text:
\section{Introduction}\label{Sec:Intro}

With the advent of new methods to evolve black-hole binaries
for several orbits~\cite{Pretorius:2005gq,Campanelli:2005dd,Baker:2005vv,Campanelli:2006gf,Baker:2006yw,Pretorius:2006tp,Scheel:2006gg}
we can now pose questions
of physical interest about the details of their interactions and
the generation of gravitational radiation, and find answers that allow
us to better understand the astrophysical properties of black-hole binaries.

The coalescence of unequal-mass black-hole binaries with randomly
oriented individual spins is expected to be one of the strongest
astrophysical sources of gravitational radiation. The first numerical
simulations of non-spinning unequal-mass black-hole binaries, along
with the calculation of the merger kicks, were reported in
Refs.~\cite{Campanelli:2004zw, Herrmann:2006ks, Baker:2006vn}.

In a preceding paper~\cite{Campanelli:2006uy} we performed the first
fully-nonlinear simulations of highly spinning black-hole binaries
and showed that the spin-orbit coupling leads to
significant changes in the orbital dynamics, as well as
gravitation radiation emitted, in the final stages of a
black-hole binary. In this paper we study in more detail the
mechanisms that transfer orbital angular momentum to spin and {\it vice versa}.

It is of astrophysical interest to know if tidal effects could lock
the spin of black-hole binaries to a corotating state,
i.e.\ a state where the spins of the individual horizons are aligned with orbital
angular momentum, and the spin magnitudes
are such that the  horizon frequency $\Omega_H$ ($\Omega_H$ is the angular
speed of locally-non-rotating observers as they pass through the horizon,
as seen by stationary observers at infinity)
 is nearly equal to
 the orbital frequency $\Omega_O$. For a Kerr hole with mass $m$ and
specific spin $a/m$
the horizon frequency takes the value
\begin{equation}
\label{eg:OmegaH}
\Omega_H=\frac{1}{2m}\left(\frac{a/m}{1+\sqrt{1-(a/m)^2}}\right).
\end{equation}
See Ref.~\cite{Caudill:2006hw} for a first post-Newtonian (1PN)
correction to the corotation condition $\Omega_O = \Omega_H$.
Note that in this paper we consider both the spin (or more accurately,
the spin angular momentum) $S$
of a black hole, which has units of $M^2$,
and the specific-spin $a/m$ ($m$ is the horizon mass),
which is dimensionless. The two are related by $S=a m$.
We will also make use of the following relationships between the spin,
specific spin, horizon frequency, and
irreducible mass ($m_{\mbox{\tiny IR}}$):
\begin{eqnarray}
  \label{eq:SOmegaH}
  S &=& \frac{4 m_{\mbox{\tiny IR}}^3 \Omega_H}{\sqrt{1 - 4 (m_{\mbox{\tiny IR}} \Omega_H)^2}}, \\
  \label{eq:aOmegaH}
  a/m &=& 4 m_{\mbox{\tiny IR}} \Omega_H \sqrt{1 - 4 (m_{\mbox{\tiny IR}} \Omega_H)^2},
\end{eqnarray}
where $16 \pi m_{\tiny \mbox{IR}}^2$ is the horizon surface area.

  There are two
competing scales in this problem: one is the time for tidal effects to
act to spin up (or down) the black holes to the orbital frequency and the
other is the rate of change of the orbital frequency due to
gravitational radiation.

In the early stages of the orbital motion of black-hole binaries, tidal
effects are small and can be described by Newtonian theory; these
effects decay strongly with the binary separation $r$ (as $\mu/r^3$,
with $\mu$ the reduced mass of the system). Tidal
effects will be stronger as the holes get closer,
but this
will also increase the orbital frequency (roughly like
$\Omega_O\sim\sqrt{\mu/r^3}$). In order to be correctly quantified, these effects
need to be evaluated dynamically with a fully non-linear numerical
simulation of the final orbital stage of black-hole binaries. Despite
this, several computations of quasi-circular sequences of initial data
for black-hole binaries have assumed corotating spins from larger
separations down to the estimated location of the innermost stable circular orbit (ISCO)
~\cite{Blanchet:2001id,Grandclement02,Caudill:2006hw}.
Corotation has a notable effect on the location of the ISCO
in all these calculations;
fortunately, recent numerical evolutions of equal-mass black-hole binaries
~\cite{Campanelli:2006gf,Baker:2006yw,Pretorius:2006tp} do not find
any particular physical effect associated with the location of the ISCO.

Bildsten and Cutler \cite{Bildsten92} (see also \cite{Kochanek:1992wk})
studied neutron star (NS) and black hole (BH) systems.  They conclude that, for
the NS-NS binary the viscosity is too low to tidally
synchronize the stars as they inspiral, and that there is simply not enough time during
the gravitational radiation driven inspiral of NS-BH binaries to tidally lock the spin
of the neutron star to the orbital period. Additionally, when the NS-BH binaries
are close enough for the tidal effects to be strong, the neutron stars are
disrupted in a few orbits.
Since these arguments
do not strictly apply to black-hole binaries, it is interesting to
explore these tidal interactions dynamically with fully non-linear numerical techniques.
This spin-up of NS-NS binaries has also been analyzed in
Ref.~\cite{Marronetti:2003gk}.

We will study in Sec.~\ref{Sec:spinup} the tidal spin-up of two equal-mass,
initially non-spinning, black holes starting from a quasi-circular
orbit (as determined by 3PN trajectories) with nearly two orbits left
before merger, as well as a similar configuration with initially
corotating black holes. These equal-mass configurations involving the very last stages of the
inspiral should nearly maximize the tidal effects.

While the transfer of orbital angular momentum to the spin of the members
of a binary system is of astrophysical relevance, we can also study the
conversion of spin into orbital angular momentum and verify the
importance of its dynamical effects.
To this end, we evolve binaries consisting of non-boosted, equal-mass black holes,
with the black holes aligned along the $y$-axis and the individual
spins aligned along the $+z$-axis, and measure the radiated
energy and angular momentum.
This simple set-up clearly displays the
transfer of spin to orbital motion.
For non-spinning holes this set-up produces
purely head-on collisions, leading to the radiation of
energy but {\it not} angular momentum.  For spinning
holes, the spin-orbit coupling causes the holes to acquire
non-head-on orbital motion, leading to the radiation of angular
momentum. Thus, an accurate measure of the angular momentum radiated
would allow us to estimate the correct magnitude of this effect.

The leading post-Newtonian spin-orbit and spin-spin corrections
to the acceleration of two equal-mass black holes (each of mass $m$),
 with spins $\vec{S}_1$ and
$\vec{S}_2$ perpendicular to the motion, in a nearly head-on collision
are:
\begin{eqnarray}
\vec{a}_{SO}&=&-\frac{16}{r^3}(\vec{v}_r\times(\vec{S}_1+\vec{S}_2)),\\
\vec{a}_{SS}&=&-\frac{6}{mr^4}(\vec{S}_1\cdot\vec{S}_2)\,\hat{r},
\end{eqnarray}
where $\vec r$ is the binary separation, $\vec v_r$ is the relative
velocity, and $SO$ and $SS$ denote spin-orbit and spin-spin coupling terms
respectively.
While the spin-spin acceleration adds an attractive term along the
radial direction (for parallel spins; repulsive for anti-parallel), the
spin-orbit acceleration is perpendicular to both the mostly-radial
(along $\hat{r}$) head-on velocity and the spin, and hence leads to a
non-vanishing orbital component of the binary motion.

In Sec.~\ref{Sec:spin-orbit} we will study this nonlinear coupling
using fully non-linear numerical evolutions and show that our method is able to extract
meaningful results for this numerically demanding relatively small effect.

\section{Techniques}\label{Sec:techniques}

We use the Brandt-Br\"ugmann puncture approach~\cite{Brandt97b} along with the
{\sc TwoPunctures}~\cite{Ansorg:2004ds} and {\sc BAM\_Elliptic}~\cite{cactus_web}
thorns to compute initial data.
In this approach the 3-metric on the initial slice has the form
$\gamma_{a b} = (\psi_{BL} + u)^4 \delta_{a b}$,
where $\psi_{BL}$ is the Brill-Lindquist conformal factor,
$\delta_{ab}$ is the Euclidean metric,
and $u$ is (at least) $C^2$ on the punctures.
The Brill-Lindquist conformal factor
is given by
$
\psi_{BL} = 1 + \sum_{i=1}^n m_i / (2 r_i),
$
where
$n$ is the total number of `punctures', $m_i$ is the mass
parameter of puncture $i$ ($m_i$ is {\em not} the horizon
mass associated with puncture $i$), and $r_i$ is the coordinate
distance to puncture $i$. In all cases below, we evolve data containing
only two punctures with equal puncture mass parameters, and we denote
this puncture mass parameter by $m_p$.
 We evolve these black-hole-binary data sets using the {\sc
LazEv}~\cite{Zlochower:2005bj} implementation of the moving puncture
approach~\cite{Campanelli:2005dd, Baker:2005vv}.  In our version of
the moving puncture approach~\cite{Campanelli:2005dd} we replace the
BSSN~\cite{Nakamura87,Shibata95, Baumgarte99} conformal exponent
$\phi$, which has logarithmic singularities at the punctures,
 with the initially
$C^4$ field $\chi
= \exp(-4\phi)$.  This new variable, along with the other BSSN
variables, will remain finite provided that one uses a suitable choice
for the gauge. An alternative approach uses standard finite
differencing of $\phi$~\cite{Baker:2005vv}. Note that both approaches
have been used successfully by several other
groups~\cite{Sperhake:2006cy, Herrmann:2006ks, Hannam:2006vv}.

We obtain accurate, convergent waveforms and horizon parameters
by evolving this system in
conjunction with a modified 1+log lapse, a modified Gamma-driver shift
condition~\cite{Alcubierre02a,Campanelli:2005dd}, and an initial lapse
$\alpha\sim\psi_{BL}^{-4}$.  The lapse and shift are evolved with
$(\partial_t - \beta^i \partial_i) \alpha = - 2 \alpha K$,
$\partial_t \beta^a = B^a$, and $\partial_t B^a = 3/4 \partial_t \tilde \Gamma^a - \eta B^a$.
These gauge conditions require careful treatment of $\chi$
near the puncture in order for the system to remain
stable~\cite{Campanelli:2005dd,Campanelli:2006gf}. In
Ref.~\cite{Gundlach:2006tw} it was
shown that this choice of gauge leads to a strongly hyperbolic
evolution system provided that the shift does not become too large.
For our version of the moving puncture approach, we find that the
product $\alpha \tilde A^{ij} \partial_j \phi$ initially has to be $C^4$ on the
puncture. In the spinning case, $\tilde A^{ij}$ is $O(r^3)$ on the
puncture, thus requiring that $\alpha \propto r^3$ to
maintain differentiability. We therefore choose an initial lapse
$\alpha(t=0) = 2/(1+\psi_{BL}^{4})$ which is $O(r^4)$ and $C^4$ on the
puncture and reproduces the isotropic Schwarzschild lapse at large
distances from the horizons.  The initial values of $\beta^i$ and $B^i$
are set to zero.

Hannam~et.~al.~\cite{Hannam:2006vv} examine the smoothness of the evolved fields at late
times at the puncture. They find that, in the case of Schwarzschild,
 $\chi$ transitions from an initially $C^4$
field to a $C^2$ field at late times. Although we require that the fields are
initially $C^4$, this late-time drop in smoothness does not
appear to leak out of the horizon (which is consistent with the
analysis in~\cite{Hannam:2006vv}).

We use a `multiple transition' fisheye transformation~\cite{Campanelli:2006gf}
to push the boundaries to $100M$, while maintaining a resolution
of up to $M/40$ in the central region,  for the head-on runs; for orbital
runs we push
 the boundaries to $215M$, with a central resolution of up to $M/31.5$.

We use a locally modified version of the {\sc AHFinderDirect}
 thorn~\cite{Thornburg2003:AH-finding}
to find the individual apparent horizon, and
we measure the angular momentum of the horizons using
our implementation of the isolated
horizon algorithm detailed in~\cite{Dreyer02a}.
(The isolated horizon formalism used here is extended to the
dynamical case in~\cite{Ashtekar03a}.) In the isolated
horizon approach, the horizon spin is given by
\begin{equation}\label{isolatedspin}
    S = \frac{1}{8\pi}\oint_{AH} (\varphi^aR^bK_{ab})\,\textrm{d}^2V .
\end{equation}
where $\varphi^a$ is an approximate Killing vector on the horizon,
$K_{ab}$ is the extrinsic curvature,  $d^2V$ is the
natural area element intrinsic to the horizon, and $R^a$ is the
outward pointing unit vector normal to the horizon on the 3D slice.
(See appendix~\ref{app:isolated_v_cir} for a comparison of the
isolated horizon technique for finding the horizon spin
to the more common horizon circumference technique.)
 We confirm that our
version of the algorithm produces the correct spin for various
coordinate distortions of Kerr, as well as for Bowen-York binary
initial data (both with and without linear momentum).

We measure the
proper distance between the horizons by shooting geodesics from the
origin to a horizon, and then finding the geodesic with the minimum
proper distance as described in Refs.~\cite{Koppitz04,Diener:2005mg} (the proper distance
between the two holes is then double this distance). This technique will
work in cases where there is (at least) Pi-symmetry on the
equatorial plane and reflection symmetry across the plane.

\section{Tidal spin-up of black-hole binaries}\label{Sec:spinup}

The spin of the individual horizons in a black-hole binary will become
tidally locked to the orbital frequency if the non-linear spin-orbit
coupling is sufficiently strong
that the timescale for tidally induced spin-up (or down) is smaller than
the timescale for change in the orbital period. In order to obtain an accurate
measure of the strength of this coupling we examine the cases of
initially non-spinning and initially corotating  close binaries, denoted
by S0 and SC respectively,
where the binary separation is small enough that the spin-orbit coupling is
large, but large enough that the binaries complete at least $\sim1.75$
orbits before merging. The initial data parameters for these
configuration are provided in Table~\ref{table:spinupID} and have
been obtained by finding quasi-circular orbits with the 3PN equations
of motion.
We originally studied the S0 non-spinning configuration in
Ref.~\cite{Campanelli:2006uy}. While in~\cite{Campanelli:2006uy}
we were interested in the spin of the remnant horizon and plunge times
as a function of the initial spin of the individual horizons, here we
are interested only in the pre-merger spin-up of the individual
horizons. It is then crucial to have an appropriate measure
of the individual spin of the holes.
We employ the isolated horizon
spin algorithm detailed in~\cite{Dreyer02a} (see Eq.~(\ref{isolatedspin})).
Strictly speaking, isolated horizon techniques are only valid for
non-interacting horizons. However, they do provide an excellent
approximation to the horizon spin even in the case of relatively
close binaries. We confirmed that our implementation gives accurate
evaluations of the horizon spins for closely separated Bowen-York
binary data. For this test we constructed a sequence of initial data using
the {\sc TwoPunctures} thorn~\cite{Ansorg:2004ds} with fixed orbital angular momentum
$L=0.876 M^2$ (the value for the S0 case above) and ADM mass of $(1.0000\pm0.0005)M$. We set the spin of the
individual holes (aligned along the $+z$-axis) to $S=0M^2$, $S=0.001M^2$, and $S=0.0001M^2$, and measured
the spins of the horizons with our implementation of the isolated horizon
algorithm. The results of these calculations are summarized in
Table~\ref{table:BYspin}. From the table we see that spins as low as
$10^{-4}M^2$ can be measured with an accuracy of better than $20\%$
even when the holes are separated by a proper distance of
$l\sim 3M$, and can be measured with an accuracy of $1\%$ or better when the holes are separated by
$l\geq 4.6M$. Interestingly, accurate measurement
of this spin can be obtain even with coordinate separations as small as $0.5M$.

\begin{table}
\caption{Initial data for quasi-circular, equal-mass black-hole binaries.
The binaries have an ADM mass of $(1.0000\pm0.0005)M$, with
orbital frequency $M\Omega$ fixed to $0.0500$, total angular momentum (orbital + spin)
given by $(0,0,J)$, and proper horizon separation $l$.
 The punctures
are located at $(0,\pm Y,0)$, with mass parameter $m_p$, momentum
$(\mp P,0,0)$, spin angular momentum $(0,0,S)$, and specific spin
$a/m$ ($m$ is the horizon mass).
}
\begin{ruledtabular}
\begin{tabular}{llllllll}\label{table:spinupID}
Name & $S/M^2$ & $Y/M$ & $P/M$ & $J/M^2$ & $a/m$ & $m_p/M$ & $l/M$\\
\hline
S0  & 0.000  & 3.280  & 0.1336  & 0.876  & 0.0000    &  0.4848   & 10.01 \\
SC & 0.025757&3.2534 & 0.1330& 0.917  & 0.1001    &  0.4831   & 9.93 \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{table}
\caption{The isolated horizon spin of a sequence of Bowen-York data with
given spins and ADM mass $(1.0000\pm0.0005)M$. Here $(\pm x,0,0)$ are the coordinate locations of the
punctures, $S$ is the measured spin (the Bowen-York spin parameter is given
in parentheses), and $l$ is the proper distance between horizons. In all cases
the linear momentum is given by $\vec P = (0,\pm 0.8764M^2/(2 x),0)$
 (to reproduce the S0 orbital angular momentum),
 and the spins are aligned with the $z$-axis. }
\begin{ruledtabular}
\begin{tabular}{lllll}\label{table:BYspin}
$x/M$ & $S/M^2(10^{-3})$ & $S/M^2(10^{-4})$ & $S/M^2(0)$ & $l/M$\\
\hline
$3.280$ & $1.000\times10^{-3}$ & $1.003\times10^{-4}$ & $3.0\times10^{-7}$ &$10.0$\\
$3.0$ & $1.011\times10^{-3}$ & $1.011\times10^{-4}$ & $1.1\times10^{-6}$ &$9.4$\\
$2.0$ & $9.998\times10^{-4}$ & $9.97\times10^{-5}$ & $3.4\times10^{-7}$ &$7.0$\\
$1.0$ & $1.001\times10^{-3}$ & $1.011\times10^{-4}$ & $1.13\times10^{-6}$ &$4.6$\\
$0.5$ & $1.01\times10^{-3}$ & $1.11\times10^{-4}$ & $1.08\times10^{-5}$ &$3.4$\\
$0.25$ & $1.02\times10^{-3}$ & $1.17\times10^{-4}$ & $1.7\times10^{-5}$ &$3.0$\\
\end{tabular}
\end{ruledtabular}
\end{table}

We evolved the S0 and SC configurations with 3 different resolutions,
$h=M/22.5$, $h=M/27$, and $h=M/31.5$, where $h$ is the central
grid-spacing, and calculated the horizon spins after every $1M$ of evolution.
The results of measuring the spin of the individual holes during the
orbital motion until the formation of a common apparent horizon (CAH) (See Table
\ref{table:spinup-results}) for the S0 configuration are displayed in
Fig.~\ref{fig:bs00_spinup}, while the results for the SC configuration are
displayed in Fig.~\ref{fig:bs0.1_spinup}.
The upper panel in both figures shows the specific spin of the individual
holes for the three resolutions. We observe that in both cases the specific-spin
increase during the evolution is small ($0.012$ for S0 and
$0.006$ for SC).
The lower panels show essentially third-order convergence as expected
from our use of third-order-accurate interpolators with {\sc AHFinderDirect}.
The late-time error in the S0 specific spin for the lowest resolution run
appears to show a very large error compared to the actual value of the specific spin.
This `error' is actually a phase error resulting from the lower resolution
runs merging sooner.
However, in a specific spin versus proper separation plot, the curves
from all three resolutions lie on top of each other
(see Fig.~\ref{fig:bs0.1_avl}) at late-times (i.e.\ at small proper distance).
This dichotomy, between the late-time convergence break-down of $a/m$
versus $t$ and the excellent late-time agreement between resolutions
in $a/m$ versus $l$, is similar to the results previously seen in the
waveforms of inspiraling
binaries~\cite{Baker:2002qf,Baker:2006yw,Campanelli:2006uy}.
In that case it was observed that $\psi_4(t)$
lost convergence due to large phase errors induced by the lower resolution
runs merging sooner, while the phase-corrected
$\psi_4$ showed excellent agreement between resolutions.
In this case the `phase error' is in $l(t)$, and by plotting $a/m$ versus
$l$ we effectively removed the phase error.

\begin{table}
\caption{Merger time, $T_{CAH}/M$, for the S0 and SC configurations versus resolution.
Horizon searches were performed every $0.3M$ and $0.2M$ for the S0 and
SC configuration respectively. }
\begin{ruledtabular}
\begin{tabular}{lll}\label{table:spinup-results}
${\rm resolution}$ & S0 & SC\\
\hline
M/22.5 & $160.7\pm0.3$ & $168.6\pm0.2$\\
M/27 & $166.0\pm0.3$ & $174.2\pm0.2$\\
M/31.5 & $168.3\pm0.3$ & $176.6\pm0.2$\\
extrapolation & $172\pm2$ & $179\pm2$\\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{bs00_spinup.eps}
\caption{The measured instantaneous specific spin of the individual horizons, for the
S0 configuration,  starting from vanishing spin at $t=0$
and up to the merger time. The top panel shows $a/m$ versus time for three resolutions
with grid-spacings $h=M/22.5$, $h=M/27$, and $h=M/31.5$,
while
the lower panel shows the differences in $a/m$ between the low and medium
resolutions and the medium and high resolutions (the latter rescaled by
1.9662 to demonstrate third-order convergence).
The differences between the low and medium resolutions and medium and high resolutions become large at
late times due to the lower resolution runs merging sooner.
The curves in the top panel have been cut off at the approximate merger value of $a/m$
(which is independent of resolution).}
\label{fig:bs00_spinup}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{bs0_1_spinup.eps}
\caption{The measured instantaneous specific spin of the individual horizons, for the
SC configuration,  starting from $a/m=0.1$ at $t=0$
and up to the merger time. The top panel shows $a/m$ versus time for three resolutions
with grid-spacings $h=M/22.5$, $h=M/27$, and $h=M/31.5$,
while
the lower panel shows the differences in $a/m$ between the low and medium
resolutions and the medium and high resolutions (the latter rescaled by
1.9662 to demonstrate third-order convergence).
The differences between the low and medium resolutions and the medium and high resolutions become large at
late times due to the lower resolution runs merging sooner.
The curves in the top panel have been cut off at the approximate merger value of $a/m$
(which is independent of resolution).}
\label{fig:bs0.1_spinup}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{bs0_1_avl.eps}
\caption{The specific spin $a/m$ for the SC configuration versus proper binary
separation $l$ for three resolutions with grid-spacings $h=M/22.5$, $h=M/27$, and $h=M/31.5$.
Note that the large late-time phase errors seen in $a/m$ versus time are
not present in this plot. The vertical line shows the proper distance at merger.
}
\label{fig:bs0.1_avl}
\end{center}
\end{figure}

The values that we obtained for the spin-up of the binary holes
are much smaller than those expected for a corotation
state
\footnote{Very rarely our isolated horizon routine fails to find the correct
approximate Killing vector and gives a very inaccurate value for the spin.
When this happens (we observes this no more than once per run for these orbital runs
and never for the head-on runs below) we drop the data point.
}.  In order to visualize this we have displayed in
Fig.~\ref{fig:a_v_l} the measured spin during the orbital motion, this
time against the proper separation of the apparent horizons.
The S0 configuration forms a common apparent horizon (CAH) at a proper separation
of $l_{CAH} = 1.75M$, while the SC configuration forms
a common apparent horizon at a proper separation of
$l_{CAH} = 1.69M$.
In order to have a measure of what the corresponding corotation
spin is, we considered the family of corotating binaries in quasi-equilibrium
 of Ref.~\cite{Grandclement02} (we have used Eq.~(\ref{eq:aOmegaH}) to convert the
$\Omega_H$ and $m_i$ provided in~\cite{Grandclement02} into
$a/m$). From the plot we see that the SC is
initially corotating, but the specific spin remains flat for most
of the run. Thus the spin-orbit coupling is far too small to keep
this system tidally locked. It is interesting to note that the
spin-up for the S0 configuration is slightly larger than the spin-up for the SC
configuration. Both configuration show essentially no spin-up above
proper separations of $\sim 4M$.

\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{a_v_l.eps}
\caption{The spin-up of the horizons versus proper binary separation.
The continuous line shows $a/m$ for the S0 configuration, the
dotted line shows $a/m$ for the SC configuration (initially
corotating), and the dot-dash line shows the corotation value
for the spin. Note that the S0 configuration shows a slightly
larger spin-up and that both spin-ups are much smaller than what is required
to tidally lock the binary.
}
\label{fig:a_v_l}
\end{center}
\end{figure}

The relatively low `spin up' that we measure during the last couple of orbits
prior to the merger indicates that the time-scale for tidal locking
is much larger than the time-scale of orbital decay due to gravitational
radiation emission. We also note that  in the leading relevant
post-Newtonian order, the change of the spin, averaged over a circular orbit,
which is given
by~\cite{Kidder:1995zr}
\begin{eqnarray}\label{Sdot}
\dot{\vec{S}}_i&=&\frac{1}{2r^3}\left\{(\vec{L}_N\times\vec{S}_i)
(4+3\frac{m_j}{m_i})+\vec{S}_j\times\vec{S}_i\right.\nonumber\\
&&\left.-3(\hat{L}_N\cdot\vec{S}_j)
\hat{L}_N\times\vec{S}_i\right\} \quad(j\neq i),
\end{eqnarray}
vanishes for all configurations with the spins aligned with the
orbital angular momentum. In Eq.~(\ref{Sdot})
$\vec{L}_N$ is the (Newtonian) orbital angular momentum, $\vec{S}_i$
is the spin of black hole $i$, $m_i$ is its mass, and $j$ denotes the other black hole.
Will~\cite{Will:2005sn} also points out
that the radiation reaction interaction does not modify the spin up to
3.5PN order. First effects of the flattening of the bodies should
actually be seen at 4.5PN order.

\section{Spin-orbit couplings in black-hole binaries}\label{Sec:spin-orbit}

We continue our study of the relevance of spin-orbit coupling to the
orbital dynamics of black-hole binaries by choosing initially head-on
(vanishing transverse linear momentum)
configurations that single out the converse transfer of spin into
orbital angular momentum.  Purely head-on collisions do not radiate
angular momentum to infinity. Thus, in these configurations, the
radiated angular momentum is an indicator of the extent to which  these
systems deviate from head-on motion (i.e.\ the extent to which they
acquire transverse motion). The energy and angular momentum radiated are
functions of the spins of the two horizons, the initial inward momentum of
each hole, and the proper separation.
To simplify the analysis, we choose head-on binaries with zero initial
linear momentum and with a fixed initial proper separation of $12.24M$. All
spins are aligned along the $z$-axis, with the binaries initially
separated
along the $y$-axis. We report the initial data parameters in
Table~\ref{table:HeadonID}.

\begin{table}
\caption{Initial data for non-boosted black-hole binaries. The binaries
have an ADM mass of $(1.0000\pm0.00005)M$, with total angular momentum $(0,0,J)$,
and proper horizon separation $l$. The punctures are located at
$(0,\pm Y,0)$, with mass parameter $m_p$, spins angular momentum $(0,0,S)$,
and specific spin $a/m$ (here $m$ is the horizon mass).}
\begin{ruledtabular}
\begin{tabular}{llllllll}\label{table:HeadonID}
Name & $S/M^2$ & $Y/M$ & $P/M$ & $J/M^2$ & $a/m$ & $m_p/M$ & $l/M$\\
\hline
HS++  & 0.200  & 4.000  & 0.000 & 0.4  &  0.7581    &   0.3444  & 12.24 \\
MS++  & 0.100  & 4.2422  & 0.000 & 0.2  & 0.3775    &  0.469755  & 12.24 \\
0S++  & 0.000  & 4.2924  & 0.000 & 0.0  & 0.0000    &  0.5000   & 12.24
\end{tabular}
\end{ruledtabular}
\end{table}

We measure the radiated energy and angular momentum by solving for
the final hole remnant horizon mass and spin, and then subtracting
these quantities from the ADM mass and angular momentum (as determined from the
initial data). To avoid confusion with the initial black-hole mass
and spins, we denote the mass and spin of the final remnant
horizon with $M_{H}$ and $J_{H}$ respectively. The radiated
mass and angular momentum then are:
\begin{eqnarray}
  E_{rad} &=& M_{ADM} - M_{H},\\
  J_{rad} &=& J_{ADM} - J_{H},
\end{eqnarray}
where in these head-on cases $J_{ADM}$ is the sum of the spins
of the two initial black holes. For notational convenience
we introduce the symbol $j_0$ defined by
\begin{equation}
j_0 = \frac{J_{ADM}}{M_{ADM}^2}.
\end{equation}
In previous papers~\cite{Campanelli:2006gf, Campanelli:2006uy}
we found that the horizon provided a very accurate
measurement of these quantities. Additionally, when using isolated
horizon techniques, we can obtain the horizon parameters accurately
within $30M$ after the merger.
In all cases the radiated energy and angular momentum are very small.
In order to accurately obtain these numbers we need to evolve all
configurations at three different resolutions (see Table~\ref{table:Headon-results}
for the grid-spacings) and then use a
Richardson extrapolation of the radiated energy and angular momentum.
The measured convergence rates for the radiated angular momentum are
$3.9$ and $3.7$ for the HS++ and MS++ configurations respectively, while the
measured convergence rates for the radiated energy are $4.0$,
$3.7$, and $3.9$ for the HS++, MS++, and 0S++ configurations respectively.
The lower convergence rates for the MS++ configuration
may simply indicate only that higher resolution is needed. Alternatively it is possible that
small third-order errors (possibly from the interpolators) are present in all
configurations, but are only observed in the MS++ configuration due
to its finer effective resolution. (The effective grid-spacing
is $h_{eff} = h/m_p$, where $h$ is the actual grid-spacing
and $m_p$ is the puncture mass parameter (see~\cite{Campanelli:2006uy} for details)).
 For the Richardson
extrapolation we use both the forms $f(h) = f_{\infty} + e\,h^4+ g\,h^5$
and $f(h) = f_{\infty} + d\,h^3 + e\,h^4$, and, using results from the
three resolutions, solve for $f_{\infty}$.
 We take the difference
between these two extrapolation to be indicative of the error in our extrapolated numbers.
In Figures~\ref{fig:VHS_PP_M_ex}~and~\ref{fig:VHS_PP_J_ex} we show the
HS++ final remnant black-hole horizon mass and spin
versus time for three resolutions, as well as the two extrapolations of
these quantities and convergence plots. It is evident from the very close
agreement between the two extrapolations that we obtain highly accurate
measurements of both the horizon mass and spin, and consequently,
the radiated mass and angular momentum.

In Table~\ref{table:Headon-results} we summarize the results for these
configurations. Note that the errors reported in the table for
a given resolution are of the
uncertainty in measurement of  $J_{H}$ and $M_{H}$ at that resolution (i.e.\ a measure of
the flatness of the late-time plateaus of $J_{H}$ and $M_{H}$). Thus the error
interval reported for e.g.\ $J$ radiated at a resolution of $M/25$ will not
necessarily contain the final extrapolated value. On the other hand,
the error intervals for the extrapolated values are our best estimate for the true value of
each quantity. A fit of the Richardson-extrapolated values of
the radiated angular momentum to the initial spins of the individual black holes
has the form:
\begin{eqnarray}
\label{eq:JradVJ}
\frac{J_{rad}}{M^2}&=& (4.8\pm0.1)\times10^{-3}j_0+\nonumber\\
&&(4.0\pm0.9)\times 10^{-3} j_0^3 +
 O\left(j_0^5\right),
\end{eqnarray}
where $j_0 = 2 S/M_{ADM}^2$ (see Table~\ref{table:HeadonID}).
The linear term is the dominant one  for these configurations (the HS++ configuration
radiates $\sim20\%$ more momentum than expected from the linear behavior alone).
Equation~(\ref{eq:JradVJ}) contains only odd powers of $j_0$ since the magnitude of
$J_{rad}$ must be independent of $j_0$ (i.e.\ the physics
would not change if the two spins were aligned along the $-z$-axis),
 and the sign of $J_{rad}$ must match the sign of  $j_0$.

\begin{widetext}

\begin{table}
\caption{The radiated mass and angular momentum as well as the
merger time ($T_{CAH}$) and final specific spin for the head-on configurations.
 The reported
error intervals in $T_{CAH}$ are a function of how often a horizon
search was performed. In all cases $T_{CAH}$ does not appear to
vary with resolution.}
\begin{ruledtabular}
\begin{tabular}{llllll}\label{table:Headon-results}
{\rm Case} & ${\rm resolution}$ & $E_{rad}/M_{ADM}$ & $J_{rad}/J_{ADM}$ & $T_{CAH}/M$ & $a/M_{H}$\\
\hline
HS++  	& M/25 & $(0.16\pm0.02)\%$ & $(2.37\pm0.02)\%$ & $36.7\pm0.2$ & $0.3918\pm0.002$\\
      	&   M/30 & $(0.136\pm0.009)\%$ & $(1.44\pm0.01)\%$ & $36.6\pm0.2$ & $0.39532\pm0.00008$\\
      	& M/40 & $(0.123\pm0.004)\%$ & $(0.832\pm0.003)\%$ & $36.7\pm0.2$ & $0.39765\pm0.00003$\\
	& extrapolation & $(0.118\pm0.002)\%$ & $(0.546\pm0.002)\%$ & $36.7\pm0.2$ & $0.39876\pm0.00002$\\
MS++  	& M/25 & $(0.065\pm0.004)\%$ & $(1.038\pm0.008)\%$ & $39.5\pm0.2$ & $0.19817\pm0.00003$\\
      	&   M/30 & $(0.063\pm0.003)\%$ & $(0.7698\pm0.0040)\%$ & $39.6\pm0.2$ & $0.19868\pm0.00003$\\
      	& M/35 & $(0.062\pm0.002)\%$ & $(0.6567\pm0.0069)\%$ & $39.5\pm0.2$ & $0.19891\pm0.00003$\\
	& extrapolation & $(0.060\pm0.002)\%$ & $(0.4986\pm0.0086)\%$ & $39.5\pm0.2$& $0.19920\pm0.00007$\\
0S++  	& M/22.5 & $0.0616\pm0.0032$ & $0$ & $40.4\pm0.2$ & $0$\\
      	& M/25 & $(0.0595\pm0.0025)\%$ &  $0$ & $40.4\pm0.2$ & $0$\\
      	& M/30 & $(0.057\pm0.002)\%$ & $0$ & $40.4\pm0.2$ & $0$\\
	& extrapolation & $(0.054\pm0.002)\%$ & $0$ & $40.4\pm0.2$ & $0$
\end{tabular}
\end{ruledtabular}
\end{table}
\end{widetext}

An analysis of the radiated energy versus spin is complicated by the
fact that the binary radiates energy in the zero-spin configuration
as well. After subtracting the zero-spin radiated energy we find that
the HS++ configuration radiates $0.064\pm0.003 M$ more energy than the zero-spin
configuration, while the MS++ configuration only radiates
$0.006\pm0.002 M$ more. Thus the HS++ configuration radiates
$12.2\pm4.6$ times as much as the MS++ configuration (after subtracting off the
zero-spin radiated energy). From these
results it appears that the radiated energy scales with $j_{0}^3$
or $j_{0}^4$. A close-limit analysis (See Eq.~(\ref{Ecl})) also indicates that there is a
small $j_{0}^2$ contribution as well. Although not obvious in the
total radiated energy, this quadratic dependence on $J$ can be seen in
the imaginary part of the $(\ell=2,m=\pm2)$ components of the radiated
energy $E(t)$ (i.e.\ the amount of radiation emitted up to time $t$),
where $E(t) = \sum_{\ell, m} (R_{\ell\,m}(t) + I_{\ell\,m}(t))$,
\begin{eqnarray}
  R_{\ell\,m}(t) = \frac{1}{4 \pi} \int_0^t Re[N_{\ell\,m}(\tau)]^2 d\tau, \\
  I_{\ell\,m}(t) = \frac{1}{4 \pi} \int_0^t Im[N_{\ell\,m}(\tau)]^2d\tau,
\end{eqnarray}
$N_{\ell\,m}(t) = (r - 2 M)/2  \int_0^t \bar \psi_{4,\,\ell\,m}(\tau)\, d\tau$,
and $\psi_4$ is calculated using the quasi-Kinnersley tetrad method~\cite{Campanelli:2005ia}.
Figure~\ref{fig:vhs_ls_l2_compare} shows $R_{\ell\,m}(t)$ and $I_{\ell\,m}(t)$
 for the $\ell=2$ components
of the waveform for the HS++ and MS++ configurations. Note that
$I_{2\,\pm2}$ (which vanishes in the zero-spin case) is 4.3 times larger
in the HS++ configuration than in the MS++ configuration. This
is consistent with a quadratic
dependence on the spin plus smaller higher-order dependencies.

Finally, if we fit the radiated energy to the polynomial dependence
$E_0 + E_2 j_0^2 + E_4 j_0^4$ we obtain
\begin{eqnarray}
\label{eq:EradVJ}
\frac{E_{rad}}{M}&=&(0.052\pm0.02)+(0.065\pm0.033)j_0^2 \nonumber\\
&&+(2.08\pm0.21)j_0^4 + O\left(j_0^6\right).
\end{eqnarray}
Note that the coefficient of the quadratic term is two orders of magnitude smaller than the
coefficient of the quartic term (and thus was not apparent in our initial
analysis above).  There are only even terms in Eq.~(\ref{eq:EradVJ}) because the $E_{rad}$ cannot
depend on the sign of $j_0$ (i.e.\ $E_{rad}$ must be the same for configurations with the spins
aligned on the $+z$-axis and with the spins aligned on the $-z$-axis).

The Bowen-York initial data family contains `spurious' radiation
that does not represent any physical history of the
binary.  In addition, the spinning case does not have the Kerr limit when
the two black holes are far apart. (For a data set with the right Kerr
limit see \cite{Dain:2000hk,Dain:2001iw}.) While this extra radiation
is of relatively small magnitude in orbiting black-hole binaries (See
~\cite{Campanelli:2006uy}), it is relevant in the highly spinning
head-on collisions considered here.  This can be observed in
Fig.~\ref{fig:vhs_ls_l2_compare}. The first plateau in the radiated
energy ($t=40M - 70M$) for the observer location at $r=35M$
is due to the radiated `spurious' energy. This `spurious' energy contributes up to
$25\%$ of the total energy radiated of the HS++ configuration, but strongly decreases
for the MS++ and 0S++ configurations.

\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{VHS_PP_M_ex.eps}\\
\includegraphics[width=3.3in]{VHS_PP_M_conv.eps}
\caption{The final remnant horizon mass versus time for the HS++ configuration.
The top panel shows the horizon mass for the three resolutions
with grid-spacings $h=M/25$, $h=M/30$, and $h=M/40$, and
the two Richardson extrapolations based on  leading third-order
and leading fourth-order errors. The bottom panel shows the differences
in the horizon mass for the $h=M/25$ and $h=M/30$ runs as well as the differences
in the mass for the $h=M/30$ and $h=M/40$ runs. This latter difference is
rescaled by 1.57052 to demonstrate 3.7-order convergence.
The accurate extrapolations (as evident by the agreement of the two
extrapolations) to infinite resolution allows for a
sufficiently precise determination of the radiated energy to model
its dependence on the initial spins.
}
\label{fig:VHS_PP_M_ex}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{VHS_PP_J_ex.eps}
\includegraphics[width=3.3in]{VHS_PP_J_conv.eps}
\caption{The angular momentum of the final remnant black hole for the
HS++ configuration measured by
the isolated horizon method.
The top panel shows the horizon spin for the three resolutions
with grid-spacing $h=M/25$, $h=M/30$, and $h=M/40$, as well as
the two Richardson extrapolations based on  leading third-order
and leading fourth-order errors. The bottom panel shows the differences
in the horizon spin for the $h=M/25$ and $h=M/30$ runs as well as the differences
in the spin for the $h=M/30$ and $h=M/40$ runs. This latter difference is
rescaled by 1.54 to demonstrate 4th-order convergence.
The  accurate extrapolations (as evident by the agreement of the two
extrapolations) to infinite resolution allows for a sufficiently precise
determination of the radiated angular momentum to model its dependence
on the initial spins.
}
\label{fig:VHS_PP_J_ex}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{vhs_ls_l2_compare.eps}
\caption{The gravitational energy radiated up to time $t$ as measured
by an observer at $r=35M$ for the $\ell=2$ modes of the HS++ and MS++ configurations
(for our configurations $R_{2\, 2} = R_{2\, -2}$, and $I_{2\, 2} = I_{2\, -2}$). The
energy from the spurious initial burst leads to the first plateaus
centered at $t=60M$.
These initial bursts of energy, while significant, are dominated by
the total emitted
radiation. We multiply $I_{2\,\pm2}$ from the MS++ configuration ($S=0.1M^2$)
by $4.3$ to demonstrate that quadratic (plus smaller higher-order terms)
dependence of  $I_{2\,\pm2}$ on $S$.
}
\label{fig:vhs_ls_l2_compare}
\end{center}
\end{figure}

One can use the lowest relevant post-Newtonian expansion to
support (qualitatively) our results.
A look at the Post-Newtonian equations of motion~\cite{Kidder:1995zr}
shows that the leading spin-orbit interaction is of 1.5PN order, while
the leading spin-spin interaction is of 2PN order. Kidder~\cite{Kidder:1995zr}
gives the radiated energy and momenta of the point-particle
spin-orbit and spin-spin couplings up to 2PN order. The relevant
equations for our nearly head-on collisions are:
\begin{subequations}\label{EdotPN}
\begin{eqnarray}
\dot{E}_N&=&-\frac{8 \bar m^4}{15r^4}v_r^2,\\
\dot{E}_{PN}&=&-\frac{2\bar m^4}{105r^4}\left\{-110v_r^4+452\frac{\bar m}{r}v_r^2\right\},\\
\dot{E}_{SO}&=&0,\quad(\vec{L}_N=0),\\
\dot{E}_{SS}&=&\frac{32 \bar m^2}{5r^6}(\vec{S}_1\cdot\vec{S}_2)v_r^2,
\end{eqnarray}
\end{subequations}
where $r$ is the binary separation, $2 \bar m=m_1+m_2$, $\vec v_{r}$ is the relative velocity, $N$ denotes Newtonian terms,
PN denotes (non-spin-orbit and non-spin-spin) post-Newtonian terms,
SO denotes spin-orbit coupling terms, and SS denotes
spin-spin coupling terms. Note that the lowest-order spin-correction to the radiated energy
is quadratic in $S$.
Likewise we can obtain the radiated angular momentum
\begin{subequations}\label{JdotPN}
\begin{eqnarray}
\dot{\vec{J}}_N&=&0,\quad(\vec{L}_N=0),\\
\dot{\vec{J}}_{PN}&=&0,\quad(\vec{L}_N=0),\\
\dot{\vec{J}}_{SS}&=&0\quad(\vec{a}_N\sim\vec{a}_{SS}\sim\hat{r}),\\
\dot{\vec{J}}_{SO}^{++}&=&-\frac{4 \bar m^2}{15r^3}\,\vec{v}_r\times(\vec{v}_r\times
(\vec{S}_1+\vec{S}_2))\nonumber\\
&&\left\{v_r^2+\frac{2 \bar m}{r}+\frac{1}{v_r^2}\frac{4 \bar m^2}{r^2}\right\}\label{JdotPNSO},
\end{eqnarray}
\end{subequations}
which displays the leading linear dependence on the spin,
and the radiated linear momentum
\begin{subequations}\label{PdotPN}
\begin{eqnarray}
\dot{\vec{P}}_N&=&0,\quad(\delta m=m_1-m_2=0)\\
\dot{\vec{P}}_{PN}&=&0,\quad(\delta m=m_1-m_2=0)\\
\dot{\vec{P}}_{SS}&=&0\quad{\rm (at\ this\ order)}\\
\dot{\vec{P}}_{SO}^{++}&=&-\frac{8 \bar m^3}{15r^5}\,{v}_r(\vec{v}_r\,\times\vec{\Delta})=0,\nonumber\\
&&(\vec{\Delta}=2(\vec{S}_1-\vec{S}_2)=0).\label{PdotPNSO}
\end{eqnarray}
\end{subequations}

Note that equation (\ref{PdotPNSO}) gives a non-vanishing
linear momentum radiated for anti-parallel spins.
The spin-orbit interaction is the only one that reverses signs
when spins change signs. This explains the different stability and
radiation properties of parallel and anti-parallel configurations.
We plan to report the results of this spin up - spin
down case leading to radiated {\it linear} momentum in a forthcoming paper.

These spinning-black-hole-binary scenarios were also studied in the complementary regime of the close
limit approximation by Gleiser and Dom\'{i}nguez
\cite{Gleiser:2001in}. They examined the behavior of  non-boosted, slowly rotating holes to second
perturbative order. In that approximation the second-order terms
proportional to the spin and the square of the proper distance, $l$,
give rise to radiation of the angular momentum, $J$, proportional to
the spin
\begin{equation}\label{Jcl}
J_{rad}\approx0.0023\, (2S)\,(l/M)^4.
\end{equation}
The analysis also predicts that the radiation of energy can be expanded as
\begin{eqnarray}\label{Ecl}
\frac{E_{rad}}{M}&\approx&7.8\times10^{-4}(2S/M^2)^4+9.8\times10^{-5}\,(l/M)^4\nonumber\\
&-&2.8\times10^{-5}\,(2S/M^2)^2(l/M)^2\nonumber\\
&+&1.3\times10^{-3}(2S/M^2)^2(l/M)^4,
\end{eqnarray}
The first term, being proportional only to $S^4$, is generated by the
radiation content of the BY initial data. The second term, proportional
to $l^4$, is the radiation for two non-spinning masses in a head-on
collision. The third term is second-perturbative-order spin-orbit
coupling term, and is the one that leads to
the radiation of angular momentum.

\section{Discussion of Results}\label{Sec:discussion}

We have studied the spin-up of initially non-spinning and initially corotating
black-hole binaries during the last
stages of the inspiral orbital motion and found that it is much
smaller than that needed to achieve tidal locking.
This means that the time scale of tidal effects to spin up the black holes
is much longer than the time scale for gravitational radiation to
drive the merger (and hence increase the orbital frequency).
This
also implies that in an astrophysical scenario, black holes with low
spin merge with essentially the same spin magnitude they had when the binary
formed (note that Eq.~(\ref{Sdot}) implies that $|\dot{\vec{S}}|=0$).
We also note that it is this short timescale for gravitational radiation
during the last few orbits that is responsible for the absence of an
ISCO (innermost stable circular orbit) when the two holes have comparable
masses. The concept of the ISCO arises naturally in the extreme mass ratio
limit, where there is a cutoff between stable and unstable orbits. In this
limit, the instability of orbits inside the ISCO is not driven by gravitational
radiation, but by non-radiative terms in the particle's equation of motion.
Thus the particle very slowly inspirals due to the emission of weak gravitational
radiation, and then plunges when it reaches the ISCO.
On the other hand, in the comparable mass regime, there is no such
cutoff because gravitational radiation dominates the dynamics during the
entire late-time inspiral and plunge phases.

In Fig.~\ref{fig:tracks} we show the tracks of the horizon centroids in
the S0 and SC configurations. Although coordinate dependent, the tracks
help illustrate the lack of an ISCO in the comparable mass regime. Rather
than showing an abrupt change near the last orbit, the tracks show a smooth
increase in plunge velocity over the entire evolution. Note that the
SC track is a tighter spiral than the S0 track, which is consistent with
the results found previously by the authors~\cite{Campanelli:2006uy}.
Additional horizon tracks for multi-orbit non-spinning binaries
can be found in Refs.~\cite{Baker:2006yw, Campanelli:2006gf},
while tracks for highly-spinning
binaries can be found in Ref.~\cite{Campanelli:2006uy}.

\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{S0SC_track.eps}
\caption{The horizon tracks for the S0 and SC configurations
with tick marks every $10M$ of evolution
(we display only one of the SC tracks for clarity). Note that the
S0 and SC tracks agree well for the first $100M$, and that after
$100M$ the SC tracks show a tighter spiral and orbit longer
before merging. The S0 configuration completes slightly less than
1.75 orbits, while the SC configuration completes slightly more.}
\label{fig:tracks}
\end{center}
\end{figure}

After the merger, the final remnant black hole formed from the S0 configuration
has a specific spin of $a/M_{H}=0.688\pm0.001$ (where $M_{H}$ is the
remnant horizon mass) while remnant formed from the
SC configuration has a specific spin of $a/M_{H}=0.717\pm0.001$. Using these
values in conjunction those found in Ref.~\cite{Campanelli:2006uy}
(see Table~\ref{tab:spindat}), we
find (from a least-squares fit) that the specific spin of the final
remnant horizon $(a/M_{H})|_R$ varies
with the initial specific spins of the two (equal-mass, equal-spin) black
holes $(a/m)|_{I}$ according to:
\begin{eqnarray}
  (a/M_{H})|_{R} &=& 0.6879 + 0.2952 \left((a/m)|_{I}\right) \nonumber \\
             &-&0.0374 \left((a/m)|_{I}\right)^2,
\end{eqnarray}
with an error of $\pm0.001$ in the range $(-0.757 < (a/m)|_{I} < 0.757)$.
From this fit we extrapolate that the remnant horizon will have a specific spin
no smaller than $0.355$ (for the case of initially-maximal, anti-aligned
spins) and no larger than
$a/M_{H} = 0.946$ (for the case of initially-maximal, aligned spins),
which supports our assertion in~\cite{Campanelli:2006uy} that
the cosmic censorship conjecture is not violated by the merger of
two black holes.
A least-squares fit of the energy radiated versus the initial
individual spins yields
\begin{eqnarray}
  \frac{E_{rad}}{M} &=& 0.0348 + 0.0297 \left((a/m)|_{I}\right) +\nonumber \\
   && 0.0170 \left((a/m)|_{I}\right)^2,
\end{eqnarray}
with an error of $\pm0.001$ in the range $(-0.757 < (a/m)|_{I} < 0.757)$.
Based on this fit we extrapolate that a quasi-circular binary
will radiated no more than $8.1\%$ of its mass during the final
few orbits and merger.

\begin{table}
\caption{The measured remnant horizon specific spin
$(a/M_{H})|_R$ and energy radiated $E_{rad}$, as well as
the predicted remnant horizon specific spin
$(a/M_{H})|_{pred}$ and energy radiated $E_{rad}/M|_{pred}$ (based on a least-squares fit) for quasi-circular,
equal-mass, equal-spin binaries with initial specific spins
$(a/m)|_I$.}
\begin{ruledtabular}
\begin{tabular}{lllll}\label{tab:spindat}
$(a/m)|_I$ & $(a/M_H)|_R$ & $(a/M_{H})|_{pred}$ & $E_{rad}/M$ & $E_{rad}/M|_{pred}$\\\hline
$-0.757$ & $0.443\pm0.001$ & $0.4430$ & $(2.2\pm.01)\%$& $2.20\%$\\
$0.000$& $0.688\pm0.001$ & $0.6878$&$(3.5\pm0.1)\%$ &$3.48\%$\\
$0.1001$& $0.717\pm0.001$ & $0.7169$&$(3.8\pm0.1)\%$&$3.79\%$\\
$0.757$ & $0.890\pm0.001$ & $0.8900 $&$(6.7\pm0.2)\%$ &$6.70\%$\\
\hline
$-1.0$ & $***$ & $0.355$& $***$ &$2.2\%$\\
$+1.0$ & $***$ & $0.946$& $***$ &$8.1\%$\\
\end{tabular}
\end{ruledtabular}
\end{table}

We have also found that the tidal coupling in black-hole binaries responsible
for the transfers of angular momentum from the individual spins to
orbital is relatively small. Nevertheless,
we have been able to compute this transfer with sufficient accuracy
to model its dependence on the initial spin.
We find that the radiated angular momentum, which is a measure of the
angular momentum transferred from spin to orbit, has a linear-plus-cubic
dependence on the initial spins; in qualitative agreement both with
the post-Newtonian (see Eq.~(\ref{JdotPNSO})) and close-limit (see Eq.~(\ref{Jcl})) predictions.
The radiated energy has a quadratic-plus-quartic dependence on the
initial spin; again, in qualitative agreement with the post-Newtonian
(see Eqs.~(\ref{EdotPN})) and close-limit (see Eq.~(\ref{Ecl})) predictions.

Unfortunately the speculations of Price and Whelan~\cite{Price:2001un} suggesting this spin-up
effect might be strong, and even dominant, are not confirmed by our computations.
We see neither the possibility of a binary stalling by this effect nor
the dominance of the $\ell=3$ mode suggested in
Ref.~\cite{Price:2001un}.

redpajama_set_name: RedPajamaArXiv


Women's Health: A Guide to Preventing Infections is a booklet for women Veterans. Please refer to and share this information with your family and friends to reduce the spread of infections.
Talk to your health care provider if you have any specific concerns or think you have an infection.


A social movement perspective on working class struggle and accommodation in capitalist societies. Historical and contemporary movements will be considered, with an emphasis on Canada.
Sociology 313 and one of 331, 333, or 365, or consent of the Department.


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CIGA and Cape Town Convention: insolvency and aviation
Vedder Price PC
The Corporate Insolvency and Governance Act 2020 shifted the focus of the UK insolvency regime from administration and liquidation to rescue and recovery and introduced numerous interesting features that apply to companies experiencing financial difficulties. This article considers how some of these features fit into the insolvency regime of the Cape Town Convention.
UK ETS – aviation emissions post-Brexit
The government aims to implement a UK emissions trading system (UK ETS) in January 2021 following Brexit. This article outlines aviation-specific considerations of the proposed UK ETS and the interaction of the proposed scheme with the Carbon Offsetting and Reduction Scheme for International Aviation, the International Civil Aviation Organisation's market-based mechanism for offsetting emissions from aviation.
Court ruling shows importance of standard representations and warranties in aircraft leases
A recent High Court case demonstrates the utility and importance of including standard representations and warranties regarding power, authority and validity in aircraft leases, including as an estoppel against a defendant's claim that a lease is void and unenforceable as a result of alleged breaches of foreign public procurement legislation and lack of authority. The decision also illustrates the benefit of choosing English law to govern aircraft lease transactions, particularly between international parties.
Court of Appeal decision on economic duress in aviation market
A recent Court of Appeal decision regarding a claim of economic duress against an airline reinforced the fundamental importance of ensuring contractual clarity and certainty as a matter of English law. For companies operating in the aviation sector, where there may be an unequal power dynamic, parties will be unable to look to economic duress to undo a lop-sided bargain unless there is some unlawful action or provable bad faith on the part of the party with greater power.
Aircraft Purchase Fleet Limited sought $260 million in damages from Compagnia Aerea Italia
In a recent High Court Case, Aircraft Purchase Fleet Limited (APFL) sought $260 million in damages from Compagnia Aerea Italia (CAI) for an alleged repudiatory breach of a framework agreement under which CAI had agreed to lease certain new Airbus A320 family aircraft. APFL had agreed to buy these aircraft from Airbus. However, CAI argued that it had become impossible for either party to perform the framework agreement following Airbus' termination of its obligations to sell aircraft to APFL.
Drones – the rise of the Basic Regulation
In July 2017 the government released proposals to regulate the use of drones in the United Kingdom. Since then, the regulation of drones has been transferred to the European Union and now falls under the EU Basic Regulation. Many of the UK government's proposals for drone operators are included in the EU Basic Regulation, which sets the groundwork for establishing rules that will require operators of drones that weigh 250kg and above to register them and ensure that they are marked for identification.
Slots can be traded for value even after insolvency
Monarch Airlines Limited's administrators have won an appeal with the Court of Appeal regarding Monarch's rights in and to certain 'slots' at Luton and Gatwick airports after it went into administration. The case is significant, as it reaffirms the value ascribed to slots by airlines and their financiers as rights of the airline and the fact that, as a result, they can be traded for value even after insolvency.
Operating lessor financings: structural overview
With competition among aircraft lessors remaining fierce, airlines are taking an increasing proportion of aircraft on operating leases. The 'wholesaling' of debt financing – where the primary recourse entity on financings is the lessor rather than the airline – is an important recent trend in the aircraft financing market that is likely to continue. Aircraft financiers should be aware of the structural items to consider in executing operating lessor financings and the pitfalls to be avoided.
Split closings: a divide in methods
The inclusion of engine pooling arrangements and rigorous maintenance requirements in operating leases frequently results in engines which formed part of a leased aircraft at delivery being off-wing. Off-wing engines create complications for transaction parties attempting to execute a sale of the aircraft. While these complications are not insurmountable, the marketplace has developed different approaches to address the off-wing engine scenario.
Necessary regulation or the government droning on?
The Department for Transport recently published its response to a public consultation concerning the safe use of unmanned aircraft systems in the United Kingdom. Both in the consultation and the response, it is clear that the government's focus is on ensuring safety, particularly relating to operational issues in the leisure market. However, the response also provides insight into the direction of the government's policy as it affects commercial operators and its determination to develop world-class systems.
Commitment letters not to be taken lightly: how committed are we?
At the outset of a transaction, parties often use a commitment letter, letter of intent or memorandum of understanding to set out the principal terms on which they wish to establish their commercial relationship. The principal terms are often non-binding in nature. The High Court recently referred to an objective test established by the Supreme Court to determine whether a party's intentions were accurately reflected in its initial documentation.
Unjustly enriched seller of aircraft asset: mistakes in sale and purchase transactions
The trading of aircraft assets between industry participants is as dynamic, legally complex and fraught with intense negotiation as ever. Although transacting parties go to great lengths to protect their pre and post-closing positions by attempting to account for all eventualities in the applicable transaction documentation, mistakes that fall outside the protective ambit of such documentation can still occur.
PK AirFinance v Alpstream: did Court of Appeal hit the mark on mortgagees' duties?
The Court of Appeal has delivered its judgment on the appeal by PK AirFinance and GECAS against the Commercial Court's decision in Alpstream v PK AirFinance. The judgment provides a welcome clarification of duties owed by mortgagees to financiers and lessors.
Cape Town Convention and Aircraft Protocol enters into force
The Cape Town Convention and Aircraft Protocol recently entered into force in the United Kingdom. The disapplication of the historically vexed lex situs rules in the context of UK Cape Town interests is a welcome step, as it means that English law and jurisdiction are likely to be the best choice for any mortgage or bill of sale on a transaction involving parties situated in the European Union.
Cape Town Convention and Aircraft Protocol set to enter into force
The Cape Town Convention and Protocol is now set to enter into force in the United Kingdom later this year. Its entry into force was confirmed following the deposit of the instrument of ratification with the International Institute for the Unification of Private Law on July 27 2015 at the UK embassy in Rome.
UK ratification of Cape Town Convention and Aircraft Protocol
The International Interests in Aircraft Equipment (Cape Town Convention) Regulations 2015 were recently laid before Parliament, marking the commencement of the final administrative steps to ratify the Convention on International Interests in Mobile Equipment and the Protocol thereto on Matters Specific to Aircraft Equipment in the United Kingdom. It could enter into force by the end of Summer 2015.
EU declarations affecting UK implementation of Cape Town Convention
The government recently published revised draft legislation designating the Convention on International Interests in Mobile Equipment and Protocol thereto on Matters Specific to Aircraft Equipment as an 'EU treaty', which will allow it to pass legislation to implement the treaty into UK law. However, the European Union made declarations when it acceded to the treaty which will affect UK implementation of the treaty.
Another hurdle cleared on path to ratification: the Cape Town Convention and Protocol
As part of the government's ongoing consultation on the ratification of the Convention on International Interests in Mobile Equipment and the Protocol thereto on Matters Specific to Aircraft Equipment, the government response to the call for evidence has been published. The response is positive, but cautious about any benefits that may accompany ratification.
Alpstream v PK Airfinance: lessons for financiers and lessors
A recent High Court decision highlights the duties of financiers in the context of aircraft repossessions and the associated power of sale in default scenarios, while also raising questions about the level of intention required on the part of a defendant in unlawful means torts. The case garnered significant mainstream media attention, principally because the claimants are controlled by Russian billionaire Alexander Lebedev.
High Court confirms enforceability of one-sided jurisdiction clauses
Following the French Court of Cassation's Rothschild decision, the High Court has ruled that one-sided jurisdiction clauses will be upheld by the English courts. This decision confirms that English courts are unlikely to follow Rothschild and is of considerable importance to aircraft lessors, lenders and airlines.
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redpajama_set_name: RedPajamaCommonCrawl


Mercury enters Jupiter's sign, Sagittarius, on November 28 for an extended transit there until February 2, 2017. This long length is thanks to a retrograde cycle of Mercury from December 19 – January 8 which keeps Mercury in Jupiter's sign for over two months. Jupiter is currently transiting through Mercury's earth sign, Virgo, which extends through September 12, 2017. When two planets occupy one another's sign, it puts them into a deep relationship which strengthens both the planets and the signs they occupy.
Mercury and Jupiter have an interesting relationship. While they are not technically "friends" they do share the distinction of being the rulers of the "dual" signs: Gemini and Virgo for Mercury and Sagittarius and Pisces for Jupiter. This implies a commonality of energy that allows for movement, growth and multifaceted awareness. This puts these two planets on either end of the teaching-learning spectrum. When in this kind of exchange of signs, it allows each of these planets to behave as if they are in their own sign, maximizing the strength and skill of both Jupiter and Mercury.
Mercury's enthusiastic and energetic desire for knowledge is greeted by Jupiter's very pure and light offering of knowledge. This is a wonderful period of time to jump onto the teaching-learning continuum and avail ourselves of this window of opportunity to both gain and digest knowledge.
This parivartana (exchange) also strengthens and protects the all of the dual signs: Gemini, Virgo, Sagittarius and Pisces. When one or two of them does well, they all do well. What a concept!
Venus moves from Jupiter's sign, Sagittarius, into Saturn's sign, Capricorn, on December 2 and spends most of the month moving through to the other side, exiting to Aquarius on December 28. The entire transit is marked by an influence of Saturn on Venus, both by owning the sign Capricorn where Venus is transiting and also by the special 3rd aspect that Saturn is projecting onto Capricorn from Scorpio.
Adding to that, until December 11, Mars is also placed with Venus in Capricorn and will be much stronger due to the exaltation of Mars. During the first third of December, we may feel, in our Venus-ruled realm of relationship, connection and creativity, some level of pressure and heat that may feel uncomfortable.
Capricorn itself is not the sweetest of signs, revealing the rocky slopes that must be traversed if we are to reach our goals and pay our prices. Venus tends to favor comfort and ease, so we may have to buck up, pitch in and help out this month. Even if there is not a reward or treat for our kindness and compassion. Willingly and gracefully walk into situations that may not be pretty or comfortable with open eyes and see what kind of fires you may put out.
Jupiter's aspect on Capricorn from Virgo does offer some hope and warmth that comes in the form of knowledge and experience that do enhance any of Venus' creations. Suffering, difficulty, perseverance and discipline may become unexpected friends as Venus pushes toward the creation of better situations.
The third round of Kala Amrita Yoga winds up on Monday, November 5 when the Moon moves into Aquarius, joining Ketu. We talked about this strong combination in October and again in November.
The basic idea with this yoga and its sister yoga, Kala Sarpa Yoga, is that all of the seven visible planets are hemmed between the two shadow planets. Both in transit and when found in a horoscope, this yoga can indicate sudden and unexpected changes, accidents, reversals, inevitable circumstances and sticky situations.
This final round of three began on November 22. The pure form of this yoga shows all of the planets in the signs on one side of the nodal axis but with none of the planets conjoining the nodes. When a planet, especially a benefic planet like the Moon, conjoins one of the nodes, it has a chance to break and transform the yoga. This will be the case when Moon enters Aquarius and moves past Ketu starting on December 5.
By the next time the Moon comes around to join Rahu in Leo on December 18, Mars will have moved into Aquarius, joining Ketu. This sets us up for a very interesting mid-month after many shifts and conjunctions occur during the first two weeks of December. As you read these next paragraphs, remember to synthesize the other sections of the forecast into this bigger picture of the Rahu-Ketu role in our current planetary framework.
Let's focus for a moment on the basic roles of the nodes, Rahu and Ketu. First, they form an axis that creates an invisible giant stirring stick whose ends are capable of extinguishing the lights of the Sun and Moon when they move through the axis. This is a fancy way of pointing out that Rahu and Ketu conjoined the Sun and Moon cause eclipses.
This eclipsing capability is rather powerful. We are talking about shadows and obfuscation. We enter the realm where it becomes difficult to discern the truth and to see clearly. When we are beginning to analyze how these nodes affect us, it is good to keep that in mind.
Rahu and Ketu are two parts of a former whole, with Rahu playing the role of the head separated from its body and Ketu playing the role of the headless body. Within the head we have the mind and the senses that fuel the desires that cause our actions. Within the body, we have our network of nerves and impulses that, without being driven by the mind, give us intuitive intelligence.
Rahu is the exploiter, the manipulator, the genius. He will urge us to accomplish our goals and fulfill our desires by whatever means we can perceive. If this means jumping across the border and coloring outside the lines, that will be done. As you can imagine, this sometimes causes problems and sometimes causes amazing inventions. Rahu realizes that diversity is the foundation of survival and revels in the chaos of difference. Rahu doesn't care about color, shape or caste and sees that all of these various differences are essential to the thriving mass of life on planet Earth.
Ketu's only motivation is freedom. Left without a head, the body's intelligence guides us toward a protected stance from whence we can seek liberation. Because of Ketu's limited range of desire, the association with Ketu tends to block the clarity that comes from objective perception; remember, Ketu has no eyes, ears, nose, tongue or mind. On one hand, this can keep us out of the trouble generated by overindulging the senses and emotions. On the other hand, we walk around acting without enough information as if we are sleepwalking, making mistakes.
The Moon moves into Rahu's space in Leo on December 18, moving past Rahu with and exact conjunction happening around midnight as December 19 comes around. December 19 is the day that the Electoral College vote happens in the US. The Moon can be used as a measure of public opinion and was conjoined Ketu on the day of the election, November 2. Association with Ketu can cause reversals; with the Moon as the planet of the public mind, it was surprising on that day when Donald Trump took the election after the public opinion polls had strongly suggested another outcome.
With Moon joining Rahu on this next level of the election process, we may expect a similar upheaval from the public mind. The most notable character in this last bout with Kala Sarpa Yoga will be Mars, in the end. After Moon moves through Leo and into Virgo on the morning of December 20, Mars alone will stand in the way of the nodal axis. Mars' methods of breaking the yoga will likely not involve negotiation or diplomacy. This could be the anger of the people coming to a head and fighting back against oppressive circumstances. We have used the US election as an example, but there are many places on the planet at this time that are seeing the rising of collective anger against the "powers that be."
Mars comes to highest exaltation at 28° Capricorn, Saturn's earth sign. The lunar mansion Dhanistha, ruled by Mars himself, overlays this position within the zodiac. Mars is very strong, climbing steadily on the heavy foundation laid by Saturn.
Before rejoicing at the use of the word "strong," remember that everything is relative on the good-bad spectrum. In individual and collective ways, we are navigating through circumstances in which the powerful energy, drive and perception of Mars is at its peak. We must think deeply about how we will wield and channel this power and fire of Mars. The master of expansion, Jupiter, and the master of contraction, Saturn, will both be aspecting Mars within the first week of December. This combination of influences upon the strong Mars is favorable in the sense that it gives balance between knowing when to act and knowing when to wait, both of which are necessary in any advancement.
Rahu and Ketu will enter into an exchange of nakshatras (lunar signs) from December 9, 2016 – April 14, 2017.
The last time this type of Rahu-Ketu nakshatra exchange occurred (Rahu in Mula, Ketu in Ardra) was from late September 2010 through January 2011. Mula nakshatra is placed in the first section of Sagittarius and is ruled by the planet Ketu. Ardra is placed within Gemini and is ruled by Rahu. Some notable world events from this time frame include: "Diplomatic Cables Release" from Wikileaks, earthquake/tsunami/volcano action in Indonesia and the Arab Spring revolution in Tunisia.
"Nakshatra" is the word for a section of the zodiac when it has been divided into twenty-seven parts known as the lunar signs, through which the Moon moves approximately one per day. The lunar signs overlay the sun signs and further define smaller sections of the zodiac. Interpreting a planetary placement by observing both the solar sign (ruled by the Sun) and the lunar sign (ruled by the Moon) allows us to more deeply understand the subtle meanings of the placement.
This time, we will have Ketu placed in Rahu's sign (Aquarius) and nakshatra (Shatabishak) with Rahu rolling in backward motion through Ketu's nakshatra (Magha) while still in the Sun's sign, Leo. For part of this time, the Sun will also be in Aquarius (Feb 12 – Mar 13, 2017), offering opportunities for eclipses while the Sun travels through the nodal axis. The first eclipse pair of 2017 happens on February 10 (lunar, full Moon) and February 26 (solar, new Moon).
If we can predict sudden and unexpected upheavals from the influence of Rahu and Ketu, with both of them in strength due to this nakshatra exchange, let us keep our eyes open for where the instability will appear.
Magha nakshatra spans the first 13°20' of the sign of Leo, ruled by the Sun. Magha's planetary ruler is Ketu and the associated diety is the Pitris, the ancestors and forefathers. Magha is associated with lineages, especially of rulership due to the influence of the Sun through Leo. The entire transit of Rahu through Leo can be said to encourage the challenge of the establishment by the hungry masses. The problem is that Rahu's penchant for obfuscation can create confusion about what should be retained and what should be destroyed. We can run around shouting for "change" but without really having an idea about what type of change would be productive and beneficial.
The mindless adherence to traditional structures of governance is very prominent in Magha nakshatra. Faith that the good king will take care of his subjects flourishes here. With Rahu moving through Magha from December 9, 2016 until he retreats into the sign of Cancer in mid-August 2017, there is a hungry force that is speaking for the masses and desires changes to the status quo. The nakshatra exchange between Rahu and Ketu extends through mid-April, but Rahu's presence in Magha holds through mid-August 2017. We must remain vigilant about monitoring the forces behind the change we purport to desire; with Rahu's influence, things will not be what they appear and the characters involved are likely wearing multiple masks.
Disclaimer. In true Rahu-Ketu form, there are two ways of tracking their progress: the mean node and the true node calculations. I use the mean node in my forecasting. Be aware in reading other forecasts that different astrologers use different calculations so the exact dates of transits may vary.
The annual meeting of the Sun and his second son, Saturn, occurs for the third consecutive year in the sign of Scorpio. The relationship between these two planets has never been smooth and their association in the cold, dark environment of Scorpio can bring forth the roots of any difficult relationship.
Sun has been transiting through Scorpio since mid-November. He glides steadily forward at his reliable average pace of one degree per day toward his meeting with Saturn, which comes on December 10 when the two meet at 25° Scorpio.
The eternal debate between the importance of Me and Not Me always flares up when Sun and Saturn associate. These types of snags are perfect opportunities for examining the virtues of a balanced and humble ego structure.
Mars departs his sign of exaltation, Capricorn, and enters into Aquarius for a five-week transit on December 11. Mars joining Ketu in Aquarius further strengthens the parivartana (exchange) between the lords of Scorpio and Aquarius as Mars joins his co-ruler of Scorpio, Ketu, and they both exchange with Saturn, co-ruler of Aquarius.
This very powerful exchange strengthens the camp of the mischievous planets, empowering Saturn, Mars and Ketu to participate in the "dirty work" that they are charged with doing. At the same time, we have a fantastic exchange between Mercury and Jupiter, bolstering the "good guys" camp. This is shaping up to be an interesting time period especially as first Sun (December) and then Saturn (January) will be crossing over the gandanta point between Scorpio and Sagittarius.
This gandanta, or knot, is a difficult juncture between a water and a fire sign. There is generally conflict when it comes to the elements of water and fire; this particular gandanta, the second of three in the zodiac, is regarded as the battlefield.
Mars' role is generally as leader of a well-resourced force, such as the general of an army. Ketu is considered the army forces itself. With the two aligned, we can see the possibility of a military advance. This could be, of course, literal or figurative. What kind of battle is brewing depends on both individual circumstances and the mood of the environment in which we find ourselves. Mars and Ketu conjoin at 13° Aquarius on December 27, adding his energy to the Rahu-Ketu nakshatra exchange.
Remember also that Venus, whose main goal is peace and comfort, would prefer diplomacy over any action that involves violence. However, Venus is somewhat caught in Saturn's grip at this time, transiting through Capricorn during the month of December.
Mars from Aquarius is also aspecting Taurus, Leo and Virgo and all planets contained therein, notably Rahu in Leo and Jupiter in Virgo. Mars' fire may be enough to shake the foundations that the fixed signs Taurus, Leo, Scorpio and Aquarius so highly value.
The Moon becomes full at 29° Taurus on the evening of Tuesday, December 13. This full Moon falls in the nakshatra of Mrigashira, which spans the signs of Taurus and Gemini under the rulership of Mars. It is known as "the Searching Star" and has the reputation for endlessly seeking, much like the unending grazing and movement of a deer.
Mars rules Mrigashira and Mars himself will be in his own nakshatra of Dhanistha; also Mars is the Lord of the Day, Tuesday. Jupiter in Virgo is also placed in a nakshatra of Mars, Chitra. Mars from Aquarius will give planetary aspect to both Moon in Taurus and Jupiter in Virgo. So while the lunar energy is at its height, we are also feeling the strong energy of Mars affecting the Moon as well as Jupiter. Strong Mars can make us act without consideration as it can promote impulsive behavior in an "act first, ask questions later (or never)" way. With the Moon also channelling that trigger-finger reaction, beware of becoming overwhelmed by emotion and acting brashly, as somewhere down the line regret for impulsive action may set in.
As mentioned earlier, the Sun travels through the gandanta between Scorpio and Sagittarius in December, specifically December 13-16, during the time of this full Moon. Please exercise caution, especially when it comes to acting on emotion during this time. We are experiencing a challenging set of planetary influences that require us to calculate our actions from a calm position if we wish to go forward with a minimum of strife.
As the Sun moves past Saturn during the first half of December and pushes through the tight gateway between Scorpio and Sagittarius, we may feel that squeeze in our own lives as the Sun represents our cherished individuality that we express on a soul level.
When we look at the instability and confusion that can come from Rahu in Leo, we have to look at strengthening the Sun as our ticket to discerning the real from the unreal. The Sun will always lead us on the righteous path, it is just that it is sometimes difficult to see the Sun when obscured by clouds or other factors. The Sun, having just traversed its debilitation sign of Libra, followed by a difficult transit through Scorpio and past Saturn, has been struggling to be our beacon of righteousness.
Generally, the Sun traveling through the friendly and positive signs of Jupiter renews our sense of what is right and true. But this time, we must recall that the added aspect of Rahu from Leo onto Jupiter's sign of Sagittarius will continue to make it difficult to know right from wrong based on the smoke signals being sent from Rahu's position.
Despite this confusion, the Sun's mission toward the good and true will be much better supported as Sun transits through Sagittarius. This is a good time to focus and get back on track, as after Sagittarius, the Sun must pass through the signs of Saturn: Capricorn and Aquarius. Gather intelligence and analyze well during this month that spans our western calendar new year. Mercury's presence in Sagittarius will be helpful in assessing the situation. The Sun will both benefit from and add to the positive nature of the Jupiter-Mercury exchange.
As mentioned earlier, Mercury will be in Sagittarius for a couple of months due to a retrograde period from December 19, 2016 – January 8, 2017. While planets in retrograde motion don't literally reverse their course in orbit around the Sun, they do appear to re-trace their path briefly from our perspective on Earth. Thus, they linger in a section of the zodiac for a spell, making us wonder what they are trying to tell us.
Retrograde planets appear so because they happen to be on a part of their orbit that brings them closer to earth. They are brighter to us and are also strong in astrological indications.
Mercury is an energetic planet. With strength, Mercury is perhaps too energetic. Without a precise plan for funneling this energy, mishaps can occur. Perhaps it is impossible to prevent these funny mishaps and frustrating miscommunications. Instead, if we see this upcoming set of waves as big and energetic, we could perhaps make a plan to harness that energy and accomplish much, especially in the way of the teaching -learning spectrum as mentioned earlier.
Mercury and Jupiter will be energizing the dual signs: Gemini, Virgo, Sagittarius and Pisces. These are the places in the zodiac where real change does happen, where things are left behind and new horizons are perceived.
No matter what the astrological framework operating at any one time, no matter what the government structure, we are still on an individual path. It is up to us to grow, learn and teach the next one. We must find our vehicle and learn to relate to new paradigms, for change is the constant in life.
Mercury retrograde sometimes makes messes, but it also opens doors wider. Go deeply into your learning and evolution. Go back to your old mistakes and learn again. Review the material until it becomes a part of your body. It is easy to collect information and data, to consume and analyze it; this is the fun job of Mercury. The problem is keeping up with digesting the material and making it a part of the soul; this is the expansive job of Jupiter. This Mercury retrograde cycle within the period of exchange with Jupiter is inviting us to digest fully such that information becomes knowledge.
When Venus enters Aquarius on December 28, she joins the Mars and Ketu combination. What an interesting confrontation, with the fiery and explosive rulers of Scorpio joined by Venus, the watery fire-fighter.
The Moon settles into the arms of the Sun just before midnight on December 28, at 14° Sagittarius, in the beginning of Purvashada nakshatra. Retrograde Mercury sits near by, bringing the Sun and Moon into the energy of the parivartana created by Jupiter and Mercury.
This is a Wednesday night, and with retrograde Mercury as the ruler of the day, Wednesday the 28th may be a great day to do the opposite of what you feel like doing, which is likely packing 10 pounds into a 5 pound package. Rahu will be aspecting the new Moon combination very strongly from Leo. Rather sit down, be quiet and listen. Think deeply about where you are putting your energy and efforts. Assure that you are on the correct path. Assess other paths and directions if you find that you are not.
The next day, step forward and make 2017 be a year filled with the reality of practical change based on personal action. Open up to inner wisdom so that right and wrong may be distinguished without fail as you continue on your path.


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redpajama_set_name: RedPajamaGithub


Gral and I had a conversation about Eidolon. Unformated because I'm lazy and with some bonus side banter.
" So, my next question is what do we do about Eidolon?
me: start developing our own plot and seize control?
06:29 Gralhruk: I pinged Cad before but he hasn't responded.
06:30 Gralhruk: Well, we've always been free.
Gralhruk: I guess my question, really, is everyone still interested.
Gralhruk: You really ought to have one of those website things with a big following.
Gralhruk: You had a minor following.
You could get a big following with a little marketing.
Gralhruk: Don't stop at pretty good.
You could have t-shirts and coffee mugs and stuff.
And then be able to quit your day job.
06:33 And you'd never have to go out in public.
Gralhruk: Okay, so you are interested in keeping Eidolon alive. . . have I got that right?
Gralhruk: I can have Rutiles open the door if you want.
Or we can wait for Cad.
It's just like chapter 2!
Except Rina was by herself then.
Gralhruk: It's a crime, really.
me: I'll turn my mighty intellect to Bards. Are we coming up with a plot for Eidolon?
Gralhruk: Well . . .
I was hoping to hear from Cad.
06:39 But I'm certainly open for ideas.
Shade and Nicos get lost in the woods and go on a wild acid trip?
06:41 I'll see if I can get Kel excited for doing something in Eidolon.
We want to close out the zombie plot.
We get seperated from the rest of the party as we run for it and something happens when we try to hook backup?
Gralhruk: So . . . Blarth, Nicos, Juni and Shade get separated?
me: Wait, did I mention she's turning into a vampire?
She's going to kill Santa, isn't she?
06:44 Gralhruk: Okay, so we get lost and find ourselves at the north pole . . .
Gralhruk: Isn't the head zombie kind of like a Blair Witch?
It's a brave choice, I'll give her that.
Kel: I'll leave it to the two of you. I'm not much for coming up with plot ideas.
itches: It's easy. Think of a film or book you like. Then steal it.
Kel: but where would we come up with all those singing children?
Kel: not to mention the nuns.
itches: There is a remote nunnery in the woods, who also run an orphanage?
I do like the nunnery/orphans idea.
Sounds like we've got the makings of a Ravenloft-esque plot here. As we finish up escaping this zombie horde, a thick fog settles on the woods, separating the party members from the remains of the caravan (and possibly each other). As the fog lifts, the (possibly divided) party finds itself either on a semi-abandoned road which leads to a ragged looking settlement (party members might also approach the settlement directly through woods). On the surface, everything seems normal and the party has a moment's reprieve after their harrowing experience. The fog, however, marked a transition from the real world into a Domain of Dread so while the party may be free of the zombie problem, they are trapped until they can figure out how to escape.
A pocket universe! - that's a pretty cool idea. I'm not a huge, huge fan of horror, but I'd be willing to give a Ravenloft setting a try.
The only problem I have with horror is all the damned vampires.
So, as far as a pocket universe, there is kind of a precedent - the first few pages of chapter 2 have Cadrius and Shade transported inexplicably from a little orchard into a sort of horror themed wood. They return without ever figuring out exactly what happened or why, but we could suppose it's along the same lines as what happens now.
When I had written the initial post my inspiration was from Fafhrd and the Grey Mouser, when they wander into the Shadowlands (basically, the afterworld) by accident.
I had forgotten about that scene, but yeah, that's about what I was thinking. I'm not familiar with the source you're referencing, but I think that we can tie together several disparate elements in doing this.
I'm not a big fan of Ravenloft, but I'm not against stealing from it for our own custom setting. Crowattic!
That said we still need a plot.
I don't know anything about Ravenloft the setting, really. I played module I6 - Ravenloft, the AD&D module that spawned the setting but that's pretty much it. So yeah, no need for us to try and use the actual setting.
I was thinking along the lines of having each of us get lost in the fog and face down some horror on our own - either something from the past, or some fear, or some other random event. Success (or maybe failure) leads us out of the fog and into the demi-plane (where we meet back up) and into our own little horror story.
The exact details of said horror story are still up for grabs. As an explanation, maybe the sudden rise of all these zombies has sort of weakened the barrier between our normal plane and this demi-plane we've slipped into. If that's true, maybe there is some way for us to use that to end the plague (and I have no idea if that makes sense or is a good idea, just kicking some ideas around).
If we do go that route, though, maybe we come across some object in the demi-plane that we bring back with us that acts as sort of a beacon for the zombies, enabling us to cast them back. But again, just brainstorming and I'm not particularly attached to any of that.
Each of the Domains of Dread (at least in their 4e incarnation, where Ravenloft isn't so much a separate setting as a feature of the Shadowfell) each have some sort of theme that characterizes why they were created: betrayal, murder, greed, and lots of variation on these and other themes. That theme, once figured out, often then provides a hint as to how to escape the domain. A domain about betrayal, for might have been formed by a particularly devious betrayal by its Lord and the trick to escaping then might be to betray the betrayer, and thus show the betrayer the cost of his betrayal, or to be steadfast in the face of near overwhelming temptation, and thus show the value of sticking to one's guns. Often both work as a means of escape, but one has the added effect of not just letting the individuals involved escape, but also freeing everyone caught in the domain (bringing the domain to an end). If we want to go this route, then I'd suggest that each of our individual horrors should be constructed in such a way as to relate to some theme, one we'd obviously need to come up with.
Along those lines, I just had an idea. Perhaps the trick that Skathros pulled on Shade to get her to kill Ricard was a trigger event that started to create a new Domain of Dread. Originally Shade, Skathros, and Ricard would have been its new lords, locked in some sort of perpetual cycle of manipulation and betrayal, but Isac's sacrifice disrupted the Domain's formation. The result is that Shade is no longer bound to the Domain as one of its Lords, but the forces that started to create the domain will not be thwarted so easily. The domain has continued to form and is now pulling the entire party in while setting in motion a series of events, which if followed, will lead to one of the party members taking the place vacated by Shade as a Domain Lord. Somewhere along the way, however, there is an opportunity to break the sequence of events, something which leads to the end of the Domain and the return of the party (and anyone trapped with them) to the Material Plane.
Not that things are much better there with the whole zombie infestation thing.
I like the theme idea a lot.
And the Shade / Skathros / Ricard thing is workable as well. I'd *really* like to find a way to tie the zombies in with that. But I'm sick and feeling too miserable to think straight right now so input from me is going to have to wait a little.
This sounds better and better. I hope I'll be able to contribute to the theme in a meaningful way.
Sorry to hear you're not feeling well, Gral. Get some rest and feel better soon!
The problem I see with trying to tie the zombies into this diversion is the timing of their appearance. Cadrius first encountered the zombies away from Tradeholm and before the whole thing with Shade, Ricard, and Skathros went down (at least, that's how I've read the timeline thus far). I suppose we could tie them to Skathros, perhaps he's developed some way of corrupting and controlling people and the hamlet where Cadrius first encountered it was some sort of test. In that case the zombies might be drawn into the Domain with us as Skathros' faction. That might also create an interesting issue regarding escape. We might find a way to escape that would release the zombies at the same time and thus be forced to choose between staying in the Domain or releasing the zombies (or finding door C, the way out without the zombies).
I'm just spit balling here. If anyone else has ideas, throw them out. Maybe they'll stick.
Well, I'm not sure where it was mentioned, but the zombies *are* related to Skathros. I'll dig around and find out what it was that we (me / J / Cad?) came up with.
I don't know how much was mentioned in the IC / OOC, as a good deal of that brainstorming stuff took place in chat.
I'd appreciate that as I seemed to have missed something then.
I'm searching through my disorganized mass of notes, scattered across all of my various electronic media. I'll post what I find as soon as I find enough of it to be coherent.
2c) This allows the goblins to scry it out, which they do, and now they want it back.
2d) To help get it back, they put a curse on it (and anybody who touches it) to return it. This curse also has the side effect of turning the person into an undead servant.
4) Shade has the theft and all removed from her memory, per standard guild procedure when dealing with magical, high profile items.
5) The church knows that both Skathros and Shade are now cursed, since they re-acquired the relic and figured out the curse.
2) The goblin witch came North to look for the reliquary himself, and he started out by using witchcraft to make himself some undead minions (or just to cause more confusion).
I thought we had come to an actual decision on how we were tying them in, but if we did I couldn't find it anywhere. So really, it's anyone's call how (or if) they are related.
Personally, I think they probably should be related as then all the zombie stuff fits better but I'm open to suggestions.
That all seems to make sense. Except for the part where the goblin witch is a he. I pictured something completely different.
Could be a she. Hard to tell with goblins .
In order for a curse to be effective at changing behavior, it has to impose an undesirable but manageable condition, especially if it the one cursed isn't aware of the curse at first. Perhaps, then, the curse is designed to first affect not the one cursed, but those around them. The person cursed becomes a sort of carrier for this zombie plauge, spreading it to those with whom they come in contact. Since none of the party contracted it, despite having regular contact with Shade while she was subject to the curse, perhaps it requires a specific form of contact: financial. Buy or sell something and the person you're trading with contracts the plague. The towns that we've encountered, then, might be places where Skathros, or one of the other middlemen that had the relic for a while have conducted a fair bit of trade. This would serve to isolate and ruin the carrier but not destroy them so that they can still undertake actions to remove the curse (i.e. return the relic to the goblins).
Since Skathros has succumbed to the plague himself, we might envision the curse as having a second stage. If becoming a plague carrier (and thus making them a pariah to normal society) isn't enough to change the person's behavior then the plague goes to work on the carrier, slowly rotting them from the inside and turning them into an undead horror to create the isolation that way. Instead of relying on purely social isolation, in creates a more physical mark of the isolation that the carrier is supposed to be experiencing.
As even that wasn't enough for Skathros, who actually quite liked the new powers he had from his changed form, the plauge has progressed to a third stage, creating the isolation through a Domain of Dread.
Alright, everyone remain calm and no one has to get hurt. I'm hijacking this hijack.
I want the next chapter to be self contained. A plot that starts and ends within the chapter without dealing with anything else. The game has been hibernating for a while now, we (I? You?) need something simple to get back into the swing of things.
The concept is that somehow (it doesn't matter story wise how) we end up in a weird location where there is a powerful spirit trapped. Think old, old style faerie story. It wants to get out by possessing one of us.
Aftermath of the zombie fight, chaos everywhere. Running into the woods at random.
Everyone who takes part suffers a weird trippy vision about some deeply held fear of the character. Vision end up falling into nothing.
Those taking part find themselves in a small abandoned town in a clearing in the woods, seems like it was abandoned a week or two ago. Everyone meets up and agrees to fort up for the night in the town.
While forting up (gathering food, building walls, searching the place) everyone sees someone walking around the town. It's a different person for everyone (Lynn for instance sees herself) but they disappear before we can speak to them. Try to make it creepy.
The sun sets and we all gather together, and I presume compare notes about what has been happening because our characters aren't stupid.
We hear something(s) moving around outside our area, but can't spot anything. Then all the fires (candles, lanterns) go out and we're attacked by something but we still can't see what it is (need a reason why folk with darkvision can't tell what it is).
The fight goes on for a bit, and then suddenly they all disappear, and we can't find any trace of our attackers (no tracks or bodies or anything). They may have just been another vision.
What I don't have is a conclusion. The plot isn't about defeating the fae, it's about surviving it. Whether we stick it out for the night or just run for the woods to take our chances, or someone actually does get possessed without (us? them) knowing it. In either of the cases we need a suitable climax for the story tension that has been building.
I'll play along, but I don't immediately have any ideas for an ending to the story.
We draw straws, the person with the short one gets eaten by a Grue.
Mine wasn't helpful either, but I meant it to be supportive.
Well, I'm game. It's more or less what we discussed earlier, except with some small purpose. We need an opening post, so I nominate whoever posts IC first.