KIC 7177553: a quadruple system of two close binaries

1.6 MB·

by  H. Lehmann

Accepted by ApJ
PreprinttypesetusingLATEXstyleemulateapjv.04/17/13
KIC 7177553: A QUADRUPLE SYSTEM OF TWO CLOSE BINARIES
H. Lehmann1, T. Borkovits2, S. A. Rappaport3, H. Ngo4, D. Mawet5, Sz. Csizmadia6, and E. Forga´cs-Dajka7
Accepted by ApJ
ABSTRACT
KIC 7177553 was observed by the Kepler satellite to be an eclipsing eccentric binary star system
with an 18-day orbital period. Recently, an eclipse timing study of the Kepler binaries has revealed
eclipsetimingvariationsinthisobjectwithanamplitudeof∼100sec,andanouterperiodof529days.
The implied mass of the third body is that of a superJupiter, but below the mass of a brown dwarf.
6
We therefore embarked on a radial velocity study of this binary to determine its system configuration
1
and to check the hypothesis that it hosts a giant planet. From the radial velocity measurements, it
0
2 became immediately obvious that the same Kepler target contains another eccentric binary, this one
with a 16.5-day orbital period. Direct imaging using adaptive optics reveals that the two binaries are
n separatedby0.4(cid:48)(cid:48) (∼167AU),andhavenearlythesamemagnitude(towithin2%). Thecloseangular
a
proximity of the two binaries, and very similar γ velocities, strongly suggest that KIC 7177553 is one
J
of the rare SB4 systems consisting of two eccentric binaries where at least one system is eclipsing.
2
Both systems consist of slowly rotating, non-evolved, solar-like stars of comparable masses. From
1
the orbital separation and the small difference in γ velocity, we infer that the period of the outer

orbit most likely lies in the range 1000 to 3000 years. New images taken over the next few years, as
]
R well as the high-precision astrometry of the Gaia satellite mission, will allow us to set much narrower
constraints on the system geometry. Finally, we note that the observed eclipse timing variations in
S
the Kepler data cannot be produced by the second binary. Further spectroscopic observations on a
.
h longer time scale will be required to prove the existence of the massive planet.
p Subject headings: stars: binaries: eclipsing, stars: binaries: spectroscopic, stars: planetary systems,
- stars: fundamental parameters
o
r
t
s 1. INTRODUCTION 2011), designed and launched for searching for transit-
a
ing planets, also delivered light curves of unprecedented
[ Most of our knowledge of stellar masses comes from
photometricqualityofalargenumberofEBs(e.g.Slaw-
the investigation of binary stars. In particular eclipsing
1 son et al. 2011). Among these EBs, multiple systems
binaries (EBs) can provide the absolute masses of their
v were also found, including triply eclipsing hierarchical
components. Precise absolute masses of stars across the
6 triplessuchasKOI-126(Carteretal.2011b),HD181068
whole Hertzsprung-Russell diagram are urgently needed
2 (Derekas et al. 2011), and KIC 4247791, a SB4 sys-
for testing the theories of stellar structure and evolution
9
tem consisting of two eclipsing binaries (Lehmann et al.
2 and, recently, more and more for establishing accurate
2012). Such unusual objects (from the observational
0 scalingrelationsforthepulsationpropertiesofoscillating
side) additionally offer the possibility for studying the
. stars in the rapidly developing field of asteroseismology
1 tidal interaction with the third body or between the bi-
(e.g. Aerts 2015). This will be especially important for
0 naries in the case of quadruple systems.
the scaling of measured planet-star mass ratios in such
6 EBsineccentricorbits,ontheotherhand,allowforthe
future space projects like PLATO (Rauer et al. 2014)
1 observationofapsidalmotionasaprobeofstellarinteri-
where the stellar masses will be asteroseismically deter-
: ors(e.g.,Claret&Gimenez1993). Moreover, suchtypes
v mined.
of EBs are interesting for searching for tidally induced
i The Kepler space telescope mission (Borucki et al.
X oscillations occurring as high-frequency p-modes in the
r 1Thu¨ringer Landessternwarte Tautenburg, Sternwarte 5, convectiveenvelopesofthecomponents(e.g.,Fulleretal.
a D-07778Tautenburg,Germany;[email protected] 2013)orastheresultofaresonancebetweenthedynamic
2BajaAstronomicalObservatoryofSzegedUniversity,H-6500 tides and one or more free low-frequency g-modes as ob-
Baja,Szegediu´t,Kt. 766,Hungary;[email protected] served in KOI 54 (Welsh et al. 2011). Tidally induced
3Massachusetts Institute of Technology, Department of
pulsations allow us to study tidal interaction that im-
Physics,77MassachusettsAvenueCambridge,MA02139-4307,
USA;[email protected] pacts the orbital evolution as well as the interiors of the
4California Institute of Technology, Division of Geological involved stars.
and Planetary Sciences, 1200 E California Blvd MC 150-21, KIC7177553(TYC3127-167-1)islistedinthecatalog
Pasadena,CA91125,USA;[email protected]
5California Institute of Technology, Astronomy Dept. MC Detection of Potential Transit Signals in the First Three
249-17, 1200 E. California Blvd., Pasadena, CA 91125, USA; QuartersofKeplerMissionData(Tenenbaumetal.2012)
[email protected] with an orbital period of 18d. Armstrong et al. (2014)
6German Aerospace Center (DLR), Institut fu¨r Planeten-
combined the Kepler Eclipsing Binary Catalog (Prˇsa
forschung, Rutherfordstraße 2, 12489 Berlin, Germany;
[email protected] et al. 2011; Slawson et al. 2011) with information from
7Astronomical Department, E¨otvo¨s University, H-1118 Bu- the Howell-Everett (HES, Everett et al. 2012), Kepler
dapest, P´azma´ny P´eter stny. 1/A, Hungary; e.forgacs- INT (KIS, Greiss et al. 2012) and 2MASS (Skrutskie
[email protected]
2 Lehmann et al.
et al. 2006) photometric surveys to produce a catalog and we saw only one to three components.
of spectral energy distributions of Kepler EBs. They es- Next, we used the least squares deconvolution (LSD,
timate the temperatures of the primary and secondary Donati et al. 1997)) technique and calculated LSD pro-
components of KIC 7177553 to be 5911±360K and fileswiththeLSDcodebyTkachenkoetal.(2013),based
5714±552K,respectively,andderiveR /R =0.89±0.28 on the same line mask as we used for our before men-
2 1
for the ratio of the radii of the components. tioned synthetic spectrum. Figure1 shows the LSD pro-
In a recent survey of eclipse timing variations (ETVs) files vertically arranged according to the JD of observa-
ofEBsintheoriginalKeplerfield(Borkovitsetal.2015, tion. The resolution is distinctly better than that of the
B15 hereafter) low amplitude (∼90sec) periodic ETVs CCFs, four components are now seen in the majority of
werefoundandinterpretedastheconsequenceofdynam- theLSDprofiles. ThemeasuredRVsaremarkedandthe
icalperturbationsbyagiantplanetthatrevolvesaround calculated orbital curves plotted in Fig.1 show that we
the eclipsing binary in an eccentric, inclined, 1.45year could assign all components in all LSD profiles to four
orbit. In order to confirm or reject the hypothesis of different stars located in two different binary systems,
the presence of a non-transiting circumbinary planet in S and S .
1,2 3,4
the KIC 7177553 system, we carried out spectroscopic The RVs of the four components, C to C , were de-
1 4
follow-up observations. This article describes the analy- terminedbyfittingtheLSDprofileswithmultipleGaus-
sis of these observations that led to an unexpected find- sians. The mean (internal) errors of the fit were 0.25,
ing,namelythatKIC7177553isaSB4starconsistingof 0.36, 0.28, and 0.37kms−1 for the RVs of C to C , re-
1 4
two binaries. This fact makes the star an extraordinary spectively. We used the method of differential correc-
object. Asapossiblequadruplesystemitspropertiesare tions(Schlesinger1910)todeterminetheorbitsfromthe
important for our understanding of the formation and RVs. Becauseoftheshorttimespanoftheobservations,
evolution of multiple systems (e.g. Reipurth et al. 2014) this could be done for the two systems separately, with-
and may be very important for the theory of planet for- out accounting for any interaction between them or for
mation in multiple systems (e.g. Di Folco et al. 2014) if light-travel-time effects. In the case of S , we fixed the
1,2
one of the binaries in KIC 7177553 hosts the suspected orbital period to the value known from the Kepler light-
giant planet. curve analysis. The RVs of the four components of KIC
7177553 determined from the multi-Gaussian fit to the
LSD profiles are listed in Table5 in the Appendix.
Figure2 shows the RVs and orbital curves folded with
2. SPECTROSCOPIC INVESTIGATION
the orbital periods. Table1 lists the derived orbital pa-
2.1. Observations and Data Reduction rameters. The times of observation of all spectra are
New spectra were taken in May to July 2015 with based on UTC. To be consistent with the Kepler DCT
the Coude-Echelle spectrograph attached to the 2-m Al- time scale and the results listed in Table3, we added
fred Jensch Telescope at the Thu¨ringer Landesstern- 68sec to the calculated times of periastron passage T.
warte Tautenburg. The spectra have a resolving power The O-C residuals after subtracting the orbital solu-
of 30000 and cover the wavelength range from 454 to tions from the RVs are shown in Fig.3, together with
754nm. The exposure time was 40 min, allowing for straightlinesresultingfromalinearregressionusing2σ-
the V =11m. 5 star for a typical signal-to-noise ratio (SN clipping to reject outliers. The slopes of the regres-
hereafter) of the spectra of about 100. The dates of ob- sion lines are −5.3 ± 1.5, −6.9 ± 2.0, 1.6 ± 1.8, and
servation are listed in Table5 in the Appendix together 2.6±1.5kms−1yr−1 for C1 to C4, respectively. These
with the measured radial velocities (RVs hereafter, see slopes describe a change in the systemic velocities of the
Sect.2.2). two binaries or additional RV components of the single
ThespectrumreductionwasdoneusingstandardESO- objects not included in our Keplerian orbital solutions.
MIDAS packages. It included filtering of cosmic rays, They are different from zero by 3.5 times the 1σ error
bias and straylight subtraction, optimum order extrac- bars for C1 and C2 but much less or not significantly
tion, flat fielding using a halogen lamp, normalization different from zero for C3 and C4. There are no outliers
to the local continuum, wavelength calibration using a anymorewhenusing3σ-clippingforthelinearregression,
ThArlamp,andmergingoftheEchelleorders. Thespec- however, and all slopes turn out to be non-significant.
tra were corrected for small instrumental shifts using a The typical accuracy in RV that we can reach with
larger number of telluric O lines. our spectrograph and reduction methods (without us-
2
ing an iodine cell) for a single-lined, solar-type star with
such sharp lines and spectra with SN of 100 is of about
2.2. Radial Velocities and Orbital Solutions
150ms−1. The rms as listed in Table1 is distinctly
In a first step, we used the cross-correlation of the ob- higher. To check for periodic signals possibly hidden in
served spectra with an unbroadened synthetic template theO-Cresiduals,weperformedafrequencysearchusing
spectrum based on the 487 to 567 nm metal lines region the Period04 program (Lenz & Breger 2005). It did not
(redwards of Hβ, where no stronger telluric lines occur) reveal any periodicity in the residuals of any of the com-
to look for multiple components in the cross-correlation ponents. Weassumethatthehigherrmsresultsfromthe
functions(CCFshereafter). Thetemplatewascalculated fact that we are dealing with an SB4 star using multi-
withSynthV(Tsymbal1996)forTeff of6200K,basedon Gaussian fits to disentangle the blended components in
a model atmosphere calculated with LLmodels (Shulyak the LSD profiles and to derive the RVs and assume that
et al. 2004). Surprisingly, there were two spectra where the listed rms stand for the measurement errors.
wecouldclearlyseefourcomponentsintheCCFs. Inall
otherspectra,thecomponentsweremoreorlessblended
The Quadruple System KIC 7177553 3
Fig. 1.— LSD profiles calculated from the 29 spectra, vertically arranged according to the JD (2457000+) of observation. Measured
RVsaremarkedbyopencirclesandcrosses,intheleftpanelforcomponentsC1 andC2 andintherightpanelforC3 andC4,respectively.
Theredandgreencurvesillustratethecorresponding,calculatedorbitalsolutions.
Fig. 2.— RVs and calculated orbital curves folded with the orbital periods. Left: C1 (open circles) and C2 (open squares). Right: C3
(opencircles)andC4 (opensquares). Phasezerocorrespondstothetimeofperiastronpassage.
Fig. 3.— O-C values in kms−1 versus BJD (2457000+) for components C1 to C4. The straight lines result from a linear regression
calculatedfromallRVsshownbyfilledcircles. Outliersareindicatedbyopencircles.
2.3. Spectrum Decomposition were removed by comparing the KOREL output spectra
withthemeancompositespectrumandapplyingcontin-
We used the KOREL program (Hadrava 1995, 2006)
uum corrections based on spline fits.
provided by the VO-KOREL8 web service (Sˇkoda &
The resulting orbital parameters are listed and com-
Hadrava2010)fordecomposingthespectra. Allowingfor
pared to those obtained in Sect.2.2 in Table1. The
timely variable line strengths of all four components, we
Fourier transform-based KOREL program does not de-
gotsmoothdecomposedspectrawithonlyslightundula-
liver the systemic velocity and also does not provide the
tionsinthesinglecontinuaastheyaretypicalforFourier
errors of the derived parameters. The parameter errors
transform-based methods of spectral disentangling (see
were calculated by solving the orbits with the method
e.g. Pavlovski & Hensberge 2010). These undulations
of differential corrections using the orbital parameters
and line shifts delivered by the KOREL program as in-
8 https://stelweb.asu.cas.cz/vo-korel/ put. Comparingtheresultswiththoseobtainedfromthe
4 Lehmann et al.
TABLE 1
Orbital solutions obtained from multi-Gaussian fits of the LSD profiles and with KOREL.
LSD KOREL
S1,2 S3,4 S1,2 S3,4
P (d) 17.996467(17) 16.5416(70) 17.996467(17) 16.5490(38)
e 0.3984(31) 0.4437(35) 0.4008(17) 0.4421(12)
ω(◦) 6.51(63) 332.30(75) 6.76(42) 331.17(27)
T (BJD) 2457184.067(27) 2457186.505(34) 2457184.039(21) 2457186.457(13)
q 0.9457(71) 0.9664(78) 0.9359(36) 0.9612(27)
C1 C2 C3 C4 C1 C2 C3 C4
K (kms−1) 54.70(26) 57.84(34) 50.06(30) 51.80(28) 54.94(17) 58.70(13) 50.148(94) 52.17(11)
γ (kms−1) −15.98(17) −15.77(22) −14.30(19) −14.43(18)
rms(kms−1) 0.873 1.17 0.936 0.871 0.45 0.35 0.26 0.30
Mmin (M(cid:12)) 1.054(14) 0.997(12) 0.663(9) 0.641(9) 1.0871(64) 1.0174(71) 0.6758(34) 0.6496(31)
Note. —ThetablelistsperiodP andorbitalelements(eccentricitye,argumentofperiastronω,timeofperiastronpassageT,andmass
ratioq)ofthetwosystems,andtheRVsemi-amplitudesK andindividualsystemicvelocitiesγ. Errorsaregiveninunitsofthelasttwo
digitsinparentheses. ThelasttworowslistthermsoftheresidualsaftersubtractingtheorbitalsolutionfromtheRVsandtheprojected
massescalculatedfromthespectroscopicmassfunction.
LSD-based RVs, we see that there is agreement within
TABLE 2
the1σerrorbars. ThermsoftheresidualsoftheKOREL Atmospheric parameters of the four stars.
solutions for the single components is distinctly lower,
however. The last row lists the projected masses calcu- C1 C2 C3 C4
lated from the spectroscopic mass functions. It can be [M/H](dex) 0.00(11) -0.10(13) -0.12(13) -0.12(13)
T (K) 5800(130) 5700(150) 5600(150) 5600(140)
seen that the minimum masses derived for system S eff
3,4 logg (c.g.s.) 4.75(38) 4.55(40) 4.63(38) 4.59(35)
are distinctly lower than those of S which implies dif-
1,2 v (kms−1) 1.76(60) 1.23(63) 1.29(65) 1.02(59)
ferent viewing angles for the two systems. turb
vsini (kms−1) 1.3(4.2) 3.9(3.7) 5.8(3.6) 2.5(4.2)
f 0.30 0.23 0.24 0.23
2.4. Spectrum Analysis
Note. — Errors are given in parentheses, in units of the last
We used the spectrum synthesis-based method as de- digits. f isthecontinuumfluxratioofthecomponents.
scribed in Lehmann et al. (2011) to analyze the decom-
malizethespectra,wesolvedEqn.A8(seetheAppendix)
posedspectraofthefourcomponents. Themethodcom-
for each combination of atmospheric parameters.
pares the observed spectra with synthetic ones using a
The results of the analysis are listed in Table2. The
huge grid in stellar parameters. A description of an
given errors where obtained from χ2 statistics as de-
advanced version of the program can also be found in
scribedinLehmannetal.(2011). Theywherecalculated
Tkachenko (2015). Synthetic spectra are computed with
from the full grid in all parameters per star, i.e. the er-
SynthV(Tsymbal1996)basedonatmospheremodelscal-
rors include all interdependencies between the different
culated with LLmodels (Shulyak et al. 2004). Both pro-
parameters of one star. We did not have enough com-
grams consider plan-parallel atmospheres and work in
puter power to include the interdependencies between
the LTE regime. Atomic and molecular data were taken
the parameters of different stars which interfere in the
from the VALD9 data base (Kupka et al. 2000).
simultaneous analysis via the flux ratios, however.
Onemainprobleminthespectrumanalysisofmultiple
systems is that programs for spectral disentangling like
3. LIGHT-CURVE ANALYSIS
KOREL deliver the decomposed spectra normalized to
the common continuum of all involved stars. To be able 3.1. Long-cadence data
to renormalize the spectra to the continua of the single For the photometric light-curve analysis we down-
stars,wehavetoknowthecontinuumfluxratiosbetween loaded the preprocessed, full Q0−Q17 Kepler long ca-
the stars in the considered wavelength range. These flux dence (LC) data series from the Villanova site10 of the
ratios can be obtained during the spectrum analysis it- Kepler Eclipsing Binary Catalog (Prˇsa et al. 2011; Slaw-
self from a least squares fit between the observed and son et al. 2011; Matijeviˇc et al. 2012). Note, the same
the synthetic spectra as we show in AppendixA. We ex- data set was used for the ETV analysis of KIC 7177553
tended our program accordingly and tested the modified which is described in detail in B15. This ∼1470d-long
version successfully on synthetic spectra. lightcurvewasfolded,binnedandaveragedfortheanaly-
The analysis is based on the wavelength interval 455- sis. Theout-of-eclipsesectionswerebinnedandaveraged
567nm that includes Hβ and is almost free of telluric equally into 0p.002-phase-length cells, while for the nar-
contributions. It was performed using four grids of at- row primary and secondary eclipses, i.e., in the ranges
mospheric parameters for the four stars. Each grid con- ofφ =[−0p.005;0p.005]andφ =[0p.737;0p.747],afour-
pri sec
sists of (step widths in parentheses) Teff (100K), logg times denser binning and averaging was applied. The
(0.1dex), v (1kms−1), vsini (1kms−1), and scaled resulting folded light curve is shown in Fig.4.
turb
solar abundances [M/H] (0.1dex). The analysis includes The light-curve analysis was carried out with the
allfourspectrasimultaneously,whicharecoupledviathe Lightcurvefactory program (Borkovits et al. 2013,
fluxratios. Toobtaintheoptimumfluxratiosandrenor- 2014). The primary fitted parameters were the initial
9 http://vald.astro.univie.ac.at/vald/php/vald.php 10 http://keplerebs.villanova.edu/
The Quadruple System KIC 7177553 5
TABLE 3
1.01
1.00 Orbital and stellar parameters of the S1,2 system.
0.99
Flux 0.98 orbitalparametersandthirdlightcontribution
elative 00..9967 PTorb (d) 157.699906.426173 ±± 00..000102017
R 0.95 TMINI 4954.545842 ± 0.00020
0.94 a(R(cid:12)) 36.76 ± 0.15
0.93
Flux 00..000024 eω (◦) 03.3.299185 ±± 00..0006110
ual 0.000 i(◦) 87.679 ± 0.055
Resid --00..000042 qspec 0.9457 ± 0.0071
-0.25 -0.20 -0.15 -0.10 -0.05 0.00 l3 0.453 ± 0.043
Phase Fixedcoefficientsanddeducedstellarparameters
Fig. 4.— Kepler light curve of KIC7177553. Upper panel: C1 C2
Folded, binned, averaged light curve (red circles) together with xbol 0.6937 0.6954
themodelsolution(blackline). Lowerpanel: Residuallightcurve. ybol 0.1603 0.1692
x 0.6662 0.6700
kep
epoch T0, orbital eccentricity (e), argument of perias- ykep 0.1923 0.1924
tron (ω), inclination (i), fractional radii of the stars A 0.50 0.50
(r = R /a), effective temperature of the secondary β 0.32 0.32
1,2 1,2
(T ), luminosity of the primary (L ), and the amount r 0.02556 ± 0.00010 0.02561 ± 0.00010
2 1
of third light (l ). The effective temperature of the pri- M (M(cid:12)) 1.043 ± 0.014 0.986 ± 0.015
mary(T )andt3hemassratio(q),aswellasthechemical R(R(cid:12)) 0.940 ± 0.005 0.941 ± 0.005
1 T (K) 5800 ± 130 5740 ± 140
abundances, were taken from the spectroscopic solution. eff
L(L(cid:12)) 0.88 ± 0.08 0.85 ± 0.08
Theotheratmosphericparameters, suchaslimbdarken-
logg (dex) 4.517 ± 0.008 4.491 ± 0.008
ing(LD),gravitybrighteningcoefficients,andbolometric
albedo were set in accordance with the spectroscopic re- ricNtohtired. l—ighTt caonndtrTiMbuINtiIona,rexBJD, y2450th0e00l+in,ela3riasntdhelopghaortiothmmetic-
bol bol
sults and also kept fixed. For LD, the logarithmic law bolometric LD coefficients, x , y the linear and logarithmic
kep kep
(Klinglesmith & Sobieski 1970) was applied, and the LD coefficients for the Kepler passband, A the coefficient for the
bolometric albedo, β the gravitational brightening exponent, and
coefficients were calculated according to the passband-
r thefractionalradius.
dependent precomputed tables11 of the PHOEBE team
(Prˇsa & Zwitter 2005; Prsa et al. 2011) which are based Finally, we allowed the amplitudes of these sinusoids
on the tables of Castelli & Kurucz (2004). Table 3 lists tobefreeparametersinthefittingprocedureinapurely
the parameters obtained from the light curve solution, mathematically way, together with the physical model-
together with some other quantities which can be cal- ingofthewell-knownellipsoidalandothereffects. These
culatedbycombiningthephotometricandspectroscopic terms have only a minor influence on the light curve so-
results. Figure4 compares our model solution with the lution, however, which is mainly based on the eclipses
observed light curve. whose amplitudes are three orders of magnitude larger.
The out-of-eclipse behavior of the light curve merits As one can see from Tables 2 and 3, the photomet-
some further discussion. The primary and secondary ric solution, and especially the third light contribution
eclipse depths are 60000 ppm and 50000 ppm, respec- to the Kepler light curve, are in good agreement with
tively. By contrast, all of the physical out-of-eclipse ef- the spectroscopic results. It confirms that the spectrum
fects, both expected and observed, are (cid:46)60 ppm. From decomposition and our derivation of spectroscopic flux
a simple Fourier series we found several residual sinu- ratios, both applied to an SB4 star for the first time,
soidal features in the folded light curve at a number of give quite reliable results.
frequencies near to higher harmonics of the orbit. The
amplitudes of these sinusoids ranged from 10 to 37 ppm. 3.2. Short-cadence data
A comparison with the amplitudes expected from dif-
The combined spectroscopic-photometric analysis re-
ferent physical effects gives the following picture, where
vealedthattheKIC7177553systemconsistsoffourvery
our estimations are based on Carter et al. (2011a): The
similar solar-like main sequence stars and thus we might
asymmetric ellipsoidal light variation amplitudes are ∼5
expect to find solar-like oscillations in the Kepler light
ppm and ∼60 ppm at apastron and periastron, respec-
curve. The frequency of maximum power, ν , can be
tively. The Doppler boosting (DB) effect has a peak max
estimated using the scaling relation
value of ∼300 ppm for each one of the stars, but their
velocities are 180◦ out of phase. Since the luminosity M/M (T /T )3.5
ν = (cid:12) eff eff,(cid:12) ν (1)
of the two stars differs by only ∼3% (see Table3), this max L/L max,(cid:12)
(cid:12)
leads to a net DB amplitude of not more than 10 ppm.
Theillumination(orreflectioneffect)rangesbetween∼4 (Brown et al. 1991). From the values given in Tables2
ppm at apastron to ∼30 ppm at periastron, after taking and3weestimatethatν shouldlieintherange2650-
max
into account that the first order terms at the orbital fre- 3700µHz, or 230-320d−1, far beyond the Nyquist fre-
quency essentially cancel because of the twin nature of quency of the LC data of 24.469d−1. Unfortunately,
the two stars. only one short-cadence (SC) run spanning 30d exists,
having a Nyquist frequency of greater than 700d−1. We
11 http://phoebe-project.org/1.0/?q=node/110 clipped off the four eclipses that appear in this segment
6 Lehmann et al.
Fig. 5.—PeriodogramoftheKeplerSCdataofKIC7177553.
and ran it through a high-pass filter with a cuton fre-
quency of 0.5d−1. Figure5 shows the result of a Fourier
transform-based frequency search. No hint to solar-like Fig. 6.— Stacked image produced from the six NIRC2 frames
showing the two well-separated binaries of KIC7177553. The in-
oscillationscouldbefound. Onlytwoisolatedpeakscor-
tensityisdisplayedonalogarithmicscale.
responding to periods of 3.678 and 3.269 min appear.
These are known artifacts of the Kepler SC light curves 0.4(cid:48)(cid:48). Foreachcalibratedframe,wefitatwo-peakPSFto
representing the 7th and 8th harmonics of the inverse of measure the fluxratio and on-skyseparations. We chose
the LC sampling of 29.4244 min (see e.g. Gilliland et al. to model the PSF as a Moffat function with a Gaussian
2010). No further prominent peaks or typical bump in component. The best-fit PSF model was found over a
the periodogram could be detected. circular area with a radius of 10 pixels around each star
Solar-like oscillations have been found for most of the (thefullwidthathalf-maximumofthePSFwasabout5
investigated solar-type and red subgiant and giant stars pixels). More details of the method can be found in Ngo
(e.g. White et al. 2011; Chaplin & Miglio 2013) and, at et al. (2015).
firstsight,thelackofsuchfindinginourdatamightseem For each image, we also computed the flux ratio by
surprising. There is a simple explanation, however. The integrating the best-fit PSF model over the same circu-
pulsation amplitude strongly decreases with increasing lar area. When computing the separation and position
νmax (Campante et al. 2014), making solar-like pulsa- angle, weappliedtheastrometriccorrectionsfromYelda
tions less detectable for stars of higher log(g) (or easier et al. (2010) to account for the NIRC2 array’s distor-
to detect for giant than for main sequence stars). The tionandrotation. Finally,wereportthemeanvalueand
detectability, on the other hand, depends on the noise the standard error on the mean as our measured val-
background,theapparentbrightnessofthestar,andthe ues of flux ratio (primary to secondary), separation, and
length of the observation cycle. Chaplin et al. (2011) position angle of the system, which are 1.018 ± 0.005,
performedaninvestigationofthedetectabilityofoscilla- 410.4±1.5mas, and 193.6±0.2deg E of N, respectively.
tions in solar-type stars observed by Kepler. From their
Figure6 it can easily be seen that our object is too faint 5. CONFIGURATION OF THE QUADRUPLE
(or the observing period too short), and the detection
As was mentioned in the Introduction, the B15 study
probability based on one SC run is close to zero.
found low amplitude, ∼1.45yr-period ETVs in KIC
7177553, which they interpreted as the perturbations by
4. DIRECT IMAGING
agiantplanet. Theshortperiodof1.45yr,togetherwith
An examination of the UKIRT J-band image of KIC a total mass of both binaries of about 4M estimated
(cid:12)
7177553 showed only a single stellar image. From the from the spectrum analysis, would yield a separation of
lack of any elongation of the image, we concluded that the two systems of only 2AU and dynamically forced
the angular separation of the two binaries is (cid:46)0.5(cid:48)(cid:48). To ETVs of the order of 1.5 hours (see Eqn. 11 in B15).
obtain better constraints, we imaged the object on 2015 It is evident from the results of direct imaging that the
October 26 UT with the NIRC2 instrument (PI: Keith 16.5d-periodbinarylocatedinsuchadistantsystemand
Matthews)onKeckIIusingKsband(centralwavelength separated by ∼0.4 arcsec cannot produce such a signal.
2.146µm) natural guide star imaging with the narrow We also conclude that there is no evidence for ei-
camera setting (10maspixel−1). To avoid NIRC2’s nois- ther light-travel-time effect or dynamical perturbations
ierlowerleftquadrant, weusedathree-pointditherpat- caused by the 16.5d binary in the ∼4year-long Kepler
tern. We obtained 6 images with a total on-sky integra- observations of KIC 7177553. This fact does not elimi-
tion time of 30 seconds. We used dome flat fields and natethepossibilitythatthetwobinariesformaquadru-
dark frames to calibrate the images and also to find and plesystem,ofcourse,butindicatesthattheperiodofthe
remove image artifacts. orbital revolution of the two binaries around each other
The resultant stacked AO image is shown in Fig.6, must exceed at least a few decades.
where we see two essentially twin images separated by Based on the quite reasonable approximation that all
The Quadruple System KIC 7177553 7
fourstarsareofspectraltypeG2Vandtakingthe(total) to (1) and (2) for the circular orbit case, except that
apparent magnitude, m =11.629±0.020, and color in- we now must introduce two more unknown quantities,
V
dex,B−V =0.741±0.028,fromEverettetal.(2012)and namely e, the outer eccentricity, and ω, the correspond-
M =4.82±0.01,(B−V) =0.650±0.01fromthesolar ing longitude of periastron. These expressions can be
V 0
values as given in Cox (2000), we estimate that the dis- written schematically as:
tancetothisquadruplesystemisD =406±10pc. Inthat (cid:112)
case, theprojectedphysicalseparationiss=167±5AU s=a f1(e,i,ω,E)+f2(e,i,ω,E) (7)
and the orbital period P must be larger than 1000yr (cid:114)
GM
(note that we assigned to the errors of D and s twice ∆γ= g(e,i,ω,E) (8)
a
the values that would follow from error propagation, ac-
counting for the approximation that all four stars have whereE istheeccentricanomalyatthetimeofourmea-
identical properties). surements. The explicit expressions for f and g are:
Becausewemeasureonlytwoinstantaneousquantities
(cid:104) (cid:112) (cid:105)2
related to the outer orbit, s, the separation of the two f = (cosE−e)cosω− 1−e2sinEsinω (9)
1
components,and∆γ,therelativeradialvelocitybetween
the two binaries (or difference in γ-velocities), we obvi- f =(cid:104)(cid:112)1−e2sinEcosω+(cosE−e)sinω(cid:105)2cos2(i10)
2
ouslycannotuniquelydeterminetheorbitalpropertiesof
√
the quadruple system. However, we can set some quite (cid:0) 1−e2cosEcosω−sinEsinω(cid:1)sini
meaningful constraints on the outer orbit. g= (11)
1−ecosE
Starting with the simpler circular orbit case, we can
show that: As we did for the circular orbit case, we choose i from
an isotropic distribution, and choose specific values for
(cid:113)
s=a cos2i+cos2φsin2i (2) both s and ∆γ based on their measured values and as-
sumed Gaussian distributed uncertainties. We choose
(cid:114)
GM the longitude of periastron, ω from a uniform distribu-
∆γ= cosφsini (3)
a tion, as is quite reasonable.
Finally,weneedtochoosearepresentativevalueofthe
where a is the orbital separation, φ the orbital phase,
outereccentricitytoclosetheequations. Forthis,weuti-
i the orbital inclination angle, and M the total system
lized the distribution of eccentricities of the outer orbits
mass. The unknown orbital phase can be eliminated to
of 222 hierarchical triple systems found in the Kepler
find a cubic expression for the orbital separation:
field (B15). This distribution has a maximum at about
(cid:18)∆γ2(cid:19) e=0.35. AlthoughtheouterorbitofKIC7177553istwo
a3 +a2cos2i−s2 =0 (4) orders of magnitude larger than is typical for the Kepler
GM
triples, we consider the derived distribution as a plausi-
bleproxyforquadruplesystemsconsistingoftwowidely
In spite of the fact that we do not know the orbital
separated close binaries. Therefore, we chose e = 0.35
inclination angle, i, we can still produce a probability
as a statistically representative value for the eccentricity
distribution for a (and hence P ) via a Monte Carlo
orb
in KIC 7177553. Equations (6) and (7) are then solved
approach. For each realization of the system we choose
numerically for a by eliminating E.
a random inclination angle with respect to an isotropic
Figures7 to 9 show the probability density functions
setoforientationsoftheorbitalangularmomentumvec-
(PDFs) resulting from 108 Monte Carlo trials, together
tor. In addition, because there are uncertainties in the
with the corresponding cumulative probability distribu-
determination of s (accruing from the uncertainty in the
tions. We obtain highly asymmetric PDFs with maxima
distance), and in ∆γ (see Table1), we also choose spe-
cific realizations for these two quantities using Gaussian at P = 1180yr, s˙ = 1.38masyr−1, and Θ˙ = 0.28◦yr−1
random errors. In particular we take s = 167±5AU for circular orbits. Allowing for non-circular orbits, the
and ∆γ = 1.5±0.28kms−1, both as 1-σ uncertainties. peakinthePDFisshiftedtoP =2060yrandthetailof
We then solve Eqn. (3) for a, and we also record the thePDFextendstomorethan10000yr. Table4liststhe
corresponding value of i. If any inclination leads to a confidence intervals obtained from the cumulative prob-
non-physical solution of Eqn. (3), we discard it, includ- ability distributions. Here, 1σ is a formal designation
ing the value of i that led to it. We repeat this process corresponding to 67.27% probability as in the case of a
some 108 times to produce our distributions. normal distribution.
In a similar fashion, for each realization of the system, We also know from the eclipsing binary light curve
wecanalsocomputetheexpectedskymotionofthevec- solution of the 18-day binary, that its orbital inclina-
tor connecting the two stars. In particular, we find: tion with respect to the observer’s line of sight is i1 =
87◦.68±0◦.06. If the constituent stars in the two binaries
s˙ =− 2π (cid:16)a(cid:17)2sinφcosφsin2i (5) are very similar, as their spectra suggest, then the or-
s P s bital inclination angle of the 16.5d binary must be close
orb
2π (cid:16)a(cid:17)2 to60◦ inorderforthemassfunctiontoyieldmassesclose
Θ˙ = cosi (6) to 1M . This implies that the two orbital planes must
P s (cid:12)
orb be tilted with respect to each other by at least ∼25-30◦.
where Θ is commonly referred to as the “position angle”
6. CONCLUSIONS
on the sky.
We can also find a P distribution for the eccentric The analysis of the radial velocity curves derived from
orb
orbit case. We do this by deriving equations analogous the LSD profiles revealed the SB4 nature of the KIC
8 Lehmann et al.
four stars are of comparable spectral type. The errors in
1
the derived atmospheric parameters are relatively large,
0.8 however. The reason is that we had to solve for the flux
y ratios between the stars as well and thus the number of
bilit 0.6 the degrees of freedom in the combined analysis is high.
a
ob 0.4 In the result, all atmospheric parameters agree within
pr 1σ oftheerrorbars. Whatwecansayisthatallcompo-
0.2
nentsofthetwosystemsaremainsequenceG-typestars
0 showing abundances close to solar, i.e., we are dealing
0 2000 4000 6000 8000 10000 with four slowly rotating, non-evolved, solar-like stars.
P (yr) Component C1 has a slightly higher mass and shows the
higher continuum flux. It seems likely that it also has a
Fig. 7.—PDFs(continuouslines)andcorrespondingcumulative
higher temperature, i.e., that the obtained difference in
probabilities(dashedlines)forcircular(black)andeccentric(red)
orbits. Forabettervisualization,wescaledthePDFsbyafactor Teff compared to the other three stars (Table2) is sig-
of100. nificant. The eclipsing binary light-curve analysis of the
Kepler long-cadence photometry confirmed the spectro-
1 scopic results.
KIC 7177553 most likely belongs to the rare known
0.8
SB4 quadruple systems consisting of two gravitationally
y
abilit 0.6 bbyoutnhdebsiimnairlaiers.spTechtirsalastsyupmesptaionnd iaspsptarroenngtlymsaugpnpitourdteeds
prob 0.4 ofthetwobinaries,aswellasbytheirsmallangularsep-
0.2 aration and similar γ velocities. In particular, several
AO surveys (see e.g. Bowler et al. 2015; Ngo et al. 2015)
0
show that for nearby objects (up to few hundred par-
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
sec)almosteverypairofstarsthatisthisbrightandthis
ds/dt (mas/yr) closelyseparatedhasbeenshowntobephysicallyassoci-
Fig. 8.— As Fig.7 but for s˙ and circular orbits. The PDF was ated. But as is the case of KIC 4247791 (Lehmann et al.
scaledbyafactorof0.5. 2012),thetimespanofactualspectroscopicobservations
istooshorttosearchforanygravitationalinteractionbe-
tween the two binaries and we neglected such effects in
1
the calculation of their individual binary orbits.
0.8 The analysis of the O-C values from fitting the RV
y curvesshowedthattherearenosignificantchangesinthe
bilit 0.6 systemic velocities during the time span of the spectro-
a
ob 0.4 scopic observations. Only when defining outliers based
pr on a 2σ-clipping did we observe a decrease of the sys-
0.2
temic γ-velocity of the 18d-binary. Though it is not
0 accompanied by a corresponding increase of the same
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 orderoftheγ-velocityoftheotherbinary,itcouldbere-
dΘ/dt (deg/yr) latedtoathirdbodyinthe18d-binarysuchasthegiant
planet we are searching for. In a similar way, the anal-
Fig. 9.—AsFig.7butforΘ˙ andcircularorbits. ThePDFwas ysis of the eclipse timing variations of the eclipsing pair
scaledbyafactorof0.1.
which is described in B15 neither proves nor refutes the
probable gravitationally bounded quadruple system sce-
TABLE 4 nario, but indicates a possible non-transiting circumbi-
1σ confidence intervals. nary planet companion around the 18-day eclipsing pair
of KIC 7177553.
P 1660–2640yr
Theouterorbitalperiodofthequadruplesystemmust
s˙ 0.67–1.21masyr−1
Θ˙ 0.14–0.25◦yr−1 be longer than 1000 yr and the corresponding probabil-
Pe 2070–3550yr ity distribution peaks at about 1200yr for circular and
Note. —Derivedfromthecumulativeprobabilitydistributions at about 2000yr when assuming eccentric orbits with
forcircularorbits(P,s˙,Θ˙)andforanorbitwithe=0.35(Pe). e = 0.35. Thus, no orbital RV variations can be mea-
sured from spectroscopy in the next decade. Further
spectroscopic observations are valuable to extend the
7177553 system and allowed us to calculate precise or- search for the hypothetical planetary companion, how-
bital solutions for the two underlying binaries. The or- ever.
bital parameters derived from the LSD profiles are con- ThepropermotionofKIC7177553inRAcos(Dec)and
sistentwiththosedeliveredbytheKORELprogram. We Dec is given in the Tycho-2 catalog (Høg et al. 2000)
find two eccentric systems having only slightly different as 6.6 and 14.5masyr−1, respectively. This is distinctly
binary periods and consisting of components of almost larger than the expected change of the projected separa-
the same masses. The systemic velocities of both sys- tionofthetwobinariesontheskyperyear,s˙,andfuture
tems differ by only 1.5kms−1. measurements will show if both binaries have the same
Theanalysisofthedecomposedspectrashowedthatall
The Quadruple System KIC 7177553 9
proper motion. But also s˙ as well as a change of the fredJenschTelescopeoftheThu¨ringerLandessternwarte
position angle, Θ˙, can be measured from speckle inter- Tautenburg. IthasmadeuseofdatacollectedbytheKe-
ferometry or direct imaging using adaptive optics in the pler satellite mission, which is funded by the NASA Sci-
next few years. The observations with the Keck NIRC2 ence Mission directorate. Some of the data presented
camera, for example, yield an accuracy of about 1.5mas herein were obtained at the W.M. Keck Observatory,
inradialseparationandabout0◦.2inpositionanglewhich which is operated as a scientific partnership among the
could be sufficient for a secure measurement of s˙ and Θ˙ CaliforniaInstituteofTechnology,theUniversityofCal-
ifornia and the National Aeronautics and Space Admin-
for an epoch difference of two to three years (see Table4
istration. The observatory was made possible by the
for a comparison). Finally, the Gaia satellite mission
generous financial support of the W.M. Keck Founda-
(Eyer et al. 2015) with its 24µas astrometric accuracy
tion. This work has made use of the VALD database,
for V = 15mag stars and spatial resolution of 0.1mas
operated at Uppsala University, the Institute of Astron-
in scanning direction and 0.3mas in cross-direction can
omyRASinMoscow,andtheUniversityofVienna. The
resolve the two binaries easily. The non-single stars cat-
project has been supported by the Hungarian OTKA
alog is scheduled to appear in the fourth Gaia release at
Grant K113117.
the end of 2018.
This work is based on observations with the 2-m Al-
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APPENDIX
SPECTROSCOPIC FLUX RATIOS FROM THE DECOMPOSED SPECTRA OF MULTIPLE
SYSTEMS
Let I be the observed, composite spectrum showing the lines of n stars, normalized to the total continuum of all n
stars, and I , k =1..n, the decomposed spectra of the single stars normalized to the same total continuum,
k
n
(cid:88)
I = I −(n−1) (A1)
k
k=1
or, in line depths with i=1−I,
n
(cid:88)
i= i . (A2)
k
k=1
The decomposed spectra shall be fitted by the synthetic spectra S , which are normalized to the individual continua
k
ofthesinglestars. Intheidealcasewehavei =f s , wheres =1−S arethelinedepthsofthesyntheticspectra,
k k k k K
(cid:80)
f is the flux ratio of star k compared to the total continuum flux, and f = 1. In the following we consider the
k k
simplest case of constant f (the derivation can easily be extended to wavelength-dependent f by developing f (λ)
k k k
into a polynomial in wavelength λ). We define
(cid:118)
χ2 =(cid:117)(cid:117)(cid:116)(cid:88)(cid:88)n (ik−fksk)2 (A3)
σ2
λ k=1 k
where σ is a weighting factor corresponding to the mean uncertainty of the observed spectrum i . Setting
k k
n
(cid:88)
f =1− f , (A4)
1 j
j=2
we minimize
    2 
(cid:88)σ12 i1−1−(cid:88)n fjs1 +(cid:88)n (ik−σf2ksk)2. (A5)
λ 1 j=2 k=2 k
Setting the partial derivatives of Eqn.A5 with respect to f to f to zero yields, with k =2..n,
2 n
     
n
(cid:88)σs12 i1−1−(cid:88)fjs1− σsk2(ik−fksk)=0. (A6)
λ 1 j=2 k
Sorting by the f and dividing by [s2]/σ2, we obtain, with k =2..n,
k 1 1
σ12[s2k]f +(cid:88)n f =1+ σ12[skik]−σk2[s1i1] (A7)
σ2 [s2] k j σ2[s2]
k 1 j=2 k 1
wherethebracketsmeanthesumoverallwavelengthbins. EquationA7isequivalenttothesystemoflinearequations
a f + f +....+ f =h
2 2 3 n 2
f +a f +....+ f =h
2 3 3 n 3 (A8)
: : : :
f + f +....+a f =h
2 3 n n n
with
σ2[s2]
a =1+ 1 k (A9)
k σ2 [s2]
k 1
σ2[s i ]−σ2[s i ]
h =1+ 1 k k k 1 1 (A10)
k σ2[s2]
k 1
ThefluxratiosfollowfromsolvingEqn.A8. Doingthisusingagridofatmosphericparametersp tocalculatedifferent
k
synthetic spectra s (p ) finally yields the optimum set of atmospheric parameters together with the corresponding
k k
optimum flux ratios.