EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2018-046 LHCb-PAPER-2018-007 29 March 2018

Measurement of the $C\\!P$ asymmetry in
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ and
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ decays

LHCb collaboration†††Authors are listed at the end of this paper.

The $C\\!P$ asymmetry in ${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ and
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ decays is measured using LHCb
data corresponding to an integrated luminosity of 3.0$\mbox{\,fb}^{-1}$,
collected in $pp$ collisions at centre-of-mass energies of 7 and 8 TeV. The
results are
${{{\mathcal{A}}^{{C\\!P}}}({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}})}=(-0.4\pm
0.5\pm 0.5)\%$ and
${{{\mathcal{A}}^{{C\\!P}}}({{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}})}=(2.3\pm
2.7\pm 0.4)\%$, where the first uncertainties are statistical and the second
systematic. This is the first measurement of
${{\mathcal{A}}^{{C\\!P}}}({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}})$
and the most precise determination of
${{\mathcal{A}}^{{C\\!P}}}({{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}})$.
Neither result shows evidence of $C\\!P$ violation.

Published in JHEP 05(2018)160

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

##  1 Introduction

Weak decays of heavy hadrons are governed by transition amplitudes that are
proportional to the elements $V_{qq^{\prime}}$ of the unitary $3\times 3$
Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2], a crucial component of the
Standard Model (SM) of elementary particle physics. Different decay rates
between heavy-flavoured hadrons and their antiparticles are possible if there
is interference between two or more quark-level transitions with different
phases. The corresponding violation of $C\\!P$ symmetry was first observed in
neutral kaon decays [3]. In $B$ decays, $C\\!P$ violation was first observed
in the interference between a decay with and without mixing [4, 5] and later
also directly in the decays of ${B}^{0}$ mesons [6, 7].

The decays of charged or neutral $B$ mesons to two charm mesons are driven by
tree-level and loop-level amplitudes, as illustrated in Fig. 1. Annihilation
diagrams also contribute, but to a lesser extent. The decays ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{D}^{+}}{{D}^{-}}$,
${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ and
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ are related by isospin
symmetry,111Unless specified otherwise, charge conjugation is implied
throughout the paper. and expressions that relate the branching fractions and
$C\\!P$ asymmetries, as well as nonfactorizable effects, have been derived [8,
9].

The $C\\!P$ asymmetry in the decay of the ${{B}^{-}}$ meson to two charm
mesons is defined as

${{\mathcal{A}}^{{C\\!P}}}({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}})\equiv\frac{\Gamma({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}})-\Gamma({{{B}^{+}}}\\!\rightarrow{{D}^{+}_{({s})}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})}{\Gamma({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}})+\Gamma({{{B}^{+}}}\\!\rightarrow{{D}^{+}_{({s})}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})}.$ (1)

Nonzero $C\\!P$ asymmetries in
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$ decays are expected [10,
11, 12, 13] due to interference of contributions from tree-level amplitudes
with those from loop-level and annihilation amplitudes. In the SM, these
$C\\!P$ asymmetries are expected to be small, ${\mathcal{O}}(10^{-2})$. New
physics contributions can enhance the $C\\!P$ asymmetry in these decays [14,
12, 15, 13].

Figure 1: Illustration of (left) tree diagram and (right) loop diagram
contributions to the decay ${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$.
Similar diagrams, with the $d$ replaced by $s$, apply to the decay
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$.

The most precise measurements of the $C\\!P$ asymmetry in
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ decays are from the Belle and
BaBar experiments, ${{\mathcal{A}}^{{C\\!P}}}=(0\pm 8\pm 2)\%$ [16] and
${{\mathcal{A}}^{{C\\!P}}}=(-13\pm 14\pm 2)\%$ [17], respectively, where the
first uncertainties are statistical and the second systematic. The $C\\!P$
asymmetry in ${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ decays has not
been measured before.

This paper describes a measurement of the $C\\!P$ asymmetry in
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ and
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ decays, using $pp$ collision
data corresponding to an integrated luminosity of 3.0$\mbox{\,fb}^{-1}$, of
which 1.0$\mbox{\,fb}^{-1}$ was taken in 2011 at a centre-of-mass energy of
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ and 2.0$\mbox{\,fb}^{-1}$ in 2012
at $\sqrt{s}=8$$\mathrm{\,Te\kern-1.00006ptV}$. Charm mesons are reconstructed
in the following decays: ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$,
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$,
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$, and
${{D}^{-}_{s}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{-}}$.

The determinations of
${{\mathcal{A}}^{{C\\!P}}}({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}})$
are based on the measurements of the raw asymmetries

$A_{\rm
raw}\equiv\frac{N({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}})-N({{{B}^{+}}}\\!\rightarrow{{D}^{+}_{({s})}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})}{N({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}})+N({{{B}^{+}}}\\!\rightarrow{{D}^{+}_{({s})}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})},$ (2)

where $N$ indicates the observed yield in the respective decay channel. The
raw asymmetries include the asymmetry in $B$ production and detection
efficiencies of the final states. If the asymmetries are small, higher-order
terms corresponding to products of the asymmetries can be neglected, and the
following relation holds

${{\mathcal{A}}^{{C\\!P}}}=A_{\rm raw}-A_{P}-A_{D},$ (3)

where $A_{P}$ is the asymmetry in the production cross-sections, $\sigma$, of
${B}^{\pm}$ mesons,

$A_{P}\equiv\frac{\sigma({{{B}^{-}}})-\sigma({{{B}^{+}}})}{\sigma({{{B}^{-}}})+\sigma({{{B}^{+}}})},$
(4)

and $A_{D}$ is the asymmetry of the detection efficiencies, $\varepsilon$,

$A_{D}\equiv\frac{\varepsilon({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}})-\varepsilon({{{B}^{+}}}\\!\rightarrow{{D}^{+}_{({s})}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})}{\varepsilon({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}})+\varepsilon({{{B}^{+}}}\\!\rightarrow{{D}^{+}_{({s})}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})}.$ (5)

##  2 Detector and simulation

The LHCb detector [18, 19] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region [20], a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes [21] placed
downstream of the magnet. The polarity of the dipole magnet is reversed
periodically throughout data-taking, to cancel, to first order, asymmetries in
the detection efficiency due to nonuniformities in the detector response. The
configuration with the magnetic field vertically upwards (downwards) bends
positively (negatively) charged particles in the horizontal plane towards the
centre of the LHC.

The tracking system provides a measurement of momentum, $p$, of charged
particles with a relative uncertainty that varies from 0.5% at low momentum to
1.0% at 200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a
track to a primary vertex (PV), the impact parameter (IP), is measured with a
resolution of $(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov (RICH)
detectors [22]. Photons, electrons and hadrons are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [23].

The online event selection is performed by a trigger [24], which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. At
the hardware trigger stage, events are required to have a muon with high
$p_{\mathrm{T}}$ or a hadron, photon or electron with high transverse energy
in the calorimeters. The software trigger requires a two-, three- or four-
track secondary vertex with a large sum of the transverse momenta of the
tracks and a significant displacement from the primary $pp$ interaction
vertices. At least one track should have
$\mbox{$p_{\mathrm{T}}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$\chi^{2}_{\text{IP}}$ with respect to any PV greater than 16, where
$\chi^{2}_{\text{IP}}$ is defined as the difference in fit $\chi^{2}$ of a
given PV reconstructed with and without the considered particle. A
multivariate algorithm [25] is used for the identification of secondary
vertices consistent with the decay of a $b$ hadron.

Simulated events are used for the training of a multivariate selection, and
for determining the shape of the invariant-mass distributions of the signals.
In the simulation, $pp$ collisions with
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$ decays are generated
using Pythia [26, *Sjostrand:2006za] with a specific LHCb configuration [28].
Decays of hadronic particles are described by EvtGen [29], in which final-
state radiation is generated using Photos [30]. The interaction of the
generated particles with the detector, and its response, are implemented using
the Geant4 toolkit [31, *Agostinelli:2002hh] as described in Ref. [33]. Known
discrepancies in the simulation for the mass scale, the momentum resolution
and the RICH response are corrected using data-driven methods.

##  3 Candidate selection

The offline selection of ${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$
candidates is a two-step process. First, loose criteria are applied to select
candidates compatible with the decay
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$. Second, a multivariate
selection is applied and optimized by minimizing the statistical uncertainty
on the asymmetry measurement.

Charm meson candidates are constructed by combining 2, 3 or 4 final-state
tracks that are incompatible with originating from any reconstructed primary
vertex ($\chi^{2}_{\text{IP}}>4$). In addition, the sum of the transverse
momenta of the tracks must exceed 1.8 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
the invariant mass must be within $\pm
25{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known charm meson mass [34]
and the tracks are required to form a vertex with good fit $\chi^{2}$.
Particle identification (PID) criteria are also applied to the final-state
particles, such that particles that have a significantly larger likelihood to
be a kaon than a pion are not used as a pion candidate, and conversely. Three-
track combinations that are compatible with both
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ and
${{D}^{-}_{s}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{-}}$ decays are
categorized as either ${D}^{-}$ or ${D}^{-}_{s}$, based on the invariant mass
of the three-track combination, the compatibility of opposite-charge track
combinations with the $\phi\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decay, and the
PID information of the final-state tracks [35].

In events with at least one ${D}^{-}$ or ${D}^{-}_{s}$ candidate and at least
one ${D}^{0}$ candidate, the charm mesons are combined to form a ${{B}^{-}}$
candidate if their invariant mass is in the range
$4.8-7.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The ${{B}^{-}}$ candidate
is required to form a vertex with good fit $\chi^{2}$, and have a transverse
momentum in excess of $4.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The resulting
trajectory of the ${{B}^{-}}$ candidate must be consistent with originating
from the associated PV, which is the PV for which the ${{B}^{-}}$ candidate
has the smallest value of $\chi^{2}_{\text{IP}}$. The reconstructed decay time
divided by its uncertainty, $\tau/\Delta\tau$, of ${D}^{0}$ and ${D}^{-}_{s}$
mesons with respect to the ${{B}^{-}}$ vertex is required to exceed $-3$,
while for the longer-lived ${D}^{-}$ meson it is required to exceed $+3$. The
tighter decay-time significance requirement on the ${D}^{-}$ eliminates
background from
${{{B}^{-}}}\\!\rightarrow{{D}^{0}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays
where the negatively charged pion is misidentified as a kaon. In the offline
selection, trigger signals are associated with reconstructed particles. Signal
candidates are selected if the trigger decision was due to the candidate
itself, hereafter called trigger on signal (TOS), or due to the other
particles produced in the $pp$ collision, hereafter called trigger independent
of signal (TIS).

The invariant-mass resolution of
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$ decays is significantly
improved by performing a constrained fit [36]. In this fit, the decay products
from each vertex are constrained to originate from a common vertex, the
${{B}^{-}}$ vertex is constrained to originate from the associated PV, and the
invariant masses of the ${D}^{0}$ and the ${D}^{-}_{({s})}$ mesons are
constrained to their known masses [34],

To reduce the combinatorial background, while keeping the signal efficiency as
large as possible, a multivariate selection based on a boosted decision tree
(BDT) [37, 38] is applied. The following variables are used as input to the
BDT: the transverse momentum and the ratio between the likelihoods of the kaon
and pion hypotheses of each final-state track; the fit $\chi^{2}$ of the
${{B}^{-}}$ candidate and of both charm meson vertices; the value of
$\chi^{2}_{\text{IP}}$ of the ${{B}^{-}}$ candidate; the values of
$\tau/\Delta\tau$ for the ${{B}^{-}}$ and for both charm meson candidates; the
invariant masses of the reconstructed charm meson candidates; and the
invariant masses of opposite-charge tracks from the ${D}^{-}_{({s})}$
candidate. Separate trainings are performed for the
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ and the
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}$${D}^{0}$ modes, and for both ${D}^{0}$
decay channels. The BDT is trained using simulated ${{B}^{-}}$ signal samples
and candidates in the upper mass sideband of the ${{B}^{-}}$ meson
($5350<m({{D}^{-}_{({s})}}{{D}^{0}})<6200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
as background. To increase the size of the background sample for the BDT
training, the charm meson invariant-mass intervals are increased from $\pm
25{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to $\pm
75{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and ‘wrong-sign’
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ candidates are also included.
Checks have been performed to verify that for all the variables used in the
BDT the simulated ${{B}^{-}}$ decays describe the observed signals in data
well, and that selections on the BDT output do not alter the shape of the
invariant-mass distribution of the combinatorial background.

The BDT combines all input variables into a single discriminant. The optimal
requirement on this value is determined by maximizing
$N_{S}/\sqrt{N_{S}+N_{B}}$, where $N_{S}$ is the expected signal yield,
determined from the initial signal yield in data multiplied by the BDT
efficiency from simulation, and $N_{B}$ is the background yield extrapolated
from the upper mass sideband to a $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ interval around the ${{B}^{-}}$
mass. This selection has an efficiency of 98% (90%) for
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}\ ({{D}^{-}}{{D}^{0}})$
decays, and a background rejection of 88% (93%).

##  4 Measurement of the raw asymmetries

After the event selection, the signal yields and the raw asymmetries are
determined by fitting a model of the invariant-mass distribution of
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$ candidates to the data.
The model includes components for the signal decays, a background from
${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{-}}{{D}^{0}}$ decays and a
combinatorial background.

The invariant-mass distribution of
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}$${D}^{0}$ decays is described by a
sum of two Crystal Ball (CB) [39] functions, with power-law tails proportional
to $[m({{D}^{-}_{({s})}}{{D}^{0}})-m({{{B}^{-}}})]^{-2}$ in opposite
directions, and with a common peak position. The tail parameters of the CB
functions, as well as the ratio of the widths of both CB components, are
obtained from simulation. The peak position of the ${{B}^{-}}$ signal and the
width of one of the CB functions are free parameters in the fits to the data.
This model provides a good description of the
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$ signals.

The Cabibbo-favoured
${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{-}}{{D}^{0}}$ decay is a
background to the ${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ channel,
despite being strongly suppressed by the invariant-mass requirement on the
${{K}^{-}}{{K}^{+}}{{\pi}^{-}}$ mass. This background is modelled by a single
Gaussian function, whose width is determined from a fit to simulated decays
and the yields determined from the ${D}^{-}_{s}$ sidebands. The yield of this
background is about 30 times smaller than that of the signal, and the shape of
the invariant-mass distribution is twice as wide. The combinatorial background
is described by an exponential function. Candidates originating from partially
reconstructed ${{{B}^{-}}}\\!\rightarrow{{D}^{*-}_{({s})}}{{D}^{0}}$ and
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{*0}}$ decays do not
contribute to the background since their reconstructed invariant mass is below
the lower limit of the fit region.

Separate unbinned extended maximum likelihood fits are used to describe the
invariant-mass distributions of candidates with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays and those with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ decays.
Figure 2 shows the fits to the invariant-mass distributions in the fit region,
$5230<m({{D}^{-}_{({s})}}{{D}^{0}})<5330{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
of the ${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ and
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ channels, separated by charge
and decay mode. The signal yields and corresponding raw asymmetries,
calculated according to Eq. 2, are listed in Table 1. No significant
dependence on the magnet polarity or data taking year is observed.
Inaccuracies in the modelling of the signal or background may result in a
small biases of the yields, but are not expected to introduce additional
asymmetries, therefore no systematic uncertainties are attributed to the
modelling of the signal and background shapes.

Figure 2: Invariant-mass distribution of ${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$ candidates, separated by charge. The top row plots are ${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ decays with ${{D}^{0}}\\!\rightarrow K^{-}\pi^{+}$, the second row with ${{D}^{0}}\\!\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$. The plots in the third row correspond to ${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ candidates with ${{D}^{0}}\\!\rightarrow K^{-}\pi^{+}$, the bottom row with ${{D}^{0}}\\!\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$. The left plots are ${{B}^{-}}$ candidates, the right plots ${{B}^{+}}$ candidates. The overlaid curves show the fits described in the text.  Table 1: Yields and raw asymmetries for ${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$ decays. Channel | $N({{{B}^{-}}})$ | $N({{{B}^{+}}})$ | $A_{\rm raw}$
---|---|---|---
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$, ${{D}^{0}}\\!\rightarrow K^{-}\pi^{+}$ | $13659\pm 129$ | $14209\pm 132$ | $(-2.0\pm 0.7)\%$
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$, ${{D}^{0}}\\!\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$ | $\phantom{0}7717\pm 103$ | $\phantom{0}7945\pm 104$ | $(-1.5\pm 0.9)\%$
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$, combined | $21375\pm 165$ | $22153\pm 168$ | $(-1.8\pm 0.5)\%$
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$, ${{D}^{0}}\\!\rightarrow K^{-}\pi^{+}$ | $\phantom{00}678\pm\phantom{0}32$ | $\phantom{00}660\pm\phantom{0}31$ | $(\phantom{-}1.3\pm 3.3)\%$
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$, ${{D}^{0}}\\!\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$ | $\phantom{00}369\pm\phantom{0}24$ | $\phantom{00}345\pm\phantom{0}24$ | $(\phantom{-}3.4\pm 4.7)\%$
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$, combined | $\phantom{0}1047\pm\phantom{0}40$ | $\phantom{0}1005\pm\phantom{0}39$ | $(\phantom{-}2.0\pm 2.7)\%$

##  5 Production and detection asymmetries

The production asymmetry between ${{B}^{-}}$ and ${{B}^{+}}$ mesons at LHCb
has been measured to be $A_{P}=(-0.5\pm 0.4)\%$ using the
${{{B}^{-}}}\\!\rightarrow{{D}^{0}}{{\pi}^{-}}$ decay [40], and no significant
dependence of $A_{P}$ on the transverse momentum or on the rapidity of the $B$
meson has been observed.

Four contributions to the asymmetry of the detection efficiencies are
considered: asymmetries in the tracking efficiency, the different $K^{\pm}$
interaction cross-sections with the detector material, and the trigger and
particle identification efficiencies.

The momentum-dependent tracking efficiency for pions has been determined by
comparing the yields of fully to partially reconstructed
${{D}^{*+}}\\!\rightarrow({{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
decays [41]. The corresponding asymmetries are summed for all final-state
tracks of simulated ${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$
events. After averaging over data-taking year and magnet polarity, the
tracking asymmetry is determined to be $(0.18\pm 0.07)\%$ for
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ and $(0.21\pm 0.07)\%$ for
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ decays, where the uncertainties
are due to the finite sample of ${D}^{*+}$ decays used for the tracking
efficiency measurement.

The interaction cross-section of ${K}^{-}$ mesons with matter is significantly
larger than that of ${K}^{+}$ mesons, resulting in a large asymmetry of the
charged kaon detection efficiency. The momentum-dependent difference in the
detection asymmetry between kaons and pions has been measured by comparing the
yield of ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ to the yield
of ${{D}^{+}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}$ decays [42]. These asymmetries, convoluted with the momentum
spectra of the final-state kaons, result in a contribution to the detection
asymmetry of $(-1.04\pm 0.16)\%$ for
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ decays, where the
uncertainty is due to the finite samples of ${D}^{+}$ decays. For
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ decays, this asymmetry cancels
to first order since it has one ${K}^{+}$ and one ${K}^{-}$ particle in the
final state, and the resulting asymmetry is $(0.02\pm 0.01)\%$.

The charge asymmetry of TIS candidates is independent of the signal decay
channel in consideration and has been measured in ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}\\!\rightarrow{{D}^{0}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$
decays [40]. After weighting by the TIS fraction, the asymmetry is found to be
0.04% and is neglected. A nonuniform response of the calorimeter may result in
a charge asymmetry of the TOS signal. Large samples of
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays have been used to
determine the $p_{\mathrm{T}}$-dependent trigger efficiencies and
corresponding charge asymmetries for both pions and kaons. After convoluting
these efficiencies with the simulated $p_{\mathrm{T}}$ spectra, averaging by
data-taking year and magnet polarity, and multiplying by the TOS fraction of
the signal, the resulting asymmetry is below 0.05%, and is considered to be
negligible.

In the candidate selection, particle identification criteria that rely on
information from the RICH detectors are used. Possible charge asymmetries in
the efficiencies of these selections are studied with samples of
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ that were selected without PID
requirements. Depending on assumptions on the correlation between the PID and
other variables in the multivariate selection, asymmetries smaller than 0.1%
are found. Therefore, no correction is applied, and a 0.1% uncertainty is
assigned.

The uncertainties of the contributions to the production and detection
asymmetry are considered to be uncorrelated and result in a value of
$A_{P}+A_{D}$ of $(-1.4\pm 0.5)\%$ for
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ and $(-0.3\pm 0.4)\%$ for
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ decays. Changes in the fit model
have a negligible effect on the measured asymmetry.

##  6 Results and conclusions

The $C\\!P$ asymmetries are determined by subtracting the production and
detection asymmetries from the measured raw asymmetry according to Eq. 3. The
obtained results are

${{\mathcal{A}}^{{C\\!P}}}({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}})=(-0.4\pm
0.5\pm 0.5)\%,$
${{\mathcal{A}}^{{C\\!P}}}({{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}})=(\phantom{-}2.3\pm
2.7\pm 0.4)\%,$

where the first uncertainties are statistical and the second systematic. The
measured value of
${{\mathcal{A}}^{{C\\!P}}}({{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}})$
provides constraints on the range of $C\\!P$ violation predicted for a new
physics model with $R$-parity violating supersymmetry [13].

In conclusion, the $C\\!P$ asymmetry in
${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{0}}$ decays has been measured for
the first time and the uncertainty on the $C\\!P$ asymmetry in
${{{B}^{-}}}\\!\rightarrow{{D}^{-}}{{D}^{0}}$ decays has been reduced by more
than a factor two with respect to previous measurements. No evidence for
$C\\!P$ violation in ${{{B}^{-}}}\\!\rightarrow{{D}^{-}_{({s})}}{{D}^{0}}$
decays has been found.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);
MinES and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU
(Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing
resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany),
INFN (Italy), SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom),
RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF
(Brazil), PL-GRID (Poland) and OSC (USA). We are indebted to the communities
behind the multiple open-source software packages on which we depend.
Individual groups or members have received support from AvH Foundation
(Germany), EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union),
ANR, Labex P2IO and OCEVU, and Région Auvergne-Rhône-Alpes (France), Key
Research Program of Frontier Sciences of CAS, CAS PIFI, and the Thousand
Talents Program (China), RFBR, RSF and Yandex LLC (Russia), GVA, XuntaGal and
GENCAT (Spain), Herchel Smith Fund, the Royal Society, the English-Speaking
Union and the Leverhulme Trust (United Kingdom).

## References

  * [1] N. Cabibbo, _Unitary symmetry and leptonic decays_ , Phys. Rev. Lett. 10 (1963) 531
  * [2] M. Kobayashi and T. Maskawa, _CP violation in the renormalizable theory of weak interaction_ , Prog. Theor. Phys. 49 (1973) 652
  * [3] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, _Evidence for the $2\pi$ decay of the $K_{2}^{0}$ meson_, Phys. Rev. Lett. 13 (1964) 138
  * [4] BaBar collaboration, B. Aubert et al., _Observation of CP violation in the $B^{0}$ meson system_, Phys. Rev. Lett. 87 (2001) 091801, arXiv:hep-ex/0107013
  * [5] Belle collaboration, K. Abe et al., _Observation of large CP violation in the neutral $B$ meson system_, Phys. Rev. Lett. 87 (2001) 091802, arXiv:hep-ex/0107061
  * [6] BaBar collaboration, B. Aubert et al., _Direct CP violating asymmetry in $B^{0}\rightarrow K^{+}\pi^{-}$ decays_, Phys. Rev. Lett. 93 (2004) 131801, arXiv:hep-ex/0407057
  * [7] Belle collaboration, Y. Chao et al., _Evidence for direct CP violation in $B^{0}\rightarrow K^{+}\pi^{-}$ decays_, Phys. Rev. Lett. 93 (2004) 191802, arXiv:hep-ex/0408100
  * [8] D. Sahoo et al., _Prediction of the CP asymmetry $C_{00}$ in $B^{0}\rightarrow{{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ decay_, JHEP 11 (2017) 087, arXiv:1709.08301
  * [9] L. Bel et al., _Anatomy of $B\rightarrow D\overline{D}$ decays_, JHEP 07 (2015) 108, arXiv:1505.01361
  * [10] R.-H. Li, X.-X. Wang, A. I. Sanda, and C.-D. Lu, _Decays of $B$ meson to two charmed mesons_, Phys. Rev. D81 (2010) 034006, arXiv:0910.1424
  * [11] H.-F. Fu, G.-L. Wang, Z.-H. Wang, and X.-J. Chen, _Semi-leptonic and non-leptonic B meson decays to charmed mesons_ , Chin. Phys. Lett. 28 (2011) 121301, arXiv:1202.1221
  * [12] L.-X. Lü, Z.-J. Xiao, S.-W. Wang, and W.-J. Li, _Double charm decays of $B$ mesons in the mSUGRA model_, Commun. Theor. Phys. 56 (2011) 125, arXiv:1008.4987
  * [13] C. S. Kim, R.-M. Wang, and Y.-D. Yang, _Studying double charm decays of $B_{u,d}$ and $B_{s}$ mesons in the MSSM with R-parity violation_, Phys. Rev. D79 (2009) 055004, arXiv:0812.4136
  * [14] Y.-G. Xu and R.-M. Wang, _Studying the fourth generation quark contributions to the double charm decays $B_{(s)}\rightarrow D_{(s)}^{(*)}D_{s}^{(*)}$_, Int. J. Theor. Phys. 55 (2016) 5290
  * [15] M. Jung and S. Schacht, _Standard model predictions and new physics sensitivity in $B\rightarrow DD$ decays_, Phys. Rev. D91 (2015) 034027, arXiv:1410.8396
  * [16] Belle collaboration, I. Adachi et al., _Measurement of the branching fraction and charge asymmetry of the decay ${{{B}^{+}}}\\!\rightarrow{{D}^{+}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ and search for ${{B}^{0}}\\!\rightarrow{{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$_, Phys. Rev. D77 (2008) 091101, arXiv:0802.2988
  * [17] BaBar collaboration, B. Aubert et al., _Measurement of branching fractions and CP-violating charge asymmetries for $B$-meson decays to ${D}^{(*)}{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)}$, and implications for the Cabibbo-Kobayashi-Maskawa angle $\gamma$_, Phys. Rev. D73 (2006) 112004, arXiv:hep-ex/0604037
  * [18] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [19] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [20] R. Aaij et al., _Performance of the LHCb Vertex Locator_ , JINST 9 (2014) P09007, arXiv:1405.7808
  * [21] R. Arink et al., _Performance of the LHCb Outer Tracker_ , JINST 9 (2014) P01002, arXiv:1311.3893
  * [22] M. Adinolfi et al., _Performance of the LHCb RICH detector at the LHC_, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [23] A. A. Alves Jr. et al., _Performance of the LHCb muon system_ , JINST 8 (2013) P02022, arXiv:1211.1346
  * [24] R. Aaij et al., _The LHCb trigger and its performance in 2011_, JINST 8 (2013) P04022, arXiv:1211.3055
  * [25] V. V. Gligorov and M. Williams, _Efficient, reliable and fast high-level triggering using a bonsai boosted decision tree_ , JINST 8 (2013) P02013, arXiv:1210.6861
  * [26] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [27] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [28] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [29] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [30] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [31] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [32] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [33] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [34] Particle Data Group, C. Patrignani et al., _Review of particle physics_ , Chin. Phys. C40 (2016) 100001, and 2017 update
  * [35] LHCb collaboration, R. Aaij et al., _Study of beauty hadron decays into pairs of charm hadrons_ , Phys. Rev. Lett. 112 (2014) 202001, arXiv:1403.3606
  * [36] W. D. Hulsbergen, _Decay chain fitting with a Kalman filter_ , Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
  * [37] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
  * [38] B. P. Roe et al., _Boosted decision trees as an alternative to artificial neural networks for particle identification_ , Nucl. Instrum. Meth. A543 (2005) 577, arXiv:physics/0408124
  * [39] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
  * [40] LHCb collaboration, R. Aaij et al., _Measurement of the $B^{\pm}$ production asymmetry and the $C\\!P$ asymmetry in $B^{\pm}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{\pm}$ decays_, Phys. Rev. D95 (2017) 052005, arXiv:1701.05501
  * [41] LHCb collaboration, R. Aaij et al., _Measurement of the ${{D}^{+}_{s}}$–${{D}^{-}_{s}}$ production asymmetry in $7$$~{}T\kern-0.50003pteV$ ${p}{p}$ collisions_, Phys. Lett. B713 (2012) 186, arXiv:1205.0897
  * [42] LHCb collaboration, R. Aaij et al., _Measurement of ${C\\!P}$ asymmetry in ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays_, JHEP 07 (2014) 041, arXiv:1405.2797

LHCb collaboration

R. Aaij43, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar59, P.
Albicocco19, J. Albrecht10, F. Alessio40, M. Alexander53, A. Alfonso Albero38,
S. Ali43, G. Alkhazov31, P. Alvarez Cartelle55, A.A. Alves Jr59, S. Amato2, S.
Amerio23, Y. Amhis7, L. An3, L. Anderlini18, G. Andreassi41, M. Andreotti17,g,
J.E. Andrews60, R.B. Appleby56, F. Archilli43, P. d’Argent12, J. Arnau Romeu6,
A. Artamonov37, M. Artuso61, E. Aslanides6, M. Atzeni42, G. Auriemma26, S.
Bachmann12, J.J. Back50, S. Baker55, V. Balagura7,b, W. Baldini17, A.
Baranov35, R.J. Barlow56, S. Barsuk7, W. Barter56, F. Baryshnikov32, V.
Batozskaya29, V. Battista41, A. Bay41, J. Beddow53, F. Bedeschi24, I.
Bediaga1, A. Beiter61, L.J. Bel43, N. Beliy63, V. Bellee41, N. Belloli21,i, K.
Belous37, I. Belyaev32,40, E. Ben-Haim8, G. Bencivenni19, S. Benson43, S.
Beranek9, A. Berezhnoy33, R. Bernet42, D. Berninghoff12, E. Bertholet8, A.
Bertolin23, C. Betancourt42, F. Betti15,40, M.O. Bettler49, M. van Beuzekom43,
Ia. Bezshyiko42, S. Bifani47, P. Billoir8, A. Birnkraut10, A. Bizzeti18,u, M.
Bjørn57, T. Blake50, F. Blanc41, S. Blusk61, V. Bocci26, O. Boente Garcia39,
T. Boettcher58, A. Bondar36,w, N. Bondar31, S. Borghi56,40, M. Borisyak35, M.
Borsato39,40, F. Bossu7, M. Boubdir9, T.J.V. Bowcock54, E. Bowen42, C.
Bozzi17,40, S. Braun12, M. Brodski40, J. Brodzicka27, D. Brundu16, E.
Buchanan48, C. Burr56, A. Bursche16, J. Buytaert40, W. Byczynski40, S.
Cadeddu16, H. Cai64, R. Calabrese17,g, R. Calladine47, M. Calvi21,i, M. Calvo
Gomez38,m, A. Camboni38,m, P. Campana19, D.H. Campora Perez40, L. Capriotti56,
A. Carbone15,e, G. Carboni25, R. Cardinale20,h, A. Cardini16, P. Carniti21,i,
L. Carson52, K. Carvalho Akiba2, G. Casse54, L. Cassina21, M. Cattaneo40, G.
Cavallero20,h, R. Cenci24,p, D. Chamont7, M.G. Chapman48, M. Charles8, Ph.
Charpentier40, G. Chatzikonstantinidis47, M. Chefdeville4, S. Chen16, S.-G.
Chitic40, V. Chobanova39, M. Chrzaszcz40, A. Chubykin31, P. Ciambrone19, X.
Cid Vidal39, G. Ciezarek40, P.E.L. Clarke52, M. Clemencic40, H.V. Cliff49, J.
Closier40, V. Coco40, J. Cogan6, E. Cogneras5, V. Cogoni16,f, L. Cojocariu30,
P. Collins40, T. Colombo40, A. Comerma-Montells12, A. Contu16, G. Coombs40, S.
Coquereau38, G. Corti40, M. Corvo17,g, C.M. Costa Sobral50, B. Couturier40,
G.A. Cowan52, D.C. Craik58, A. Crocombe50, M. Cruz Torres1, R. Currie52, C.
D’Ambrosio40, F. Da Cunha Marinho2, C.L. Da Silva73, E. Dall’Occo43, J.
Dalseno48, A. Danilina32, A. Davis3, O. De Aguiar Francisco40, K. De Bruyn40,
S. De Capua56, M. De Cian41, J.M. De Miranda1, L. De Paula2, M. De Serio14,d,
P. De Simone19, C.T. Dean53, D. Decamp4, L. Del Buono8, B. Delaney49, H.-P.
Dembinski11, M. Demmer10, A. Dendek28, D. Derkach35, O. Deschamps5, F.
Dettori54, B. Dey65, A. Di Canto40, P. Di Nezza19, S. Didenko69, H.
Dijkstra40, F. Dordei40, M. Dorigo40, A. Dosil Suárez39, L. Douglas53, A.
Dovbnya45, K. Dreimanis54, L. Dufour43, G. Dujany8, P. Durante40, J.M.
Durham73, D. Dutta56, R. Dzhelyadin37, M. Dziewiecki12, A. Dziurda40, A.
Dzyuba31, S. Easo51, U. Egede55, V. Egorychev32, S. Eidelman36,w, S.
Eisenhardt52, U. Eitschberger10, R. Ekelhof10, L. Eklund53, S. Ely61, A.
Ene30, S. Escher9, S. Esen12, H.M. Evans49, T. Evans57, A. Falabella15, N.
Farley47, S. Farry54, D. Fazzini21,40,i, L. Federici25, G. Fernandez38, P.
Fernandez Declara40, A. Fernandez Prieto39, F. Ferrari15, L. Ferreira Lopes41,
F. Ferreira Rodrigues2, M. Ferro-Luzzi40, S. Filippov34, R.A. Fini14, M.
Fiorini17,g, M. Firlej28, C. Fitzpatrick41, T. Fiutowski28, F. Fleuret7,b, M.
Fontana16,40, F. Fontanelli20,h, R. Forty40, V. Franco Lima54, M. Frank40, C.
Frei40, J. Fu22,q, W. Funk40, C. Färber40, E. Gabriel52, A. Gallas Torreira39,
D. Galli15,e, S. Gallorini23, S. Gambetta52, M. Gandelman2, P. Gandini22, Y.
Gao3, L.M. Garcia Martin71, B. Garcia Plana39, J. García Pardiñas42, J. Garra
Tico49, L. Garrido38, D. Gascon38, C. Gaspar40, L. Gavardi10, G. Gazzoni5, D.
Gerick12, E. Gersabeck56, M. Gersabeck56, T. Gershon50, Ph. Ghez4, S. Gianì41,
V. Gibson49, O.G. Girard41, L. Giubega30, K. Gizdov52, V.V. Gligorov8, D.
Golubkov32, A. Golutvin55,69, A. Gomes1,a, I.V. Gorelov33, C. Gotti21,i, E.
Govorkova43, J.P. Grabowski12, R. Graciani Diaz38, L.A. Granado Cardoso40, E.
Graugés38, E. Graverini42, G. Graziani18, A. Grecu30, R. Greim43, P.
Griffith16, L. Grillo56, L. Gruber40, B.R. Gruberg Cazon57, O. Grünberg67, E.
Gushchin34, Yu. Guz37,40, T. Gys40, C. Göbel62, T. Hadavizadeh57, C.
Hadjivasiliou5, G. Haefeli41, C. Haen40, S.C. Haines49, B. Hamilton60, X.
Han12, T.H. Hancock57, S. Hansmann-Menzemer12, N. Harnew57, S.T. Harnew48, C.
Hasse40, M. Hatch40, J. He63, M. Hecker55, K. Heinicke10, A. Heister9, K.
Hennessy54, L. Henry71, E. van Herwijnen40, M. Heß67, A. Hicheur2, D. Hill57,
P.H. Hopchev41, W. Hu65, W. Huang63, Z.C. Huard59, W. Hulsbergen43, T.
Humair55, M. Hushchyn35, D. Hutchcroft54, P. Ibis10, M. Idzik28, P. Ilten47,
K. Ivshin31, R. Jacobsson40, J. Jalocha57, E. Jans43, A. Jawahery60, F.
Jiang3, M. John57, D. Johnson40, C.R. Jones49, C. Joram40, B. Jost40, N.
Jurik57, S. Kandybei45, M. Karacson40, J.M. Kariuki48, S. Karodia53, N.
Kazeev35, M. Kecke12, F. Keizer49, M. Kelsey61, M. Kenzie49, T. Ketel44, E.
Khairullin35, B. Khanji12, C. Khurewathanakul41, K.E. Kim61, T. Kirn9, S.
Klaver19, K. Klimaszewski29, T. Klimkovich11, S. Koliiev46, M. Kolpin12, R.
Kopecna12, P. Koppenburg43, S. Kotriakhova31, M. Kozeiha5, L. Kravchuk34, M.
Kreps50, F. Kress55, P. Krokovny36,w, W. Krupa28, W. Krzemien29, W.
Kucewicz27,l, M. Kucharczyk27, V. Kudryavtsev36,w, A.K. Kuonen41, T.
Kvaratskheliya32,40, D. Lacarrere40, G. Lafferty56, A. Lai16, G. Lanfranchi19,
C. Langenbruch9, T. Latham50, C. Lazzeroni47, R. Le Gac6, A. Leflat33,40, J.
Lefrançois7, R. Lefèvre5, F. Lemaitre40, P. Lenisa17, O. Leroy6, T. Lesiak27,
B. Leverington12, P.-R. Li63, T. Li3, Z. Li61, X. Liang61, T. Likhomanenko68,
R. Lindner40, F. Lionetto42, V. Lisovskyi7, X. Liu3, D. Loh50, A. Loi16, I.
Longstaff53, J.H. Lopes2, D. Lucchesi23,o, M. Lucio Martinez39, A. Lupato23,
E. Luppi17,g, O. Lupton40, A. Lusiani24, X. Lyu63, F. Machefert7, F. Maciuc30,
V. Macko41, P. Mackowiak10, S. Maddrell-Mander48, O. Maev31,40, K. Maguire56,
D. Maisuzenko31, M.W. Majewski28, S. Malde57, B. Malecki27, A. Malinin68, T.
Maltsev36,w, G. Manca16,f, G. Mancinelli6, D. Marangotto22,q, J. Maratas5,v,
J.F. Marchand4, U. Marconi15, C. Marin Benito38, M. Marinangeli41, P.
Marino41, J. Marks12, G. Martellotti26, M. Martin6, M. Martinelli41, D.
Martinez Santos39, F. Martinez Vidal71, A. Massafferri1, R. Matev40, A.
Mathad50, Z. Mathe40, C. Matteuzzi21, A. Mauri42, E. Maurice7,b, B. Maurin41,
A. Mazurov47, M. McCann55,40, A. McNab56, R. McNulty13, J.V. Mead54, B.
Meadows59, C. Meaux6, F. Meier10, N. Meinert67, D. Melnychuk29, M. Merk43, A.
Merli22,q, E. Michielin23, D.A. Milanes66, E. Millard50, M.-N. Minard4, L.
Minzoni17, D.S. Mitzel12, A. Mogini8, J. Molina Rodriguez1,y, T. Mombächer10,
I.A. Monroy66, S. Monteil5, M. Morandin23, G. Morello19, M.J. Morello24,t, O.
Morgunova68, J. Moron28, A.B. Morris6, R. Mountain61, F. Muheim52, M.
Mulder43, D. Müller40, J. Müller10, K. Müller42, V. Müller10, P. Naik48, T.
Nakada41, R. Nandakumar51, A. Nandi57, I. Nasteva2, M. Needham52, N. Neri22,
S. Neubert12, N. Neufeld40, M. Neuner12, T.D. Nguyen41, C. Nguyen-Mau41,n, S.
Nieswand9, R. Niet10, N. Nikitin33, A. Nogay68, D.P. O’Hanlon15, A.
Oblakowska-Mucha28, V. Obraztsov37, S. Ogilvy19, R. Oldeman16,f, C.J.G.
Onderwater72, A. Ossowska27, J.M. Otalora Goicochea2, P. Owen42, A.
Oyanguren71, P.R. Pais41, A. Palano14, M. Palutan19,40, G. Panshin70, A.
Papanestis51, M. Pappagallo52, L.L. Pappalardo17,g, W. Parker60, C. Parkes56,
G. Passaleva18,40, A. Pastore14, M. Patel55, C. Patrignani15,e, A. Pearce40,
A. Pellegrino43, G. Penso26, M. Pepe Altarelli40, S. Perazzini40, D.
Pereima32, P. Perret5, L. Pescatore41, K. Petridis48, A. Petrolini20,h, A.
Petrov68, M. Petruzzo22,q, B. Pietrzyk4, G. Pietrzyk41, M. Pikies27, D.
Pinci26, F. Pisani40, A. Pistone20,h, A. Piucci12, V. Placinta30, S.
Playfer52, M. Plo Casasus39, F. Polci8, M. Poli Lener19, A. Poluektov50, N.
Polukhina69, I. Polyakov61, E. Polycarpo2, G.J. Pomery48, S. Ponce40, A.
Popov37, D. Popov11,40, S. Poslavskii37, C. Potterat2, E. Price48, J.
Prisciandaro39, C. Prouve48, V. Pugatch46, A. Puig Navarro42, H. Pullen57, G.
Punzi24,p, W. Qian63, J. Qin63, R. Quagliani8, B. Quintana5, B. Rachwal28,
J.H. Rademacker48, M. Rama24, M. Ramos Pernas39, M.S. Rangel2, F.
Ratnikov35,x, G. Raven44, M. Ravonel Salzgeber40, M. Reboud4, F. Redi41, S.
Reichert10, A.C. dos Reis1, C. Remon Alepuz71, V. Renaudin7, S. Ricciardi51,
S. Richards48, K. Rinnert54, P. Robbe7, A. Robert8, A.B. Rodrigues41, E.
Rodrigues59, J.A. Rodriguez Lopez66, A. Rogozhnikov35, S. Roiser40, A.
Rollings57, V. Romanovskiy37, A. Romero Vidal39,40, M. Rotondo19, M.S.
Rudolph61, T. Ruf40, J. Ruiz Vidal71, J.J. Saborido Silva39, N. Sagidova31, B.
Saitta16,f, V. Salustino Guimaraes62, C. Sanchez Mayordomo71, B. Sanmartin
Sedes39, R. Santacesaria26, C. Santamarina Rios39, M. Santimaria19, E.
Santovetti25,j, G. Sarpis56, A. Sarti19,k, C. Satriano26,s, A. Satta25, D.
Savrina32,33, S. Schael9, M. Schellenberg10, M. Schiller53, H. Schindler40, M.
Schmelling11, T. Schmelzer10, B. Schmidt40, O. Schneider41, A. Schopper40,
H.F. Schreiner59, M. Schubiger41, M.H. Schune7,40, R. Schwemmer40, B.
Sciascia19, A. Sciubba26,k, A. Semennikov32, E.S. Sepulveda8, A. Sergi47,40,
N. Serra42, J. Serrano6, L. Sestini23, P. Seyfert40, M. Shapkin37, Y.
Shcheglov31,†, T. Shears54, L. Shekhtman36,w, V. Shevchenko68, B.G. Siddi17,
R. Silva Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S. Simone14,d, N.
Skidmore12, T. Skwarnicki61, I.T. Smith52, M. Smith55, l. Soares Lavra1, M.D.
Sokoloff59, F.J.P. Soler53, B. Souza De Paula2, B. Spaan10, P. Spradlin53, F.
Stagni40, M. Stahl12, S. Stahl40, P. Stefko41, S. Stefkova55, O. Steinkamp42,
S. Stemmle12, O. Stenyakin37, M. Stepanova31, H. Stevens10, S. Stone61, B.
Storaci42, S. Stracka24,p, M.E. Stramaglia41, M. Straticiuc30, U. Straumann42,
S. Strokov70, J. Sun3, L. Sun64, K. Swientek28, V. Syropoulos44, T. Szumlak28,
M. Szymanski63, S. T’Jampens4, Z. Tang3, A. Tayduganov6, T. Tekampe10, G.
Tellarini17, F. Teubert40, E. Thomas40, J. van Tilburg43, M.J. Tilley55, V.
Tisserand5, M. Tobin41, S. Tolk40, L. Tomassetti17,g, D. Tonelli24, R.
Tourinho Jadallah Aoude1, E. Tournefier4, M. Traill53, M.T. Tran41, M.
Tresch42, A. Trisovic49, A. Tsaregorodtsev6, A. Tully49, N. Tuning43,40, A.
Ukleja29, A. Usachov7, A. Ustyuzhanin35, U. Uwer12, C. Vacca16,f, A. Vagner70,
V. Vagnoni15, A. Valassi40, S. Valat40, G. Valenti15, R. Vazquez Gomez40, P.
Vazquez Regueiro39, S. Vecchi17, M. van Veghel43, J.J. Velthuis48, M.
Veltri18,r, G. Veneziano57, A. Venkateswaran61, T.A. Verlage9, M. Vernet5, M.
Vesterinen57, J.V. Viana Barbosa40, D. Vieira63, M. Vieites Diaz39, H.
Viemann67, X. Vilasis-Cardona38,m, A. Vitkovskiy43, M. Vitti49, V. Volkov33,
A. Vollhardt42, B. Voneki40, A. Vorobyev31, V. Vorobyev36,w, C. Voß9, J.A. de
Vries43, C. Vázquez Sierra43, R. Waldi67, J. Walsh24, J. Wang61, M. Wang3, Y.
Wang65, Z. Wang42, D.R. Ward49, H.M. Wark54, N.K. Watson47, D. Websdale55, A.
Weiden42, C. Weisser58, M. Whitehead9, J. Wicht50, G. Wilkinson57, M.
Wilkinson61, M.R.J. Williams56, M. Williams58, T. Williams47, F.F.
Wilson51,40, J. Wimberley60, M. Winn7, J. Wishahi10, W. Wislicki29, M.
Witek27, G. Wormser7, S.A. Wotton49, K. Wyllie40, D. Xiao65, Y. Xie65, A. Xu3,
M. Xu65, Q. Xu63, Z. Xu3, Z. Xu4, Z. Yang3, Z. Yang60, Y. Yao61, H. Yin65, J.
Yu65, X. Yuan61, O. Yushchenko37, K.A. Zarebski47, M. Zavertyaev11,c, L.
Zhang3, Y. Zhang7, A. Zhelezov12, Y. Zheng63, X. Zhu3, V. Zhukov9,33, J.B.
Zonneveld52, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
7LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Universita e INFN, Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69National University of Science and Technology MISIS, Moscow, Russia,
associated to 32
70National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
32
71Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 38
72Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43
73Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 61

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vIligan Institute of Technology (IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xNational Research University Higher School of Economics, Moscow, Russia
yEscuela Agrícola Panamericana, San Antonio de Oriente, Honduras
†Deceased


# The Long Night: Modeling the Climate of Westeros

Adiv Paradise Department of Astronomy & Astrophysics ,University of Toronto
50 St. George Street, Toronto, ON, M5S 3H4, Canada  Centre for Planetary
Sciences, University of Toronto, Scarborough
1265 Military Trail, Toronto, ON, M1C 1A4, Canada  Alysa Obertas  Department
of Astronomy & Astrophysics ,University of Toronto
50 St. George Street, Toronto, ON, M5S 3H4, Canada  Centre for Planetary
Sciences, University of Toronto, Scarborough
1265 Military Trail, Toronto, ON, M1C 1A4, Canada  Canadian Institute for
Theoretical Astrophysics, University of Toronto
60 St. George Street, Toronto, ON, M5S 3H8, Canada  Anna O’Grady Department of
Astronomy & Astrophysics ,University of Toronto
50 St. George Street, Toronto, ON, M5S 3H4, Canada  Dunlap Institute for
Astronomy & Astrophysics, University of Toronto
50 St. George Street, Toronto, ON, M5S 3H4, Canada  Matthew Young Department
of Astronomy & Astrophysics ,University of Toronto
50 St. George Street, Toronto, ON, M5S 3H4, Canada  Dunlap Institute for
Astronomy & Astrophysics, University of Toronto
50 St. George Street, Toronto, ON, M5S 3H4, Canada

(Received April 1, 2019)

###### Abstract

Many previous authors have attempted to find explanations for Westeros’s
climate, characterized by a generally moderate, Earth-like climate punctuated
by extremely long and cold winters, separated by thousands of years. One
explanation that has been proposed is that the planet orbits in a Sitnikov
configuration, where two equal-mass stars (or a star and a black hole) orbit
each other on slightly eccentric orbits, and the planet moves along a line
through the barycenter perpendicular to the primaries’ orbital plane
(Freistetter & Grützbauch, 2018). We modify an intermediate-complexity GCM to
include the effects of such an orbit and integrate it for thousands of years
to determine whether such an orbit can a) be habitable and b) explain the
climatic variations observed by the inhabitants of Westeros, in both double-
star and star-black hole configurations. While configurations with low primary
eccentricity and initial conditions that permit only small excursions from the
ecliptic plane are habitable, these orbits are too stable to explain Westerosi
climate. We find that while orbits with more bounded chaos are able to produce
rare anomalously long and cold winters similar to Westeros’s Long Night, huge
variations in incident stellar flux on normal orbital timescales should render
these planets uninhabitable. We note that the presence of an orbital
megastructure, either around the planet or the barycenter, could block some of
the sunlight during crossings of the primaries’ orbital plane and preserve
Westeros’s habitability. While we find that bounded chaotic Sitnikov orbits
are a viable explanation for Westeros’s Long Night, we propose that chaotic
variations of the planet’s axial tilt or semimajor axis, potentially due to
torques from nearby planets or stars, may be a more realistic explanation than
Sitnikov orbits.

habitability – climate – exoplanets – chaos – orbital dynamics – alien
megastructures

††journal: Acta Prima Aprilia

##  1 Introduction

A major concern among Westerosi people, especially those living at high
latitudes, is the belief that Winter is Coming, at some unknown time in the
future (Martin, 1996). Here we distinguish Westerosi Winters (with a capital
‘W’) from winters (with a lowercase ‘w’), such that ‘winter’ refers to normal
decreases in temperatures experienced in Westeros frequently, every few months
to years, and ‘Winter’ refers to the Long Night, a catastrophic life-
threatening event characterized by cold lasting a generation, kings freezing
on their thrones, and White Walkers advancing and threatening the Kingdoms of
Men (Martin, 1996).

Westerosi Winters are believed unpredictable by the maesters and pose a severe
threat to health and safety in Westeros. This makes understanding their cause
a major priority. Discovering their geophysical cause may even lead to the
ability to predict when the next Winter will occur, allowing the Seven
Kingdoms ample time to set aside their squabbles and prepare food stores and
armies to face the coming Winter. Understanding the climate of Westeros also
provides a case study for a habitable climate whose time-evolution and orbital
environment is very different from Earth’s.

A great deal of work has already gone into explaining the climate of Westeros.
Kostov et al. (2013) proposed that Westeros could inhabit a P-type
circumbinary system, where the planet orbits both stars in a binary, noting
that the orbit could vary chaotically, leading to variable and unpredictable
lengths of Westerosi winters. Freistetter & Grützbauch (2018) however noted
that there is no evidence of multiple suns in Westerosi skies, and that the
variations in a P-type circumbinary orbit are too small and too regular to
explain the severity, duration, and infrequency of the Long Night, instances
of which can be separated by as many as 8,000 years (Martin, 1996). They
proposed that Sitnikov orbits, which can vary much more chaotically, are
instead a more plausible explanation, noting that the single-sun problem can
be resolved by having one of the primaries be a black hole. Freistetter &
Grützbauch (2018) however did not actually calculate how the amount of
incident sunlight, or insolation, would vary throughout a Sitnikov orbit, or
make any attempt to actually model the climate’s response to such an orbit.

Sitnikov orbits, a special case of the restricted three-body problem, are
characterized by an eccentric stellar binary with the planet orbiting on a
straight line through the system’s barycenter, such that the axis of motion is
perpendicular to the primaries’ semimajor axis (Sitnikov, 1961). The duration
of excursions away from the plane is chaotic, such that there exist initial
conditions which lead to any integer sequence of primary orbits between
ecliptic crossings, though finding those initial conditions may be difficult
(Moser, 1973). Sitnikov orbits can be stable (with small chaotic variations),
bounded but chaotic (with larger variations), and unbounded and chaotic (where
the planet is eventually ejected) (Kovács & Érdi, 2007). Some orbits in
particular can be “sticky”, such that a planet appears stable for a large
number of orbits before becoming unbound (Dvorak et al., 1998). We note that
the impact of three-body chaos on planetary climate is habitability is also
explored in considerable depth in Liu (2014).

While our work focuses on equal-mass binaries, approximates the planet as a
massless particle, and restricts the planet to orbiting on the perpendicular
axis, other studies have shown that these properties of Sitnikov orbits also
apply to systems with unequal-mass binaries (Perdios & Markellos, 1987),
planets with mass and influence on the parent binary (Dvorak & Sui Sun, 1997),
and systems where the planet’s orbital motion is not exactly on the
perpendicular axis (Perdios & Kalantonis, 2012).

Examining astrophysical explanations for Westeros’s unusual climate is
interesting from an exoplanet science point of view. To date, a huge number of
exoplanets has been discovered (Mullally, 2017). Within this sample we find an
enormous diversity of planets, many of which defy easy explanations. The very
first exoplanet ever discovered, PSR B1620-26b, belongs to a binary system
with a pulsar and white dwarf (Backer, 1993). The second exoplanet ever
discovered, 51 Pegasi b, is a hot Jupiter, orbiting much closer to its star
than Mercury orbits our own Sun (Mayor & Queloz, 1995). While it is not clear
how a Sitnikov system could form, the large number of planets discovered and
expected to be discovered makes it worthwhile to explore the fringes of the
exoplanet parameter space.

Another motivation for studying this problem is the similarity between the
Long Night and snowball climates, where the oceans are completely covered by
sea ice (Pierrehumbert, 2005; Abbot, 2014; Hoffman et al., 2017). These
climates are stable at the same greenhouse gas concentrations and insolations
as temperate climates, and the two states are separated by sharp transitions
to a runaway ice-albedo feedback (Budyko, 1969; Sellers, 1969; Deser et al.,
2000). This has happened at least twice on Earth (Hoffman & Schrag, 2002;
Condon et al., 2002; Tajika, 2007), and is expected to happen on at least some
Earth-like exoplanets (Menou, 2015; Abbot, 2016; Paradise & Menou, 2017). It
is not expected to be very common however on tidally-locked planets (Checlair
et al., 2017). Westeros therefore represents an opportunity to investigate
another regime in which snowball glaciations may be possible, and orbital
chaos may represent another mechanism for triggering them.

##  2 Methods

###  2.1 Sitnikov Orbit

We assume the orbits of the primaries are Keplerian and unaffected by the
planet. The equation of motion for a massless particle on a Sitnikov orbit is
given by Sitnikov (1961):

$\ddot{z}(t)=-\frac{2GMz(t)}{\left(r^{2}(t)+z^{2}(t)\right)^{3/2}}$ (1)

where $G$ is the gravitational constant and $M$ is the combined mass of the
primaries. The distance of the primary from the barycenter, $r(t)$, can be
easily calculated through

$r(t)=a\frac{1-e^{2}}{1+e\cos\nu(t)}$ (2)

where $e$ is the primary’s eccentricity, $a$ is the primary’s semimajor axis,
and $\nu$ is the primary’s true anomaly. We calculate $\nu$ from the primary’s
mean anomaly and eccentric anomaly, the latter of which is calculated with a
Newton-Raphson iterator, run until the result is converged to within machine
error ($<10^{-15}$). We integrate the orbit of the planet using a 4th-order
Yoshida integrator (Yoshida, 1990). We integrated several configurations for
several thousand orbits with both a Yoshida integrator and a leapfrog
integrator, and found negligible differences in most cases, and in cases where
the planet was ejected, the main difference was the direction in which it was
ejected. We therefore conclude that with more than $10^{4}$ timesteps per
orbit, a 4th-order Yoshida integrator has sufficient accuracy for our
purposes.

###  2.2 Planetary Rotation

We assume two possible geometries for the planet’s rotation relative to the
direction of motion, which we term the spinning-top and tennis-ball
configurations, shown in Figure 1. In the spinning-top configuration, the
planet’s rotation axis is aligned with the direction of motion (similar to a
spinning top gently levitating above a magnetic base), such that day and night
progress from East to West, and only the declination of the primaries changes
as the planet moves throughout its orbit. This is analogous to having 90∘
obliquity in a Keplerian orbit. In the tennis-ball configuration, the planet’s
rotation axis is perpendicular to the direction of motion (the semimajor
axis). This is analogous to the low obliquity of most Solar System planets,
and is similar to the rotation of a tennis ball tossed in the air.

We choose to orient the rotation axis of the tennis-ball configuration
perpendicular to both the planet’s semimajor axis and the primaries’ semimajor
axis, such that in the case of primaries on eccentric orbits, day and night
progress longitudinally when the planet is above or below the plane, and the
primaries are at low declinations at apoapse, where they spend most of their
time. The formulae for the stellar angles in each configuration are given in
Appendix A

Figure 1: Diagrams of the Sitnikov orbital configuration with the two rotation
states we consider. The spinning-top configuration (left) has the planet’s
rotation axis parallel to the axis of motion, while the tennis-ball
configuration (right) has the planet’s rotation axis perpendicular to both its
own semimajor axis and the primaries’ semimajor axis.

###  2.3 Climate Model Configuration

To model the climate of Westeros, we use PlaSim, an intermediate-complexity 3D
global climate model described in Fraedrich et al. (2005). PlaSim solves a set
of primitive equations on a discretized grid using a spectral transform
method. PlaSim represents the ocean as a 50-meter mixed slab without
horizontal advection (as in, heat is transported only through diffusion),
models sea ice thermodynamically (ice forms when it is cold enough to do so),
and couples the atmosphere to the land and ocean through latent and sensible
heat transport (shallow and deep convection and precipitation/evaporation),
drag, and radiation. PlaSim incorporates a 3-band radiation model, with two
shortwave bands and one longwave band. Absorption is by water, CO2, and ozone,
while scattering is computed based on the total mass of the atmosphere and
cloud optical depth. Clouds are computed diagnostically based on critical
relative humidities for each vertical layer, and precipitation includes both
convective and large-scale precipitation, and models all three phases of
water—precipitation can fall as rain or snow, and water can be re-evaporated
out of precipitation before it reaches the ground (Fraedrich et al., 2005).

PlaSim has been used to study a number of Earth-like climates, including
snowball climates, where sea ice extends to the equator (e.g. Lucarini et al.,
2010; Boschi et al., 2013; Paradise & Menou, 2017), high-obliquity climates
(Linsenmeier et al., 2015), tidally-locked planets (Checlair et al., 2017;
Abbot et al., 2018), and planets undergoing a moist greenhouse (Gomez-Leal et
al., 2019). To capture the unique climate dynamics of a planet on a Sitnikov
orbit, we introduce modifications to PlaSim to allow for chaotically-evolving
incident stellar flux and multiple suns in the sky. We integrate the positions
of the primaries and the planet each timestep, and then compute the solar
zenith angle independently for each primary. Because some of the scattering
and absorption parameterizations depend on the angle of sunlight propagating
through the atmosphere, we compute shortwave radiative transfer independently
for each primary, and then combine the fluxes from each at the layer
boundaries (Paradise et al., 2019).

We run PlaSim in its T21 resolution, corresponding to 32 latitudes and 64
longitudes ($5.6^{\circ}\times 5.6^{\circ}$ at the equator), with 10 vertical
levels spaced mostly-linearly in pressure. We run our Yoshida integrator for
104 years for several hundred different initial conditions, with varying
initial mean anomalies, eccentricities, and initial $z$, and then from the
output select 2 initial conditions that yield bounded chaotic orbits, with
infrequent extremely cold and long winters. We test whether a spinning-top
configuration (see subsection 2.2) can be habitable at the initial elevations
necessary for significant chaotic variations by using generic initial
conditions ($e=0.1$, initial mean anomaly $M=0.8$) at several initial
elevations, ranging from 0.3 to 0.8 AU. Because the spinning-top configuration
at initial elevations much beyond this range plunge the antistellar hemisphere
into darkness for a significant portion of the orbit, while delivering
weakened light to the near hemisphere, the spinning-top configuration plus a
strongly-chaotic orbit results in model-crashing cold temperatures during each
orbit. We cannot rule out the role of White Walkers on such planets, but to be
safe we use the tennis-ball configuration for the simulations with chaotic
orbits. Because the chaotic configurations lead to increases in stellar flux
at ecliptic-crossing of a factor of 5–6 relative to maximum excursion above
the plane, in some models we place an upper limit on the flux that the planet
can receive, choosing 1400 W/m2. We run each simulation for 2000 years, with a
timestep of 45 minutes.

This approach of first only integrating the orbits to find promising initial
conditions and then re-integrating during the climate simulation itself turned
out to be a mistake, since the chaotic nature of Sitnikov orbits and the
finite precision of machine arithmetic meant that the final orbits differed
often significantly from the initial calculations, with 2 of our 4 2000-year
simulations being overcome by White Walkers and not emerging from the Long
Night within the timescale of the simulation (may Jon Snow rest in peace). Due
to time constraints, we were unfortunately not able to run more simulations to
rectify this problem, but note that a more intelligent approach would have
been to pre-calculate the orbits to find promising initial conditions, and
then feed the calculated orbit into PlaSim as a prescribed time series,
interpolating between the larger timesteps of the initial calculation rather
than re-integrating everything. A hybrid method could also be possible, where
the prescribed timesteps, of which there may be one per day or fewer, are used
to ‘guide’ the integration of the non-prescribed timesteps.

Figure 2: Land-Sea mask for our Westeros-like model setup, shown here in a
Mollweide projection (such that the entire planet is shown). Oceans are shown
as blue/dark, while land is beige/light. Note that Sothoryos, Ultos, and the
Sunset Sea are completely speculative, as are latitudes of various locations.

Our Westeros planet is mostly Earth-like, with a surface atmospheric pressure
of 1 atm, 360 $\mu$bars of CO2, and a rotation rate of 24 hours. Due to the
absence of modern meteorology equipment among the Maesters of Westeros, when
it comes to the true surface pressure and pCO2, we know about as much as Jon
Snow, which is to say, nothing (Martin, 1996). For simplicity, here we assume
them to be Earth-like. In the name of making the model as realistic as
possible, however, we attempt to replicate the known land distribution of
Westeros, shown in Figure 2. This land distribution is based on published maps
of the known world (e.g Martin, 2012; Whitehead, 2016; u/selvag, 2018), but we
note that because the world is not fully explored (Vineyard, 2014), the
southern hemisphere and Sunset Sea are purely speculative. We note that we
assume that the North beyond the Wall extends to the north pole, such that the
Arctic pole is land, rather than ocean. We do not however expect large
differences between our map and reality to have a large effect on the model’s
outcomes.

##  3 Results

Figure 3: Global mean surface temperature averaged over 25-year intervals, and
monthly sea ice extent for planets in spinning-top configuration (rotation
axis aligned with the direction of motion), for initial elevations above the
parent stars’ orbital plane ranging from 0.3 AU to 0.8 AU, in 0.5 AU
increments. Each planet that goes more than 0.3 AU or so away from the
ecliptic plane enters a snowball state, and for our purposes is considered not
habitable. Note that 0.3 AU is not a sufficiently large elevation to result in
significantly chaotic climates, as seen in the monthly sea ice.

Assuming a primary insolation of 0.95 L⊙, orbits with initial elevations
greater than approximately 0.3 AU result in orbital variations in insolation
of more than 600 W/m2. At 0.3 AU elevation and a primary eccentricity of 0.1,
planets in the spinning-top configuration experience total darkness for
latitudes above 15-18∘ at maximum excursion. Lower latitudes experience
sunlight reduced by $\cos(\psi-\delta)$, where $\psi$ is the latitude and
$\delta$ is the declination of the primary relative to the planet’s equatorial
plane. For example, locations at 45∘ latitude on the far hemisphere experience
a further reduction in sunlight intensity of 45-50%, for a minimum insolation
of 450 W/m2. This is comparable to the intensity of sunlight at similar
latitudes during Earth winter. An initial elevation of 0.6 AU, however,
results in a minimum insolation of 233 W/m2 at the same latitude. We tested
initial elevations ranging from 0.3 to 0.8 AU in 0.5 AU increments, with a
0.95 L⊙ primary and equal-mass black hole, and as shown in Figure 3, found
that all but 0.3 AU resulted in snowball planets, confirming our hypothesis
that only planets with initial elevations around 0.3 AU or smaller can be
habitable in spinning-top configurations, ruling out this rotation geometry
for the chaotic orbits possible at larger initial elevations.

Figure 4: Monthly average temperatures averaged across the planet, and at
Winterfell and King’s Landing, for each of the four 2,000-year simulations.
Note that machine precision can result in significant orbital deviations
between two simulations with the same initial conditions. In two of our
simulations, we do observe the planet both entering and exiting a Long Night,
although the change that triggers this in one appears to be smaller than the
truly brutal excursions seen in some of the other models, in which decades can
pass between crossings.

With the planet in a tennis-ball spin configuration and initial elevations
large enough for chaotic orbits, we find no luminosities for which the planet
remains habitable, either due to plunging into model-crashing cold severe
enough to chill a White Walker, or due to ‘pulses’ of very high incident
sunlight as the planet passes through the primaries’ orbital plane. When the
primaries are near periapse during this conjunction, incident light is
maximized, and in each case where the stellar luminosity is high enough to
avoid lethal cold during excursions away from the plane, surface temperatures
reach an excess of 100 ∘C during ecliptic crossing.

Figure 5: Temperature averaged on generation (25-year) baselines, compared to
the daily insolation received by the planet for each of the two orbits which
produce a Long Night. We note that the Long Night in Orbit #1 appears to be a
snowball episode triggered by a relatively small decrease in insolation, while
Orbit #2 is caused by large excursions to lower insolations.

When we artificially place an upper bound on incident flux, we find that we
are able to maintain mostly-habitable conditions with tennis-ball
configurations at initial elevations high enough to cause significant chaotic
variations in excursion distance and duration. We ran four simulations in this
configuration, for two different sets of initial conditions: in each case we
model both the single-sun case, where one primary is a black hole, and the
double-sun case, where both primaries are luminous. In the double-sun case, we
set each star’s luminosity to be half that of the star in the single-sun case,
so that total incident light remains the same. The single-sun case results in
overall warmer temperatures, likely due to stronger immediate sunlight
reducing daytime sea ice, thereby lowering the planet’s overall albedo,
increasing the total energy absorbed, and therefore increasing the heat stored
for the night. The results are shown in Figure 4, with the comparison to
insolation shown in Figure 5.

Figure 6: Surface Temperatures in the depth of a snowball-like Long Night.
Hatched areas indicate lands held by the White Walkers. Note that this would
be a catastrophic outcome for the humans of Westeros, and represents one of
the more extreme Winters in our simulations.

During Long Nights, the single-sun case reaches above-freezing temperatures
during ecliptic crossings, with small reductions in sea ice, but the double-
sun case remains in a snowball state for the duration of the Long Night. An
example of such a snowball climate is shown in Figure 6. Looking at 25-year
averages, consistent with the experiences of different generations, it does
appear that both have characteristics common to snowballs, as shown in Figure
7. We find that in both sets of initial conditions, the planet can enjoy
centuries of temperate conditions, punctuated at chaotic and random intervals
by Winters lasting anywhere from decades to centuries or even millennia
between thaws. This suggests that with a means of limiting excess sunlight
during ecliptic-crossing, Sitnikov orbits with the planet rotating in a
tennis-ball configuration can indeed explain White Walkers.

Figure 7: 25-year average temperatures and sea ice cover for Orbit #1. The
sharp transitions to and from total sea ice cover indicate that the Long Night
in this case is a snowball event.

##  4 Discussion

Our results suggest that Sitnikov orbits can explain the climate of Westeros,
but only if specific conditions are met, each of which we will discuss in more
detail: [enumerate] The planet is in a tennis-ball configuration, reaches a
sufficiently high elevation above the plane, and the planet has some means of
limiting the incident sunlight during the otherwise-deadly crossing of the
ecliptic plane. The planet is in a spinning-top configuration with small
elevation above the plane, but has extremely specific initial conditions such
that the number of primary orbits between crossings can be described by a
series containing a very large number of consecutive integers of $O(1)$,
punctuated by a relatively small number of larger integers (Moser, 1973). We
note that in the regime of small primary eccentricity and small initial
elevation above the plane, the energetics of the system make such an orbit
unlikely. Some other element of the planet, such as its axial tilt, also
varies chaotically.

###  4.1 Limiting sunlight during ecliptic-crossing

We have found that in order for planets to survive in more-chaotic orbits with
greater elevation above the plane without lethally cold temperatures every few
months, they must experience lethally hot temperatures during crossing of the
ecliptic plane. This can be prevented if the planet has some means of limiting
the sunlight reaching the planet. One reasonably-feasible solution might be a
space-based megastructure, such as an orbital sunshade, which has been
proposed for climate engineering on Earth (Early, 1989; Angel, 2006). This
megastructure could take the form of a large swarm of smaller sunshades which
orbit the barycenter at various inclinations and semimajor axes to shield the
planet throughout the period of greatest danger, or they could orbit the
planet and re-orient themselves to let sunlight through during excursions away
from the plane. Conversely, if stellar luminosity is low enough to survive
ecliptic-crossing, large space-based mirrors could redirect additional
sunlight to the planet during excursions, keeping the planet habitable.

Proposing that Westeros is accompanied by orbital megastructures may sound
like an outlandish departure from known Westerosi history, but we argue that
this is actually entirely consistent with what is known about the history of
Westeros’s inhabitants. The Wall, standing 700-900 feet tall and stretching
for 300 miles, is described by the Westerosi as imbued with powerful magical
spells that suffice to keep White Walkers at bay (Martin, 2003). While the
Wall itself was built by Bran the Builder, the supposed ‘magical’ properties
were provided by the Children of the Forest (Martin, 2014), a race of beings
described as possessing powerful magic. We note here that “any sufficiently
advanced technology is indistinguishable from magic” (Clarke, 1973), which
suggests that it is quite possible that the Children of the Forest were in
fact an extremely technologically-advanced race, capable of large-scale
engineering and therefore certainly able to construct orbital megastructures.
We note however that that this shifts the balance in favor of higher
luminosity and sun shades rather than lower luminosity and sun mirrors, as the
latter would suggest that the White Walkers could always be easily and
permanently defeated by increasing the intensity of redirected sunlight, and
would preclude the necessity of experiencing a Long Night. Alternatively, we
note that the Long Night in our simulations lasts much longer than a single
generation, so perhaps the Children of the Forest did intervene in this way to
drive back the White Walkers. If this is the case, then Jon Snow and the
humans of Westeros are surely doomed in the coming Winter.

###  4.2 A chaotic spinning-top configuration

As we have noted, orbits with small initial elevations and small eccentricity
tend to be more stable than orbits with higher primary eccentricity or higher
initial elevation, which makes them poor candidates for triggering the Long
Night as frequently as every 8,000 years. Nonetheless, a spinning-top
configuration with small eccentricity and lower initial elevation is
attractive for characterizing Westero’s orbital dynamics, as Westerosi seasons
are mild enough to be Earth-like, and the Long Night is described as not just
cold, but dark—as if the planet suddenly traveled far enough above the plane
for the primaries to sink below the horizon at Westerosi latitudes for a
generation. In our tennis-ball configuration, we must accept that sometimes
the declination of the Sun(s) will change dramatically, particularly during
ecliptic crossing, and that the ‘darkness’ of the Long Night is merely
metaphorical, or in reference to the reduced intensity of sunlight. As
Freistetter & Grützbauch (2018) point out, Moser’s Theorem does posit that a
solution should exist which mixes the more-appealing diurnal characteristics
of the spinning-top configuration with the dramatic Long Night excursions of
our more-chaotic orbits (Moser, 1973). However, large excursions are the
consequence of the planet receiving a large amount of momentum from near-
periapse primaries during a crossing, and then experiencing comparatively less
deceleration at maximum excursion due to the primaries moving to apoapse. In
the case of low eccentricity and low elevation, the size of any individual
‘kick’ is necessarily small, and a very large number of orbits are required to
build up significant momentum, and that build-up ought to be gradual. Moser’s
Theorem states that this must not necessarily be the case, and sudden
excursions should be possible, but we argue that these energetics suggest it
is at the very least unlikely, occupying a small portion of the phase space.

###  4.3 Chaotic variations in axial tilt

A third possibility which requires neither orbital megastructures nor unlikely
orbital configurations is that the planet’s axial tilt is in neither the
spinning-top nor tennis-ball configuration, but instead varies chaotically. In
a system where the planet’s angular momentum vector is mis-aligned with the
primaries’ angular momentum vector, the orientation of the planet’s rotation
axis will precess chaotically. The presence of a moon can stabilize this
precession to some extent, but if the planet’s orbital characteristics vary in
time, then that variation and torques from other planets or nearby stars can
serve to destabilize the rotation axis (Vinson & Hansen, 2017; Makarov et al.,
2018; Saillenfest et al., 2019). A major Westerosi religion invokes ‘the light
of The Seven’, which could be understood to refer to 7 other planets in the
system, which may be large, close, or both (Martin, 1996). It is therefore
likely that a planet could be in a low-eccentricity, low-elevation
configuration with an axial tilt similar to the spinning-top configuration,
and then occasionally chaotically precess to a very different axial tilt,
moving Westeros from temperate latitudes to polar, darker latitudes,
especially if the new tilt points Westeros away from a single luminous
primary’s apoapse, similar to how the Arctic Circle on Earth is plunged into
months of darkness during Northern Hemispheric winter. Further work will be
needed to model the effect of such an additional source of chaos.

###  4.4 Other alternatives

While we find that with some caveats, Sitnikov orbits could be the correct
explanation, there are still other explanations that could work as well. We do
not believe that Kostov et al. (2013)’s circumbinary proposal can yet be
dismissed. The non-double-sun problem, as we have seen, is easily solved by
invoking a black hole as one of the primaries. In addition, other studies have
shown that chaos in circumbinary systems can be significant (Batygin et al.,
2015), and the climatic consequences non-negligible (May & Rauscher, 2016;
Popp & Eggl, 2017). We recommend further work to identify parts of the chaotic
circumbinary parameter space which could replicate the Long Night phenomenon.
Furthermore, we still cannot rule out the role that magic may play in bringing
about White Walkers.

###  4.5 Model Caveats

Our model comes with a few caveats. Because our machine-level precision is
finite, and Sitnikov orbits are chaotic, the orbital solutions represented by
our simulations may differ from reality, as we are only approximating
positions and velocities at each timestep. Additionally, PlaSim does not
include oceanic heat transport through ocean currents (Fraedrich et al.,
2005), which could significantly change the habitability and climate dynamics
of the planet (Yang et al., 2019). We also note that our land distribution is
approximate and speculative.

##  5 Conclusion

We have used a modified 3D climate model to simulate the climate of Westeros
assuming a Sitnikov orbital configuration, following the proposal in
Freistetter & Grützbauch (2018). We find that there do exist configurations
which result in both habitability and long, cold periods characteristic of the
Long Night. We note however that this explanation is substantially less likely
if the Children of the Forest have not constructed orbital megastructures.
Introducing chaotic variations in axial tilt may make a Sitnikov explanation
more likely, but we argue that a chaotic, vanilla P-type circumbinary orbit
cannot be ruled out.

Furthermore, we note that some of our models resulted in transitions to
snowball climates. This suggests that chaotic variations in orbital
characteristics or axial tilt, particularly in unusual planetary systems,
could serve as a trigger for snowball glaciation on terrestrial exoplanets
whose orbital environment differs significantly from Earth’s. For these
planets, we predict that Winter is Coming.

## Acknowledgments

We would like to thank George R.R. Martin for crafting such an inspiring yet
confusing world, and HBO for inspiring us with amazing visuals. We would also
like to extend gratitude to the system administrators of our local computing
cluster, for not noticing (or turning a blind eye to) this egregious misuse of
resources. J.J. Zanazzi provided valuable comments on the evolution of a
planet’s spin axis orientation in mis-aligned systems. Michael B. Lund, editor
of Acta Prima Aprilia, provided advice on submission guidelines and deadlines.
Ariel Amaral provided necessary emotional support and enthusiasm. We
acknowledge that this work could not have been performed without a large
amount of resources produced by the Muskoka Roastery Coffee Co. This work was
performed during the authors’ free time and, beyond computing resources, did
not make use of public tax dollars in the form of any kind of grant money.

We would furthermore like to acknowledge that our work was performed on land
traditionally inhabited by the Wendat, the Anishnaabeg, Haudenosaunee, Métis,
and the Mississaugas of the Credit First Nation.

## References

  * Abbot (2014) Abbot, D. S. 2014, Journal of Climate, 27, 4391.  http://journals.ametsoc.org/doi/10.1175/JCLI-D-13-00738.1
  * Abbot (2016) —. 2016, The Astrophysical Journal, 827, 117.  http://arxiv.org/abs/1606.03030http://dx.doi.org/10.3847/0004-637X/827/2/117http://stacks.iop.org/0004-637X/827/i=2/a=117?key=crossref.137ec89f7b157f1ff7301ee50ca31be3
  * Abbot et al. (2018) Abbot, D. S., Bloch-Johnson, J., Checlair, J., et al. 2018, The Astrophysical Journal, 854, 3.  http://arxiv.org/abs/1801.10551http://dx.doi.org/10.3847/1538-4357/aaa70fhttp://stacks.iop.org/0004-637X/854/i=1/a=3?key=crossref.e0325e68e507678dff38fffd59736583
  * Angel (2006) Angel, R. 2006, Proceedings of the National Academy of Sciences, 103, 17184.  http://www.pnas.org/cgi/doi/10.1073/pnas.0608163103
  * Backer (1993) Backer, D. C. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 36, Planets Around Pulsars, ed. J. A. Phillips, S. E. Thorsett, & S. R. Kulkarni, 11–18
  * Batygin et al. (2015) Batygin, K., Deck, K. M., & Holman, M. J. 2015, The Astronomical Journal, 149, 167\.  http://stacks.iop.org/1538-3881/149/i=5/a=167?key=crossref.b6b4c70889965a956fca0ea064966dbb
  * Boschi et al. (2013) Boschi, R., Lucarini, V., & Pascale, S. 2013, Icarus, 226, 1724
  * Budyko (1969) Budyko, M. I. 1969, Tellus, 21, 611
  * Checlair et al. (2017) Checlair, J., Menou, K., & Abbot, D. S. 2017, The Astrophysical Journal, 845, 132\.  http://arxiv.org/abs/1705.08904http://dx.doi.org/10.3847/1538-4357/aa80e1http://stacks.iop.org/0004-637X/845/i=2/a=132?key=crossref.0179b453ba8de0d24a234f4256873009
  * Clarke (1973) Clarke, A. C. 1973, Profiles Of The Future (MACMILLAN).  https://www.xarg.org/ref/a/0330236199/
  * Condon et al. (2002) Condon, D. J., Prave, A. R., & Benn, D. I. 2002, Geology, 30, 35
  * Deser et al. (2000) Deser, C., Walsh, J. E., & Timlin, M. S. 2000, Journal of Climate, 13, 617
  * Dvorak et al. (1998) Dvorak, R., Contopoulos, G., Efthymiopoulos, C., & Voglis, N. 1998, Planetary and Space Science, 46, 1567 , second Italian Meeting on Celestial Mechanics.  http://www.sciencedirect.com/science/article/pii/S0032063397002031
  * Dvorak & Sui Sun (1997) Dvorak, R., & Sui Sun, Y. 1997, Celestial Mechanics and Dynamical Astronomy, 67, 87
  * Early (1989) Early, J. T. 1989, Journal of the British Interplanetary Society, 42, 567
  * Fraedrich et al. (2005) Fraedrich, K., Jansen, H., Kirk, E., Luksch, U., & Lunkeit, F. 2005, Meteorologische Zeitschrift, 14, 299
  * Freistetter & Grützbauch (2018) Freistetter, F., & Grützbauch, R. 2018, 1.  http://arxiv.org/abs/1803.11390
  * Gomez-Leal et al. (2019) Gomez-Leal, I., Kaltenegger, L., Lucarini, V., & Lunkeit, F. 2019, Icarus, 321, 608.  https://linkinghub.elsevier.com/retrieve/pii/S0019103517307820http://arxiv.org/abs/1901.02901http://dx.doi.org/10.3847/1538-4357/aaea5f
  * Hoffman & Schrag (2002) Hoffman, P. F., & Schrag, D. P. 2002, Terra Nova, 14, 129.  http://doi.wiley.com/10.1046/j.1365-3121.2002.00408.x
  * Hoffman et al. (2017) Hoffman, P. F., Abbot, D. S., Ashkenazy, Y., et al. 2017, Science Advances, 3, e1600983.  http://advances.sciencemag.org/lookup/doi/10.1126/sciadv.1600983
  * Kostov et al. (2013) Kostov, V., Allan, D., Hartman, N., Guzewich, S., & Rogers, J. 2013, arXiv:1304.0445.  http://arxiv.org/abs/1304.0445
  * Kovács & Érdi (2007) Kovács, T., & Érdi, B. 2007, Astronomische Nachrichten, 328, 801.  http://doi.wiley.com/10.1002/asna.200710799
  * Linsenmeier et al. (2015) Linsenmeier, M., Pascale, S., & Lucarini, V. 2015, Planetary and Space Science, 105, 43
  * Liu (2014) Liu, C. 2014, The Three-Body Problem (Remembrance of Earth’s Past Book 1) (Tor Books).  https://www.xarg.org/ref/a/B00IQO403K/
  * Lucarini et al. (2010) Lucarini, V., Fraedrich, K., & Lunkeit, F. 2010, Quarterly Journal of the Royal Meteorological Society, 136, 2
  * Makarov et al. (2018) Makarov, V. V., Berghea, C. T., & Efroimsky, M. 2018, ApJ, 857, 142
  * Martin (1996) Martin, G. R. R. 1996, A Game of Thrones, A Song of Ice and Fire (Bantam)
  * Martin (2003) —. 2003, A Storm of Swords (A Song of Ice and Fire, Book 3) (Bantam).  https://www.xarg.org/ref/a/B000FBFN1U/
  * Martin (2012) —. 2012, The Lands of Ice and Fire (A Game of Thrones): Maps from King’s Landing to Across the Narrow Sea (A Song of Ice and Fire), media tie in edn., A Song of Ice and Fire (Bantam)
  * Martin (2014) —. 2014, The World of Ice & Fire: The Untold History of Westeros and the Game of Thrones (A Song of Ice and Fire) (Bantam).  https://www.xarg.org/ref/a/B00EGMGGVK/
  * May & Rauscher (2016) May, E. M., & Rauscher, E. 2016, The Astrophysical Journal, 826, 225.  http://dx.doi.org/10.3847/0004-637X/826/2/225http://stacks.iop.org/0004-637X/826/i=2/a=225?key=crossref.a44b5e9710d80f445fec4128ecccafbd
  * Mayor & Queloz (1995) Mayor, M., & Queloz, D. 1995, Nature, 378, 355
  * Menou (2015) Menou, K. 2015, Earth and Planetary Science Letters, 429, 20
  * Moser (1973) Moser, J. 1973, Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics (AM-77) (Princeton Landmarks in Mathematics and Physics) (Princeton University Press).  https://www.xarg.org/ref/a/0691089108/
  * Mullally (2017) Mullally, S. E. 2017, in AGU Fall Meeting Abstracts, Vol. 2017, P51E–01
  * Paradise et al. (2019) Paradise, A., Borth, H., Lunkeit, F., & Kirk, E. 2019, alphaparrot/GPlaSim: Sitnikov Orbits, , , doi:10.5281/zenodo.2611088.  https://doi.org/10.5281/zenodo.2611088
  * Paradise & Menou (2017) Paradise, A., & Menou, K. 2017, The Astrophysical Journal, 848, 33
  * Perdios & Markellos (1987) Perdios, E., & Markellos, V. V. 1987, Celestial Mechanics, 42, 187
  * Perdios & Kalantonis (2012) Perdios, E. A., & Kalantonis, V. S. 2012, Celestial Mechanics and Dynamical Astronomy, 113, 377
  * Pierrehumbert (2005) Pierrehumbert, R. 2005, Journal of Geophysical Research, 110, 1
  * Popp & Eggl (2017) Popp, M., & Eggl, S. 2017, Nature Communications, 8, 14957.  http://dx.doi.org/10.1038/ncomms14957http://www.nature.com/doifinder/10.1038/ncomms14957
  * Saillenfest et al. (2019) Saillenfest, M., Laskar, J., & Boué, G. 2019, A&A, 623, A4
  * Sellers (1969) Sellers, W. D. 1969, Journal of Applied Meteorology, 8, 392.  https://doi.org/10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2
  * Sitnikov (1961) Sitnikov, K. 1961, Soviet Physics Doklady, 5, 647
  * Tajika (2007) Tajika, E. 2007, Earth, Planets and Space, 59, 293
  * u/selvag (2018) u/selvag. 2018, Westeros, Essos and The Known World of Game of Thrones, Google Maps-style (Updated version). [3400 x 2267] [OC], , .  https://www.reddit.com/r/MapPorn/comments/6qqwwh/westeros_essos_and_the_known_world_of_game_of/
  * Vineyard (2014) Vineyard, J. 2014, George R.R. Martin on What Not to Believe in Game of Thrones, , .  https://www.vulture.com/2014/11/george-rr-martin-new-book.html
  * Vinson & Hansen (2017) Vinson, A. M., & Hansen, B. M. S. 2017, MNRAS, 472, 3217
  * Whitehead (2016) Whitehead, A. 2016, Atlas of Ice and Fire, , .  https://atlasoficeandfireblog.wordpress.com/2016/03/06/terra-incognita-beyond-the-known-world/
  * Yang et al. (2019) Yang, J., Abbot, D. S., Koll, D. D. B., Hu, Y., & Showman, A. P. 2019, The Astrophysical Journal, 871, 29.  http://arxiv.org/abs/1902.02103{%}0Ahttp://dx.doi.org/10.3847/1538-4357-aaf1a8
  * Yoshida (1990) Yoshida, H. 1990, Physics Letters A, 150, 262

##  Appendix A Stellar Angles

  * •

In the tennis-ball configuration, the hour angle of one of the primaries in
Westeros’s sky is 0 or $\pi$ when the planet’s elevation above the plane
$z(t)=0$, and when $z(t)\neq 0$, is given by

$h(t)=\tan^{-1}\left(\frac{r(t)\cos\theta(t)}{z(t)}\right)-\frac{\pi}{2}$ (A1)

  * •

$\theta(t)=\nu(t)-\pi$ for the western primary, and $\theta(t)=\nu(t)$ for the
eastern primary.

  * •

The primary’s declination is given by

$\delta(t)=\tan^{-1}\left(\frac{r(t)\sin\theta(t)}{\sqrt{z^{2}(t)+r^{2}(t)\cos^{2}\theta(t)}}\right).$
(A2)

  * •

In the spinning-top configuration, the hour angle is given by

$h(t)=\pi-\theta(t)$ (A3)

and the declination is given by

$\delta(t)=\frac{z(t)}{|z(t)|}\left[\frac{\pi}{2}-\left|\,\tan^{-1}\left(\frac{r(t)}{z(t)}\right)\right|\right]$
(A4)

  * •

A local hour angle $h^{\prime}(t)$ is computed from $h(t)$ by adding the
planet’s angular rotation rate, $\Omega$, and local longitude $\varphi$ such
that

$h^{\prime}(t)=h(t)+\Omega{t}+\varphi$ (A5)

  * •

With these angles, the cosine of the stellar zenith angle $\Phi_{z}$ can be
calculated for any latitude $\psi$ and longitude $\varphi$ with

$\cos\Phi_{z}=\sin\psi\sin\delta(t)+\cos\psi\cos\delta\cos{h^{\prime}(t)}$
(A6)

  * •

Hodor Hodor, Hodor Hodor Hodor, Hodor:

$\text{Hodor}=\text{Hodor}$ (A7)