have assumed the bodies involved have rigidities that dominate their self-gravitational stress
(characteristic of bodies less than roughly 200~km in radius).  Because higher-order terms cause tidal
evolution to proceed faster, choosing to ignore them produces upper limits on tidal evolution timescales
and lower limits on material properties in terms of the product of rigidity and the tidal dissipation
function.  However, we have shown that the correction for close orbits has only a minor integrated effect
on outward tidal evolution and the calculation of material properties, comparable to or less than the
effect of uncertainties in measurable properties such as density, mass ratio, and semimajor axis (scaled
to the radius of the primary component).  In the case of outward evolution, the binary system evolves
rapidly through the range of separations where the close-orbit correction is strongest, so one can safely
ignore the correction to obtain order-of-magnitude estimates of timescales and material properties using
the classical equations for tidal evolution.  Accounting for higher orders is more applicable to studying,
famously in the case of Phobos, observed secular accelerations and the infall of a secondary to the surface
of its primary where the higher-order terms instead gain strength.

Though we have presented the expansion of the gravitational potential and the resulting equations of tidal
evolution in the context of two asteroids in mutual orbit, the essence of this work could be generalized
for use in the determination of the Roche limit and the study of close flybys.  The use of a higher-order
expansion of the gravitational potential in terms of Legendre polynomials is warranted whenever the
separation of two bodies is within five times the radius of one of the bodies\footnote{The potential felt
by the primary requires higher-order terms with $\ell>2$ if the separation is less than  5$R_{\rm p}$;
the potential felt by the secondary requires higher-order terms with $\ell>2$ if the separation is less than
5$R_{\rm s}$.} (see Table~\ref{tab:legendre}).  Historically, in the context of disruption of a body at the
Roche limit or due to a close flyby of a larger body~\citep{srid92,rich98,hols06,hols08,shar06,shar09},
stresses are only considered in the much smaller secondary while the primary is assumed to be rigid.  For
small secondaries, the cohesionless Roche limit of 1.5--2$R_{\rm p}$ is much larger than 5$R_{\rm s}$ such
that higher-order terms in the potential expansion are not necessary.  However, as larger secondaries are
considered ($R_{\rm s}/R_{\rm p}>0.1$), higher-order terms in the gravitational potential will further stress
the secondary near the Roche limit.  Also, with components of increasingly similar size, the assumption of a
rigid primary is not appropriate; the tidal stress on the primary will deform it from a spherical shape and
produce an external potential as in Section~\ref{sec:extpotl} that will in turn further stress the secondary.
If the components are not spin-locked, tidal torques will also play a role in stressing the secondary.  Thus,
if evaluating the Roche limit for components of similar size and/or components that are not spin-locked,
one must consider the description presented here.  For disruption during a close flyby, or simply modification
of the spin state of the passing body~\citep{sche01,sche00,sche04}, one must consider the proximity of the
flyby in terms of the expansion of the gravitational potential and whether or not tidal bulges can be raised
on the components that would produce torques capable of further altering the spin state of either component.

It is also important to remember that the higher-order theory presented here has implicitly assumed initially
spherical bodies.  Extension of this work from spheres to ellipsoids or to arbitrary shapes would affect the
mutual gravitational potential, linear and angular momentum balance, and orbital equations as described
by~\citet{sche09} and~\citet{shar10}.  Once the shape is made nonspherical in the absence of a tidal
potential, the system is subject to a ``direct'' torque that naturally occurs from the changing gravitational
pull felt by the orbiting component due to the nonspherical shape of the other component.  Accounting for the
tidal potential introduces the ``indirect'' torque described here due to the deformation of one component
by the gravitational presence of the other component.  Because the amplitude of the tidal bulge on asteroids,
the parameter $\lambda$ in this work, can be very small due to its direct dependence on the ratio of
self-gravitational stress to rigidity, its direct dependence on the mass ratio, and its inverse dependence
on the separation raised to the third (or higher) power, natural deviations from a spherical shape may exceed
the amplitude of the tidal bulge.  However, one must recall that the direct torques due to a nonspherical
shape will change direction as the body rotates under the orbiting component tending to cancel the pre- and
post-encounter effects of the torque as opposed to the indirect torque that is in a consistent direction so
long as the bulge always leads or lags the orbiting component.  It may be important to consider direct torques
due to natural departures from a spherical shape via the use of shape models:  oblate or prolate spheroids,
triaxial ellipsoids, or vertex models such as those made for the components of the 1999 KW4 binary system and
other asteroids.

\section*{Acknowledgements}

The authors are indebted to the two referees whose detailed reviews and insightful suggestions improved the
clarity and quality of the manuscript.  The authors are especially grateful to Michael Efroimsky for many
discussions on the finer points of tidal theory and celestial mechanics.  This work was supported by NASA
Planetary Astronomy grants NNG04GN31G and NNX07AK68G to Jean-Luc Margot.

\bibliographystyle{icarus}

redpajama_set_name: RedPajamaArXiv


Shyneeta and Carl know firsthand what a difference blood donors make in the lives of patients. They've had family members receive blood transfusions while battling sickle cell disease. And it inspires the work they do hosting blood drives.
On behalf of them, thank you for giving blood.
For more information on sickle cell disease and how you can make a difference, click here.


On November 19, 2016, 46 men were initiated into the Theta Mu Provisional Chapter at Northern Illinois University. Members from the Delta Chi Chapter at Elmhurst College assisted Coordinator of Chapter Development Samir Hafiz, Pi Kappa Alpha Fraternity, and their Grand Chapter Advisor Scott Woodfill, Grand Valley '08, with the initiation. The Silent Dinner took place in the Campus Life Building, room 100. The Initiation Ceremony took place in Dusable Hall, rooms 204, 206 and 228. The following men were present: Erik Marato, Elmhurst '13, Lucas Barnard, Elmhurst '14, Andrew Connet, Elmhurst '15, Nick Schulberg, Elmhurst '15, Thomas Sullivan, Elmhurst '15, Carlos Cantu, Elmhurst '15, and Luis Aburto, Elmhurst '16.
On November 20, 2016, Theta Mu was installed as a Provisional Chapter of Alpha Sigma Phi and Niel Patel, NIU '16, was the Installing President. The Installation Ceremony took place in Dusable Hall, room 176 on the NIU campus.
On February 17, 2018, the group officially chartered as the Theta Mu Chapter of Alpha Sigma Phi. The Chartering Ceremony took place at the Barsem Alumni and Visitors Center on campus. Grand Senior President Matt Maurer, Bowling Green '03, served as the Installing Officer. John Drummond, Illinois State '88, and Evan Geiselhart, Illinois '82, were also in attendance.


text:
Some clouds in the morning will give way to mainly sunny skies for the afternoon. Very hot. High near 95F. Winds SW at 10 to 20 mph..
Clear. Low 77F. Winds SW at 10 to 15 mph.
Brad Strom is the owner of the Crown Theater on Crown Point's square. The theater is in the middle of extensive renovations.
The Crown Theater on Crown Point's square in in the midst of extensive renovations.
The Crown Theater on Crown Point's square in in the midst of extensive renovations. This is the ticket booth.
The Crown Theater on Crown Point's square in in the midst of extensive renovations. A popcorn machine rests against the theater's seats.
The Crown Theater on Crown Point's square in in the midst of extensive renovations. Old projectors will be replaced by a digital projector.
The Crown Theater on Crown Point's square in in the midst of extensive renovations. A notice from the past remains.
The Crown Theater on Crown Point's square in in the midst of extensive renovations. The old doors have been saved for future use.
Doors to Crown Theater open for renovation
Times Staff
Robert Earnshaw
Lake County reporter
CROWN POINT — The public has not walked through the doors of the Crown Theater in half a decade, but since early May, workers have been inside, sharing space with old film posters, discarded projector reels and rolls of unused movie tickets, feverishly working to restore the venue.
The Times was recently given a tour inside Crown Theater by Brad Strom, its new owner. Strom plans to restore the movie house into a setting for live theater, a comedy club and small concert venue. Movies will still be shown in the form of independent films and film festivals.
Strom said he is also open to ideas from the community.
"It's a blank canvas," Strom said. "People can come in and use this for whatever they want. If people have stuff they want to try, that is what we are here for. I like the arts, I like theater, I like all of it. So whatever we can do to help the community. I've had people reach out for weddings and fundraisers. There is a lot."
Part of the restoration includes raising the ceiling in the lobby where above it sits the projector room. That bulky reel-to-reel film equipment will no longer be needed for movies as Strom is converting to digital projection.
"You can't get a lot of parts for them anymore," Strom said of the reel-to-reel projectors.
Seats have been torn out of the rear of the theater where Strom plans to build a balcony or raised platform for seating as well as a VIP area. An old mural sits to the right of the lobby where a bar will go. Vintage theater doors rest against one of the walls in the theater, ready to be refitted and opened again by customers. And across from them lies the the large crown that was a familiar part of the theater's facade.
Workers removed the crown as they prepare to restore the facade to what it was about a century ago based on photos Strom said he received from a resident. Strom said they will try to repurpose the crown into the design, but if not, it will "go into the right hands," - perhaps in a charity auction to allow someone to preserve it.
"It is still here and doing OK," Strom said.
Strom said they removed rotted wood from the facade and anything that was not original.
"We are restoring this building to what it was 100 years ago," he said.
Strom, who plans to be open by the first of the year, if not before, has no doubts that Crown Theater will become a cherished part of the downtown square.
New Crown Theater owner hopes to fill the entertainment void left in Region by closing of Star Plaza Theatre
CROWN POINT — The Star Plaza Theatre has been closed for nearly half a year, but one of its …
Once lights get turned on at Crown Theater, they could shine economic light on neighboring businesses
CROWN POINT — The new owner of the Crown Theater hopes the arts he brings to the long-vacant…
The Crown Theater facade is an artist's retrospective show
CROWN POINT — A work crew has drawn back the curtain on the Crown Theater's new facade, whic…
Theater owner kick-starts fund drive
CROWN POINT — The new owner of the Crown Theater is kicking it up a notch as he invites the public to get involved in the restoration of the l…
UPDATE: Crown Theatre's grand opening pushed back; opening act moves to Bulldog Park
CROWN POINT — A little more than a week before the historic 102-year-old doors of the Crown Theatre were set to officially open once again, bu…
Brad Strom
Crown Theater
Downtown Square
Film Projector
Rob covers urban affairs and other matters in Crown Point, St. John, Winfield and beyond. Previously he covered Valparaiso, Hammond, Gary and East Chicago. He's also written for various magazines and pens a culture blog for The Times.
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