Spin-orbit resonance, transit duration variation and possible secular perturbations in KOI-13

0.57 MB·English

by  Gy. M. Szabó

Mon.Not.R.Astron.Soc.000,1–5(2010) Printed10January2012 (MNLATEXstylefilev2.2)
Spin–orbit resonance, transit duration variation and
possible secular perturbations in KOI-13
Gy. M. Szabo´1,2⋆, A. P´al 1,3, A. Derekas1, A. E. Simon1,2, T. Szalai4, L. L. Kiss1,5
2 1Konkoly Observatory of the Hungarian Academy of Sciences, PO. Box 67, H-1525 Budapest, Hungary
1 2Department of Experimental Physics and Astronomical Observatory, University of Szeged, H-6720 Szeged, Hungary
0 3Department of Astronomy, E¨otv¨os University, P´azm´any P´eter s´eta´ny 1/A, 1117 Budapest, Hungary
2 4Department of Opticsand Quantum Electronics, Universityof Szeged, D´om t´er 9., H-6720 Szeged, Hungary
5Sydney Institute for Astronomy, School of PhysicsA28, Universityof Sydney, NSW2006, Australia
n
a
J
Accepted Received;inoriginalform
9

]
R ABSTRACT
S
. KOI-13is the first knowntransiting system exhibiting light curve distortions due
h
togravitydarkeningoftherapidlyrotatinghoststar.Inthispaperweanalysepublicly
p
availableKeplerQ2–Q3short-cadenceobservations,revealingacontinuouslightvari-
-
o ationwithaperiodofProt =25.43±0.05hourandahalf-amplitude of21ppm,which
r islinkedtostellarrotation.Thisperiodisinexact5:3resonancewiththeorbitofKOI-
t
s 13.01, which is the first detection of a spin-orbit resonance in a host of a substellar
a companion.Thestellarrotationleadstostellaroblateness,whichis expectedtocause
[
secular variations in the orbital elements. We indeed detect the gradual increment of
2 the transit duration with a rate of (1.14±0.30)×10−6day/cycle. The confidence of
v this trend is 3.85-σ, the two-sided false alarm probability is 0.012%. We suggest that
1 the reason for this variation is the expected change of the impact parameter, with a
3 rateofdb/dt=−0.016±0.004/yr.Assuming b≈0.25,KOI-13.01may becomea non-
2 transitingobjectin75−100years.Theobservedrateis compatiblewiththe expected
4
secular perturbations due to the stellar oblateness yielded by the fast rotation.
.
0
Key words: planetary systems
1
1
1
:
v
i 1 INTRODUCTION On23September,2011,newKepler photometry(ShortCa-
X
denceQ3 data) became available. Here we suggest that the
KOI-13 (KIC 009941662) is a unique astrophysical labora-
r photometric data reveal the stellar rotation, and give ob-
a tory of close-in companions in an oblique orbital geometry.
servational evidence for transit duration variations (TDV)
The system consists of a widely separated common proper
which are a sign of secular perturbations.
motion binary of A-type stars, one hosting a highly irra-
diated planet candidate with Porb 1.7626 day (Borucki
≈
2011). The host star of KOI-13.01 is the brighter compo-
2 ROTATION OF THE HOST STAR
nent,KOI-13A,androtatesrapidly(vsini 65 70km/s,
≈ −
Szab´o et al. 2011). The transit curves show significant dis- Afterremovingtheorbitalphase-dependentlightvariations
tortion that is stable in shape, and it is consistent with a fromtheKepler photometryofKOI-13,wedetectedanaddi-
companion orbiting a rapidly rotating star exhibiting grav- tionalperiodicsignalthatwearguebelowisduetorotation
ity darkening byrotation (Barnes 2009). Barnes et al. 2011 of the host star. The light curve was corrected with a di-
derivedaprojectedalignmentofλ=23◦ 4◦ andthestar’s lution factor of 1.818 (Szab´o et al. 2011). The folded light
tnhoerrtehfoproelethisetisltteeldlaarwianyclfirnoamtiotnh,eio∗bs=erv4e3±r◦by4ψ◦=(a4s8s◦u±mi4n◦g, wcuervseeewtihtehePlloirpbsoisidpalloattneddbineatmhienugpvpaerriaptiaonnesl(oMf Fazigeh. 1etanald.
±
M∗ = 2.05 M ). The mutual inclination is ϕ = 54–56◦. 2011, Shporeretal.2011). Thereferencefluxvalueistaken
Companion is⊙determined to have a mass of 9.2 1.1 MJ tobethecentreofthesecondaryeclipse,whenonlyKOI-13
±
(Shporer et al. 2011) and between 4 2–6 3 MJ (Mazeh Aisvisible.Wesmoothedandsubtractedthisvariationfrom
± ±
et al. 2011), and a radius of 1.44 RJ (Barnes et al. 2011). theout-of-transitlightcurve,andde-projectedtheresiduals
into the time domain (upper middle panel). A prominent
frequency was detected in these residuals at 0.9437 1/day,
⋆ E-mail:[email protected] i.e.at25.43 0.05hourperiod(Fig.1,lowermiddlepanel).
±
(cid:13)c 2010RAS
2 Gy. M. Szab´o et al.
Freq /3
orb
250
m
s) 3
200 nit
u
y
150 ar m
pm bitr 2
p 100 ar
e (
d
50 u
mplit m 1
0 A
−50
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.00 0.05 0.10 0.15 0.20 0.25
Orbital phase
Frequency (1/d)
100
Figure 2. Fourier-transforms of the light curve shape moments
50
µ1–µ3 show modulations in the transit shape with a period of
m
p 0 3Porb. The peaks around 0.189 1/day frequency (= 1/Porb/3)
p have≈4-σ confidence.
−50
−100
200 250 300 350
The semi amplitude of this variation is 21ppm, well above
the average noise level of 1ppm in the Fourier spectrum.
Time (Kepler days)
Harmonics of this frequency can also be detected up to the
20 fourth order. This period has been found by Shporer et al.
m) 2011andMazehetal.2011independently.Thebinnedphase
p 15 diagram of the residuals with 25.43 hour period is plotted
p
e ( in the bottom panelof Fig. 1.
ud 10 Both Mazeh et al (2011) and Shporer et al (2011)
plit suggested a probable pulsational origin. Contrary to their
m
a 5 proposition,weinterpretitastherotationalperiodofKOI-
13A.Theargumentsforarotationaloriginarethefollowing:
0
10 100 The period is perfectly compatible with the expected
•
Period (h) rotational rate. Barnes et al. 2011 predicted a 22–22.5 hour
rotational period for KOI-13 A, depending on the stellar
60 mass. They did not estimate the error, but it is easy to
calculate that errors in vsini and in the stellar inclination
40 results in 3.9 hours. This prediction is in good agreement
±
with therotational origin of the new period.
20 Balona (2011) detected signs of stellar rotation in 20%
•
m of all A–F stars in the Kepler field caused by the granula-
pp 0 tionnoise,and8%ofthemalsoexhibitstarspot-likefeatures
in the light curves. Typically, these stars exhibit a domi-
−20 nant peak with 10–100 ppm amplitude at periods less than
3days,themedian is around1dayfor stars in therangeof
−40 7500–10,000K.Anotherdiagnosticisthescatterlevelatfre-
quenciesbelow<50/day,whichexponentiallyincreasesbya
factorof 1.6towardlowfrequencies/highperiods.Thepe-
≈
riodogram of KOI-13 looks exactly as described by Balona
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
(2011), and the folded light curve (Fig. 1, lower panel) is
Rotational phase also compatible with a rotating A-type star with magnetic
activity.
Figure 1. Top: out-of-transit ligh tcurve of KOI-13, corrected
Starssimilar toKOI-13Acanexhibitporg modepul-
for the third light. The reference flux level is KOI-13 A, i.e. the
•
flux measured at eclipse. After subtracting this signal from the sations(δSctandγDor)orboth(Uytterhoevenetal.2011).
timeseries(middleupperpanel),asignalwith25.432hourperiod However,theobservedpulsationshaveperiodslessthanone
is detected with high significance (lower middlepanel). Bottom: day,andseveralmodesareobservedwithsimilaramplitudes.
Thephasediagramwith25.432hperiod,afterremovingtheorbit- The general appearance of the frequency spectrum of KOI-
dependent variations. 13 does not resemble a pulsating star.
(cid:13)c 2010RAS,MNRAS000,1–5
KOI-13: spin-orbit resonance, TDV and secular perturbations 3
Mazeh et al 2011 detected four harmonics of the 25.4-
• 0.1210
hourfrequencyinQ3data,whichweconfirmhere.Thepres- s)
y
enceofsuchharmonicsistypicalforstellarrotation(Balona a
d
et al. 2011). n ( 0.1205
o
ati
It should benoted thatboth weand Mazeh et al. 2011 dur 0.1200
detected other frequencies between 1.5–2 c/day which are sit
separated about equidistantly and have amplitudes of 4–7 n
a
ppm. Their harmonics do not appear in Q2+Q3 data, and Tr 0.1195
these peaksdo look compatible with pulsation.
Thesourceofthe25.4-hourperiodisthehoststar,KOI- 16.8
13 A. This is seen from the systematic modulation of the
transit shapes. Because this period is in 3:5 ratio with the
16.7
orbitalperiod,everythirdtransitoccursinfrontofthesame
stellar surface, and a modulation of the light curve shape R*
/
is expected with aperiod of 3 transits. Sincethe individual z
16.6
lightcurvesaretoonoisyforadirectcomparison,wehaveto
combine many data points and analyse their moments. Let
usdefineµn,then-thlightcurvemomentofeachindividual 16.5
transit as er
et 0.15
m
µn := ti−DCi n∆fi (1) para 0.10
i∈{Xtransit}(cid:16) (cid:17) act
p
m 0.05
whereti andfi arethetimes and occulted fluxesbelonging e i
tcoalceuaclahteddatmaipdo-tinimtse, Dofitshtehteratnrasnits,itbadsuerdatoionntahnedepChieimsetrhies of th 0.00
in Borucki 2011. Once the moments are assigned to the in- are
dividualtransits, a time series of moments can beanalyzed qu -0.05
S
inthestandardfashion.InFig.2,weplottheperiodograms 0 20 40 60 80 100
of the first three lightcurve moments (note that the ampli- Transit number
tudes are in relative units). The periodograms show a de-
tection with 5.27 day period, with a significance of 3 4-σ Figure 3. Variations in the transit duration (top panel), the
foreachlightcurvemoments,confirmingthatthe25.4−-hour reciprocaltransitduration(middlepanel)andinthesquareofthe
impactparameterderivedfromtransitshape(b2,lowestpanel),as
period modulates the shape of the transit with a period of
thefunctionofthecyclenumber.Theplotofthereciprocaltransit
3Porb=5Prot. durationissuperimposedbythemodelofthebest-fitlineartrend
together withthe3σ errorsofthelinearregression.
3 TRANSIT DURATION VARIATIONS
2.1 The possible 5:3 spin–orbit resonance of
The analysis of the light curve moments suggested a long-
KOI-13 A
periodtime-dependentvariationinthetransitcurveshapes.
The 25.4-hour period is very close (within 0.1%) to the 5:3 By fitting each transits individually, we concluded that the
spin-orbit resonance with KOI-13.01. Because of the large duration of the transit is gradually increasing (Fig. 3, top
mutualinclinationof59◦ (Barnesetal.2011),thelongitude panel).Thesignificanceofthisfindingis3.85σ, withafalse
of the companion varies with changing velocity. Interest- alarm probability of 0.012%, based on an MCMC estimate.
ingly, the sinodical longitude of KOI-13.01 remains exactly The 3-σ confidence interval of thefitted linear regression is
constant (within 1% fluctuations) on 1/8 orbital arc sur- plotted in the middle panel of Fig. 3, onto the distribution
≈
roundingthepositions whenthecompanion isat extremely of the reciprocal transit length (which is the actual output
high/lowlatitudes.Thus,KOI-13.01andKOI-13Amoveas ofthelightcurvefitalgorithm).Wenowdescribethedetails
if they were in exact 1:1 resonance for 3 hours. of how thelight curveswere analyzed.
To date, theories of spin-orbit resonance cover the fol- First, every transit was fitted independently using the
lowingcases:(i)spatialapproximationwithrigidbodies(e.g. analytic models of Mandel & Agol (2002). This fit invokes
Makarov2011),wherethebodyinresonancehaslittlemass; symmetrictemplateswhichcannotfittheknownasymmetry
(ii): resonant orbits of massive stars (e.g. Witte & Savonije of the light curves; but since the asymmetry appears simi-
2011); (iii) resonances in systems of compact bodies (e.g. larly in each transits, we expect no time-dependent bias in
Schnittman2004).Thecaseofresonancebetweenthestellar theresults.Thefreemodelparametersare:thetransittime,
spinandtheorbitofasubstellarcompanionisyettobeex- the relative radius of the planet, the reciprocal of the half
ploredtheoretically but,ifconfirmed,itwillbeasignificant duration,ζ/R⋆,andthesquareoftheimpactparameter,b2.
findingin relation to theevolution of planetary systems. Theseparametersarealmostuncorrelatedtoeachother(Pa´l
(cid:13)c 2010RAS,MNRAS000,1–5
4 Gy. M. Szab´o et al.
2008). Indeed, our analysis shows that the inverse duration GM ∞ R n
ζ/R⋆ follows a secular trend (see Fig. 3). V(r,θ)=− R 1− Jn r Pn(cosθ) , (4)
" #
Thederivedvaluesoftheseparametersaredisplayedin Xn=2 (cid:16) (cid:17)
Fig. 3. Since KOI-13 did not exhibit a transit timing vari- where M is the total mass, R is the equatorial radius, Jn
ation, we can rule out short-term secular variations in the are constants and n are the Legendre polynomials. The
P
orbital semimajor axis of the transiting companion. There- most prominent perturbation is caused by J2, due to the
fore, we conclude that the decrease in ζ/R⋆, indicating a oblateness of the host body. MacCullagh’s Theorem allows
lengtheningof thetransit duration,is duetotheincreasing uscompute J2 using
orbital inclination (i.e. due to decreasing impact parame-
t(er3).1.T6he8o.2b)ser1v0e−d5ldin−e1acryctlree−n1d=in(ζ/1R7.⋆9 is4d.6(ζ)/R10⋆−)/5ddt−=2. J2 = M1R2 Θzz− Θxx+2 Θyy ≈ ΘzMz−RΘ2xx, (5)
S−inceth±e relat·ion between ζ/R⋆, a/−R⋆ and±b2 is· where Θxx =(cid:16)Θyy 6Θzz are th(cid:17)e principal moments of iner-
a √1 b2 ζ tia.Itisknownthatanon-zeroJ2 resultsinsecularpertur-
= − , (2) bationsintheangularorbitalelements.Namely,thesecular
R⋆ n R⋆ term in Ω (argument of ascending node) is computed as
(cid:16) (cid:17) (cid:16) (cid:17)
(see also Pa´l 2008), we can compute the time derivative of
dΩ 3 a −2 cosϕ
b as dt =−2J2n R (1 e2)2. (6)
db 1 b2 ζ −1 d ζ −
b˙ = = − . (3) (cid:16) (cid:17)
dt b R⋆ dt R⋆ Herendenotestheorbitalmeanmotion,aisthesemi-major
(cid:16) (cid:17) (cid:16) (cid:17) axis and e is theorbital eccentricity.
This was obtained by re-ordering equation (2) and assum-
It is known from vector geometry, that if an unit vec-
bing= t0h.2a5t3a/R0.⋆02i0s icnotnostthanist.eqBuyatisounb,swtiteuhtiandgb˙o=ur(be4s.t4-fit tor n precesses around the unit vector p with an angular
1.2) 10−5±d−1=( 0.016 0.004)y−1.Thisisatiny−chang±e frequency of ω0, the time derivativeof n will be
× − ±
in thetransit duration,an order of magnitudesmaller than n˙ =ω0(p n). (7)
×
theprecisionwithwhichbcanbedeterminedfromthelight
Inourcase,nistheunitvectorparalleltotheorbitalangular
curveshapein thefittingprocedure(Fig.3,bottom panel).
momentumof thetransitingbody.Sincethecomponentsof
Moreprecisely,theplanetappearedtomovedbyonly 15%
≈ nare
of its radius during the observed 100 transits, having no
≈
detectableeffectonthelightcurveshape.Consequently,we sinicosΩ
reasonably neglected the oblate shape of the rotating star n= sinisinΩ (8)
when converting transit duration to b. The stellar oblate- cosi !
ness of a 2 M star with radius 1.7 R is 3% (Murray &
Dermott,1999⊙),andlessinaninclinedp⊙rojection.Therefore andtheonlyobservablequantityisnz ≡cosi,wecanwrite
thebiasduetostellar oblatenessis1–2% inthedetermined n˙z =ω0(pxny pynx). (9)
valueofb˙,muchbelowtheaccuracyofitsdeterminedvalue. −
InSection4,wewillshowthatb˙ iscompatiblewiththesec- Bysubstitutingpx=sinipcosΩp andpy =sinipsinΩp and
ular perturbations caused by the oblateness of KOI-13 A, thecomponents nx and ny from equation (7), we obtain
therotating host star. dcosi
=ω0sinisinipsinλ, (10)
dt
4 INTERPRETATION where Ωp is the ascending node of the stellar equator, and
λisthelongitudeoftheplanet’sascendingnode,relativeto
AssumingMp =9.2MJ (seealsoBarnesetal.2011),thean- that of the ascending node of the stellar equator by defini-
gularmomentuminthestarandthecompanionaresimilar. tion.
Thus, theaxes of both theplanet orbit and thestellar spin
areprecessingaroundthetotalangularmomentumaxis,op-
posingeach other.Todate,thereisnoexacttheoryforthis 4.2 The inferred oblateness of the host star
case. In the observed examples, angular momentum of the
Assuminga circular orbit for thetransiting companion and
orbit(doublestars)orstellarspin(e.g.Mercuryorlunisolar
substitutingtheaboverelationandequation(??)intoequa-
precession) dominates theother. Onecan studythepreces-
tion (6), and by taking ω0 = dΩ/dt as the precession rate
sion of the orbital plane and the stellar spin under slightly
inductedby J2, we finally obtain
different assumptions. Wewill show here that these lead to
compatible results, and give a satisfactory estimate of the dcosi 3 a −2
= J2n (11)
stellar oblateness. dt −2 R∗ ×
(cosic(cid:16)osip(cid:17)+sinisinipcos∆Ω)
× ×
4.1 Secular J perturbations sinisinipsin∆Ω.
2 ×
Asknownfromthetheoryofsatellitemotions(Kaula1966), Since transits are observed, we can say here that cosi
≪
higherordermomentsofthegravitationalpotentialofahost sini 1.Inaddition,b=(a/R⋆)cosi,thustheaboveequa-
body yield periodic and secular perturbations in the orbits tion≈can be rearranged to give b˙ as
of nearby companions. The external gravitational potential 3 a −1
of an extended body can be expressed as b˙ = J2n sin2ipsinλcosλ. (12)
−2 R⋆
(cid:16) (cid:17)
(cid:13)c 2010RAS,MNRAS000,1–5
KOI-13: spin-orbit resonance, TDV and secular perturbations 5
For J2 we obtain ularvariations can be detected more easily in TDV than in
transittimingsforshorttimecoverage(Pa´l & Kocsis2008).
J2sin2ipsinλcosλ=(3.8±1.0)·10−5. (13) An argument against the reality of TDV can be that we
Substituting the derived stellar parameters we find J2 = fittedsymmetrical templates toa transit with knownslight
(2.1 0.6) 10−4, and dΩ/dt=(3.4 0.9) 10−5/day. asymmetry.However,weconsiderthatthedetectedtrendis
± × ± ×
Is this result compatible with a model star with the not an artifact of this kind, because (i) there is no visible
same mass, radius and rotational rate as KOI-13 A? To time-dependent trend in the asymmetric part of the light
check this, we calculated the moment of interia of n = 3 curves; (ii) the errors contain the ambiguity introduced by
polytrope models that were distorted uniform stellar ro- the asymmetric part, while the detection is still significant,
tation. The distortion of a rotating star is (R r)/R = and (iii) it is rather unlikely that the varying asymmetries
Ω2R3(2GM)−1 + 3J2/2 (Stix 2004), where R −and r are affect only thetransit duration,and leavetheotherparam-
the radii at the equator and the poles, respectively. The eters unchanged.
J2 term we derived for KOI-13 is two orders of magnitude Ininterpretingthecause oftransit duration variations,
less than the rotational term and can be neglected. The ourhypothesisofsecularJ2 perturbationsisaprobablesce-
local distorions were calculated at at each internal point nario, but other processes cannot be excluded (e.g. pertur-
and then a z-contraction was applied. We assumed dz/z = bations from an unreported outer planet; the presence of a
Ω2r3(2GMr)−1 3Ω2/(8πG ρ r), where Mr and ρ r moon around KOI-13.01). However, the stellar rotation is
− ≡ − h i h i
arethemassandthemeandensityinsidetheradiusr.This the most plausible reason for TDV, because it does not in-
modelisafirstapproximationthatneglectshigher-orderdis- voke other bodies into the explanation, and the inferred J2
tortions,butisenoughpreciseforaroughestimatebecause isfullycompatiblewith theexpectedvalue.Inotherwords,
thedistortionsareeverywheresmallerthan3%.Theresult- one would predict secular J2 perturbations of KOI-13.01 at
ing values are Θzz = 0.07760 MR2, Θxx = 0.07743 MR2, the rate detected, taking into account the rapid rotation of
J2 = 1.7 10−4. Thus, the expected value of J2 is in the the host star and the close proximity of its substellar com-
×
rangesuggestedbythetheoryofsecularperturbations,sup- panion.
porting our interpretation.
ACKNOWLEDGMENTS
4.3 Precession rate
This project has been supported by the Hungarian OTKA
Applyingthe framework developed for lunisolar precession,
GrantsK76816,K83790 andMB08C 81013, the“Lendu¨let”
theprecession rate of the star can beformulated as
Program of the Hungarian Academy of Sciences, the ESA
ddΩtp = 3GMp(Θ2azz3Θ−zΘznxx)cosϕ = 23J2GMa3pncβo.sϕ (14) garrashnitpPoEfCthSe9H8u07n3gaarniadnbAyctahdeeJm´aynoosfBScoileynacieRs.eTseiamrcBheSdcdhionlg-
is acknowledged for his comments.
HereMp isthemassoftheplanetandβ =0.07760forKOI-
13 A, as we derived in the previous section. Evaluating Eq
14, and applying the value of J2 =2.1 10−4 we find that
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