Discovery of a red giant with solar-like oscillations in an eclipsing binary system from Kepler space-based photometry

0.34 MB·English

by  S. Hekker

Draft version January 3, 2010
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
DISCOVERY OF A RED GIANT WITH SOLAR-LIKE OSCILLATIONS IN AN ECLIPSING BINARY SYSTEM
FROM KEPLER SPACE-BASED PHOTOMETRY
S. Hekker1, J. Debosscher2, D. Huber3, M. G. Hidas3,4,5, J. De Ridder2, C. Aerts2,6, D. Stello3, T.R. Bedding3,
R. L. Gilliland7, J. Christensen-Dalsgaard8, T. M. Brown4, H. Kjeldsen8, W. J. Borucki9, D. Koch9, J. M.
Jenkins10, H. Van Winckel2, P. G. Beck2, J. Blomme2, J. Southworth11, A. Pigulski12, W. J. Chaplin1, Y. P.
Elsworth1, I. R. Stevens1, S. Dreizler13, D. W. Kurtz14, C. Maceroni15, D. Cardini16, A. Derekas3,17, M. D.
Suran18
Draft version January 3, 2010
0
1 ABSTRACT
0
Oscillating stars in binary systems are among the most interesting stellar laboratories, as these can
2
provide information on the stellar parameters and stellar internal structures. Here we present a red

n giant with solar-like oscillations in an eclipsing binary observed with the NASA Kepler satellite. We
a compute stellar parameters of the red giant from spectra and the asteroseismic mass and radius from
J theoscillations. Althoughonlyoneeclipsehasbeenobservedsofar,wecanalreadydeterminethatthe

3 secondary is a main-sequence F star in an eccentric orbit with a semi-major axis larger than 0.5AU
and orbital period longer than 75days.

]
R Subject headings: binaries: eclipsing - stars: oscillations - stars: individual (KIC8410637)
S
h. 1. INTRODUCTION data have already revealed the presence of nonradial
p The prospects for asteroseismology of red-giant stars oscillation modes in red giants (De Ridder et al. 2009)
- have increased significantly with the launch of Ke- and have given rise to a population synthesis study
o (Miglio et al.2009). ThecurrentlyavailableKepler data
pler (Borucki et al. 2009) in March 2009 and CoRoT
r
reveal clear solar-like high radial overtone p-mode oscil-
t (Baglin et al. 2006) in December 2006. These satellites
s lations in a large sample of red giants, extending in lu-
a provide uninterrupted long time series of high-precision
minosity from the red clump to the bottom of the giant
[ photometry for a large sample of stars. Among these
branch (Bedding et al. 2009).
stars are many red giants, allowing for both statistical
1 For in-depth studies, including detailed stellar model-
analyses as well as detailed individual studies. CoRoT
v ing, accurate stellar parameters are needed. Red giants
9 [email protected] of different masses occupy a narrow region in the H-R
9 1School of Physics and Astronomy, University of Birmingham, diagram, whichresultsinrelativelylargeerrorswhenes-
3 EdgbastonB152TT,UnitedKingdom timating the mass from measured classical parameters,
0 2Instituut voor Sterrenkunde, Katholieke Universiteit Leuven, such as effective temperature, surface gravity and lumi-
Celestijnenlaan200D,B-3001Leuven,Belgium
1. 3Sydney Institute for Astronomy (SIfA), School of Physics, nosity, and from evolutionary tracks. A better estimate
0 UniversityofSydney,NSW2006,Australia can be obtained from the characteristic frequencies of
0 4Las Cumbres Observatory Global Telescope, Goleta, CA solar-like oscillations in these stars and their frequency
93117,USA
1 5Department of Physics, University of California, Santa Bar- separations, either through modeling or through scal-
: bara,CA93106,USA ing relations (Kjeldsen & Bedding 1995). Here we use
v 6IMAPP, Department of Astrophysics, Radbout University a modeling approach and scaling relations to compute
Xi Ni7jmSepgaecne,PTeOleBscooxpe90S1c0i,e6n5c0e0IGnsLtitNuitjem,eg3e7n0,0thSeanNeMthaerrtlianndDsrive, an asteroseismic mass and radius.
Eclipsing binary systems provide the primary method
r Baltimore,MD21218,USA
a 8Department of Physics and Astronomy, Aarhus University, for measuring masses and radii of stars. However, the
DK-8000AarhusC,Denmark probability of observing an eclipsing system is low, be-
9NASA Ames Research Center, MS 244-30, Moffet Field, CA cause of the geometrical restriction that the system has
94035,USA
10SETI Institute/NASA Ames Research Center, MS 244-30, to be viewed close to the orbital plane. We report
MoffetField,CA94035,USA here the discovery of a pulsating red giant in an eclips-
11AstrophysicsGroup,KeeleUniversityNewcastle-under-Lyme, ing binary system observed by Kepler. This star was
ST55BG,UK
12Instytut Astronomiczny Uniwersytetu Wroclawskiego, foundusingautomatedvariabilityclassificationmethods
Kopernika11,51-622Wroclaw,Poland to the targets of the asteroseismology program of Kepler
13Georg-AugustUniversita¨t,Institutfu¨rAstrophysik,Friedrich- (Debosscher et al. 2009: Blomme et al. 2009).
Hund-Platz1,D-37077G¨ottingen
14Jeremiah Horrocks Institute of Astrophysics, University of 2. OBSERVATIONS
CentralLancashire,PR12HE,UK
15INAF-Osservatorio Astronomica di Roma, via Frascati 3, Photometric data for KIC841063719 (TYC 3130-2385)
I-00040MonteporzioC.(RM),Italy have been taken with Kepler during the commissioning
16IASF-Roma, INAF, V. del Fosso del Cavaliere 100, 00133 run (Q0) from 2009 May 2 through 11, and the follow-
Roma,Italy
17Konkoly Observatory, Hungarian Academy of Sciences, ing first science run (Q1) of 33.5d. We present results
H-1525Budapest,P.O.Box67,Hungary based on these combined time series of about 40d. This
18Astronomical Institute of the Romanian Academy, Str.
CutituldeArgint5,RO40557,Bucharest,RO 19 KIC=Kepler Input Catalog,Brownetal.(2005).
2 S. Hekker et al.
TABLE 1
Stellar parameters of KIC8410637.
KIC Ofek(2008) spectroscopy
Teff [K] 4680±150 4650 4650±80
log(g)(c.g.s.) 2.8±0.3 2.70±0.15
[Fe/H][dex] −0.3±0.3 −0.38 0.0±0.1
vsini[kms−1] 5.0±0.7
ξmicro [kms−1] 1.3±0.1
star has a magnitude of 10.77 mag in the Kepler band-
pass and was selected by the Kepler Asteroseismic Sci-
ence Consortium (KASC, Gilliland et al. 2009) for long-
cadence observations. The time series data have a 29.4-
min cadence and show both solar-like oscillations in the
frequencyregimeexpectedforaredgiant,andaneclipse
(see Fig. 1).
This star has also been part of the TrES-Lyr1 ground-
basedsurvey20 (O’Donovan et al.2006)fromwhichtime
series photometry with a time span of 75d are at our
disposal. In these data no eclipse is visible, while the
primary and possibly the secondary eclipse (depth pre-
dicted in Section 6) would have been detected, had they
occurred during the time span of the observations. The
factthattheprimaryisnotseenintheTrESdataimplies Fig. 1.— Top: time series data from Kepler of KIC8410637.
Middle: asectionofthelightcurvecenteredontheeclipsewiththe
that the period is longer than the duration of the TrES
best-fit model (solid line). Bottom: the residuals after correcting
observations. fortheeclipsemodel,clearlyshowingthesolar-likeoscillations.
ThisstarwasalsoobservedphotometricallybytheSu-
October 13 and one on 2009 October 15. These spec-
perWASP cameras (Pollacco et al. 2006) for 100 days in
tra have a resolution of about 85000 and a wavelength
2007 and for 90 days in 2008. There is a suspicion of a
secondary eclipse, but no primary eclipse has been ob- range of 4000 9000˚A. We analyzed the average spec-
−
served during these periods, which have some gaps due trum,whichhasatypicalsignal-to-noiseratioof 90at
to weather conditions. Observations for KIC8410637 6000 ˚A, with methods described by Hekker & M∼el´endez
are also available from the All Sky Automated Sur- (2007) and list the results in the last column of Ta-
vey (ASAS) (Pojman´ski 1997), in which there are three ble 1. This method does not directly provide uncer-
points that have lower flux than the majority. These tainties on the individual parameters. Here, we adopted
points might be due to the eclipse, which would imply the total of the difference and scatter between the val-
anorbitalperiodofabout380dorasubmultiplethereof. ues of Hekker & Mel´endez (2007) and the values of
Unfortunately, therearetoofewpoints, andtheiruncer- Luck & Heiter (2007) (see Hekker & Mel´endez 2007, for
tainty is too large to draw firm conclusions. more details). For vsini, we computed the uncertainty
Additionally, we checked whether we could put more from a 10% uncertainty in the total FWHM of the lines,
constraints on the period from the time stamps of the which is combined with the σ on the macro turbulence.
datawehaveatourdisposal. Wedeterminedwhichperi- TheseresultsareinagreementwiththoseintheKICand
odswouldhaveledustomissprimaryeclipsesbecauseof the literature. They confirm the red-giant nature of the
alackofobservations. Wewouldhavemissedalleclipses primary star.
foranorbitalperiodofabout135,220,253,261,270and Weinvestigatedthespectrumforsignaturesofthesec-
295 days. All other possible periods are longer than the ondary star, in the form of additional absorption lines
380 days, mentioned above. and cross-correlation profiles, but no clear signatures
could be detected. The three spectra obtained over a
3. STELLARPARAMETERS time span of 3 days are not sufficient to see radial veloc-
For KIC8410637 the Kepler Input Catalogue ity variations due to the binary orbit.
(Brown et al. 2005) provides a set of stellar pa-
4. ASTEROSEISMICANALYSIS
rameters, as listed in Table 1. Tycho-2 (Høg et al.
2000) and 2MASS (Skrutskie et al. 2006) photometry Solar-like oscillations are visible both during and out-
are also available. Ofek (2008) fitted these colors with side the eclipse (Fig. 1). To analyze these we adopted
syntheticphotometryofstellartemplates(Pickles1998). two approaches. In the first approach, we corrected the
Using two methods, Ofek (2008) classified the star as a light curve including the eclipse by fitting and subtract-
metal-weak K2III star with an effective temperature of ing trends for different parts of the time series using a
4650K or a K4V star with a temperature of 4350K. linear polynomial for the commissioning data, a second-
We also obtained three spectra with the HERMES order polynomial for the part of the Q1 data before the
spectrograph (Raskin & Van Winckel 2008) mounted on eclipse, the mean value for the data in full eclipse, and a
the Mercator Telescope, La Palma, Spain: two on 2009 linear polynomial forthe data obtained after the eclipse.
We omitted the observations during ingress and egress.
20http://nsted.ipac.caltech.edu/NStED/docs/holdings.html The resulting time series is shown in the top panel of
Red giant in an eclipsing binary system from Kepler 3
power spectrum. Octave2 computed ν and A
max ℓ=0
from a Gaussian fitting to the oscillation envelope, and
∆ν from an autocorrelation of the oscillation frequen-
cies. The latter have been determined with a Bayesian
method, where oscillation frequencies are those with a
posterior probability of less than 0.1% of being noise.
The Sydney pipeline produced ν from a smoothed
max
powerspectrum. Thecurrentdatadonotyetallowusto
measure the life times of the solar-like oscillations. For
more details on the pipelines we refer to Hekker et al.
(2009a) and Huber et al. (2009).
The resulting parameters are listed in Table 2. Note
that A is computed for a power spectrum with mean
ℓ=0
squared scaling, which is a factor √2 different with
respect to the scaling relations of Kjeldsen & Bedding
(1995). With the effective temperature and the scaling
laws from Kjeldsen & Bedding (1995) we computed the
radius,massandluminosityoftheprimary(seeTable2).
We also used the RADIUS pipeline described by
Stello et al.(2009)tocomputethestellarradius. Thein-
putparametersforthispipelinearethelargeseparation,
T and metallicity, for which we used the spectroscopic
eff
values listed in Table 1. The results from this analysis
are listed in Table 2 and lead to a value for log(g) of 2.6
0.1, which is in agreement with our spectroscopically
±
derived value.
The values computed with different methods and
approaches are in good agreement. The weighted
mean values of the results from the different meth-
ods are used throughout the rest of the paper,
i.e., RR = 11.8 0.6 R⊙, LR = 58 6 L⊙,
± ±
MR = 1.7 0.3 M⊙ (suffix ‘R’ refers to the red gi-
±
ant) together with the spectroscopically determined ef-
Fig. 2.— Top: flux as a function of time, after removing the
binarysignatureasdescribedinthetext. Centre: thepowerspec- fective temperature. We point out that the estimates of
trumof thered giant oscillations. Thepowerexcessduetosolar- these fundamental parameters rely on the input physics
like oscillations is present in the range 30−60µHz. The excess of stellar models, through the interpretation of observ-
below 20µHz might be due to granulation in the primary’s at-
ables (e.g. ν and A ) in terms of the properties of
mosphere. Bottom: H-R diagram with BASTI (Pietrinfernietal. max l=0
2004)evolutionarytracksfordifferentmassesandsolarmetallicity. the convection and thermodynamics, which we assumed
The filled circle shows the position of the red giant. The shaded to be known and error-free. This implies that there may
area shows the secondary position constrained by two values de- be unknown systematic uncertainties. Nevertheless, the
rived from the light-curve fit: the radius ratio (dashed lines) and
radius and mass have values in good agreement with the
the bolometric luminosity ratio (dashed-dotted lines; taking into
accounttheKepler bandpass). Theasteriskindicatestheposition expected range for red-clump stars. For the radius there
oftheSun. is also agreement between the result from scaling rela-
tions and from models (RADIUS).
Fig.2. Inthesecondapproach, weomittedthecomplete
eclipse from the time series. 5. BINARYANALYSIS
The Fourier spectrum shows a clear power excess (see Theveryflatbottomoftheeclipse, exceptforthered-
the second panel of Fig. 2). The power excess is present giant oscillations, seems to be indicative of a smaller ob-
between 30 60µHz, which is the dominant range of ject occulted by a considerably larger object. This is
−
ν (frequency of maximum oscillation power) found underlined by the fact that for an annular eclipse, i.e.,
max
for a large sample of red giants observed by CoRoT transit,limbdarkeningeffectsplayamoreprominentrole
(Hekker et al. 2009b). This reflects the high population andwouldhavecausedamoregraduallychangingshape
density of red-clump stars, compared to stars on the as- of the eclipse. Because we have only one eclipse, a full
cending branch (Miglio et al. 2009). No power excess is analysisofthebinaryisnotyetpossible,butweareable
present at frequencies between 70 µHz and the Nyquist toputseveralconstraintsonthenatureofthesecondary
frequency of 280µHz. We analyzed the power spec- and on the orbit.
∼
trum obtained from the corrected time series with two FromtheKepler data,wedirectlymeasuredthedepth
methodsoftheOctavepipeline(Hekker et al.2009a: Oc- of the eclipse to be about 9%. This depth is determined
tave1 and Octave2) and the power spectrum of the light by the luminosity ratio of the two stars, i.e., l = L /L
2 R
curve where the eclipse had been omitted with the auto- =0.099 0.005. Theuncertaintyisestimatedtakingthe
±
matedpipelinedevelopedinSydney(Huber et al.2009). shape of the light curve outside the eclipse into account.
Octave1 computed ν and A (maximum mode In addition to the depth, we measured the time dur-
max ℓ=0
amplitude)fromabinnedpowerspectrum,and∆ν(large ing which the secondary is fully obscured: 1.64 0.01d.
±
frequency separation) using the power spectrum of the The ingress and egress are symmetric and each lasts for
4 S. Hekker et al.
TABLE 2
Oscillation parameters of KIC8410637computed with different methods.
Octave1 Octave2 Sydney RADIUS
— measured — model
νmax [µHz] 45.0±4.8 45.2±1.3 44.2±0.5
∆ν [µHz] 4.64±0.23 4.5±0.1 4.51±0.37
δν02 [µHz] 0.6±0.1 0.69±0.09
Aℓ=0 [ppm] 57.6±7.3 61.8±6.2 53.4±1.9
— derived —
R[R⊙] 11.2±1.6 11.8±0.7 11.4±1.9 13.1±1.7
L[L⊙] 53±16 58±8 55±19 72±19
M [M⊙] 1.7±0.7 1.8±0.3 1.7±0.9
0.276 0.007d. The symmetry of the ingress and egress The physical parameters in our model are r, v, l, and
±
indicatesthattherelativeaccelerationofthebinarycom- b = sinδ (the impact parameter). In order to avoid
ponentsbetweeningressandegressisequaltozero,i.e. it strong correlations among these parameters (in partic-
is justified to assume the sky-projected relative velocity ular between r, v, and b), when performing the fit we
of the two stars to be constant during the eclipse. replace v with v2/(1 b2) and r with r/v2. The values
Theeclipsetimedependsontheradiiofbothstarsand of the physical param−eters resulting from this fit are b=
the geometry of the system. Knowing that the eclipse 0.0+0.15, r = 0.1435+0.0008, v = 1.0338+0.0005R /day,
−0.0027 −0.011 R
il.aes.t,soonnllyyaduvreinryg∼sm2a%llosreglemssenotftohfetthoetaolrobribtitiaslcpoevreiroedd, and l = 0.10060−+00..0000000066, consistent with the values ob-
tained from the geometric analysis.
during the eclipse, and that the system is viewed close
Totestourassumptionthatv isconstant,werepeated
totheorbitalplane,impliesthatthetimeoftotaleclipse
the fit with allowing v to change linearly with time. The
(τ ) can be expressed as:
total result was a deceleration of 0.015+0.005R /day2 (i.e.
−0.004 R ∼
P(R cosδ R ) 3% slower at egress than at ingress). This may be a real
R 2
τ = − , (1)
total 4aE(ǫ) asymmetry in the eclipse, implying an eccentric orbit.
However, it could also be due to the pulsation signal
with P the orbital period, a the semi-major axis, E the altering the apparent shapes of the ingress and egress.
complete elliptic integral, ǫ the eccentricity and δ the We converted the fitted value of l into a bolometric
latitude of the eclipse on the stellar disc and RR and R2 luminosity ratio by folding blackbody curves with the
the radius of the red giant and the secondary, respec- Kepler spectral response curve. This is a function of the
tively. The total time of ingress, total eclipse and egress secondary’stemperature, asshowninFig.2(dot-dashed
(τin+total+eg)fromthesamegeometricconsiderationscan lines). With the constraint from the radius ratio and
be expressed as:
L R2T4
P(R cosδ+R ) 2 = 2 eff,2 , (5)
τin+total+eg = R4aE(ǫ) 2 . (2) LR RR2Te4ff,R
wefindL2 =5.2 0.7L⊙. Thesecondaryisthusproba-
DividingEq.(1)byEq.(2)providesanestimateofR2 bly an F main-seq±uence star. Using the mass-luminosity
as a function of cosδ: relation for stars with masses between 0.5 and 2.0M⊙,
τtotal RRcosδ R2 cosδ r i.e., (L/L⊙)=(M/M⊙)4.5, we found a mass for the sec-
τin+total+eg ≡x= RRcosδ+−R2 = cosδ−+r, (3) uonndcearrtyaiMnt2yooff1t.h4e4m±a0s.s0-5luMm⊙in,owsihtyerreewlaetidonidinntootatacckoeutnhte.
with r = R /R . In this way we found that We also computed the surface gravity for the secondary
2 R
and found log(g ) = 4.2 0.1.
2
R2 1 x The binary also has to±obey Kepler’s third law:
=cosδ − , (4)
R 1+x
R 4π2a3
M +M = , (6)
where x = 0.75 0.03. Using δ = 0 (central eclipse) R 2 GP2
±
and the value of the radius of the red-giant star derived
in which we still have a as unknown, and we take a min-
in the previous section, we then find an upper limit of
imum value of 75d for P. With this value we computed
R2 ≤1.7±0.1R⊙. a minimum semi-major axis for the system of 0.5AU.
To confirm this geometric analysis we computed a ba-
On the other hand, if we assume a circular orbit, we
sic model of the eclipse signal. We neglected limb dark-
cancomputetheperiod. Fromtheeclipsefitweknowthe
ening and computed the fractional overlap between the
orbital velocity and combining this with Kepler’s third
two discs, taking into account the relative luminosity of
law, we get:
the star being eclipsed. As explained above, we assumed
2πG(M +M )
the sky-projected relative velocity (v) of the two stars P = R 2 . (7)
to be constant. This model has been fitted to the por- v3
tion of the light curve near the eclipse using a Markov- Using the masses of both stars, we obtain P = 32 6
±
ChainMonteCarloapproach(seee.g.Torres et al.2008, days. This is considerably shorter than the minimum
and references therein), which yields the joint posterior orbitalperiodweinferredfromcurrentlyavailableobser-
probabilitydensityofalltheparameters, giventhedata. vations, which implies an eccentric orbit.
Red giant in an eclipsing binary system from Kepler 5
longer than the time span of the TrES data, i.e., at least
TABLE 3
parameter overview of eclipsing binary KIC8410637. 75d. Also,aninitialinspectionoftherawKepler dataof
Q2(additional3monthsofdatafollowingtheQ1phase)
primary secondary does not seem to reveal another eclipse. In the case that
R[R⊙] 11.8±0.6 1.7±0.1 we missed an eclipse due to a lack of observations, the
L[L⊙] 58±6 5.2±0.7 minimum orbital period would be 135 days.
M [M⊙] 1.7±0.3 1.44±0.05 From the orbital velocity of the secondary, we com-
Teff [K] 4650±80 6700±200 puted the period for a circular orbit, which would be
log(g)(c.g.s.) 2.70±0.15 4.2±0.1
shorter than half the total time span of the TrES data
orbitalparameters
and we should have seen an occultation in those data.
Porbit [days] >75
Therefore we concluded that the orbit is eccentric. The
a[AU] >0.5
possible annular eclipse in the SuperWASP data has a
6. DISCUSSION lengthof 7daysanddepthof 0.03mag. Ifconfirmed
∼ ∼
this would indeed imply a large orbital eccentricity.
From observations obtained with Kepler, we found a
Fromtheeclipsetimesanddepthwewereabletocom-
pulsating red giant in an eclipsing binary. Once a full
pute the luminosity and the radius of the secondary. All
orbit has beenobserved, the orbital parameters will pro-
values seem to be compatible with an F main-sequence
vide an independent measure of the stellar parameters,
star. The mass-luminosity relation for main-sequence
such as mass and radius, with respect to the asteroseis-
stars then leads to a mass estimate of the secondary as
micvalues. Theseadditionalconstraintsmakethisavery
well as its surface gravity. These results are summarized
interesting case and the star is therefore being followed
inTable3andthepositionofbothstarsareindicatedin
up by the Kepler Mission as well as with ground-based
an H-R diagram in Fig. 2.
spectroscopy.
KIC8410637isanextremelyinterestingbinaryforfur-
Stellar parameters of the primary red giant have been
ther follow-up. Longer time series from Kepler will im-
computed from a spectroscopic analysis and from the
provedramaticallythedetectionthresholdforthederiva-
solar-like oscillations. We did not find evidence of the
tion of the properties of the oscillations of the red giant.
secondarycomponentinthespectra. Thecurrentlyavail-
able spectra are not yet sufficient to put constraints on
the orbit.
From the single eclipse observed by Kepler and addi-
Funding for this Discovery mission is provided by
tionalindependentobservationsfromTrES,SuperWASP
NASA’s Science Mission Directorate. We would like to
and ASAS, we inferred constraints on the orbit and sec-
acknowledgetheentireKepler teamfortheireffortsover
ondary star. The annular eclipse, i.e., the secondary
manyyears. Withouttheseeffortsitwouldnothavebeen
passing in front of the red giant, would cause an eclipse
possible to obtain the results presented here. SH, WJC,
with a depth of 1.7%, assuming a circular orbit and
∼ YPE, IRS and DWK acknowledge support by the UK
without taking limb-darkening into account. Such a dip
Science and Technology Facilities Council. The research
would be observable, but has not been detected by Ke-
leading to these results has received funding from the
pler, TrES or ASAS. Only the SuperWASP data show
European Research Council under the European Com-
a feature that might be due to an annular eclipse. The
munity’s Seventh Framework Programme (FP7/2007–
SuperWASP and ASAS data have gaps due to weather
2013)/ERCgrantagreementn◦227224(PROSPERITY),
conditions. Although there are also three gaps (4.7, 2.9
fromtheResearchCouncilofK.U.Leuven,fromtheFund
and 1.7d) in the TrES time series data with a duration
for Scientific Research of Flanders (FWO), and from the
longer than the duration of the eclipse, we expect the
BelgianFederalScienceOffice. DSacknowledgessupport
chance that an eclipse falls exactly during one of these
from the Australian Research Council.
gaps to be low. Therefore, we infer that the orbit is
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