On the (im)possibility of testing new physics in exoplanets using transit timing variations: deviation from inverse-square law of gravity

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by  Yi Xie

Mon.Not.R.Astron.Soc.000,1–??(2002) Printed30January2014 (MNLATEXstylefilev2.2)
On the (im)possibility of testing new physics in exoplanets
using transit timing variations: deviation from
inverse-square law of gravity
4
1 Yi Xie1,3⋆ and Xue-Mei Deng2,3
0 1School of Astronomy and Space Science, Nanjing University,Nanjing 210093, China
2 2Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China
n 3Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University,Ministry of Education, Nanjing 210093, China
a
J

9 Accepted .Received;inoriginalform
2

] ABSTRACT
P
Ground-based and space-borne observatories studying exoplanetary transits now and
E
in the future will considerably increase the number of known exoplanets and the
.
h precisionofthe measuredtimes of transitminima. Variationsin the transit times can
p not only be used to infer the presence of additional planets, but might also provide
- opportunities for testing new physics in the places beyond the Solar system. In this
o
work, we take deviation from the inverse-square law of gravity as an example, focus
r
t on the fifth-force-like Yukawa-type correction to the Newtonian gravitational force
s
a whichparameterizesthisdeviation,investigateitseffectsontheseculartransittiming
[ variationsandanalyzetheirobservabilityinexoplanetarysystems.Itisfoundthatthe

mostoptimisticvaluesofYukawa-typeseculartransittimingvariationsareatthelevel
1 of ∼ 0.1 seconds per year. Those values unfortunately appear only in rarely unique
v
casesand, mostimportantly,they are stillat leasttwo ordersofmagnitude below the
7
current capabilities of observations. Such a deviation from the inverse-square law of
0
4 gravity is likely too small to detect for the foreseeable future. Meanwhile, systematic
7 uncertainties, such as the presence of additional and unknown planets, will likely be
. exceptionally difficult to remove from a signal that should be seen.
1
0 Key words: gravitation– celestial mechanics – planetary systems
4
1
:
v
i
X 1 INTRODUCTION planets (e.g. Holman & Murray 2005; Agol et al. 2005;
Heyl& Gladman2007;Nesvorny´ et al.2012)andstudythe
r Currently, more than 880 exoplanets have been discov-
a dynamics of multiple planets systems (e.g. Holman et al.
ered and about 300 of them are in the transiting sys-
2010; Lissauer et al. 2011; Fabryckyet al. 2012; Ford et al.
tems.1 Now and in the future, ground-based and space-
2012; Steffen et al. 2012; Nesvorny´ et al. 2013). Recently,
borne observatories used for studying transits of exo-
the Kepler mission (Koch et al. 2010; Borucki et al. 2010)
planets will considerably increase the number of known
released a catalog of transit timing measurements of the
exoplanets and the precision of the observed times of
first twelve quarters, which identifies the Kepler objects of
transit minima.2 The measured transit timing variations
interest with significant TTVs (Mazeh et al. 2013). Such a
(TTVs) can be used to infer the presence of additional
largeamountofconfirmedandpotentialtransitingexoplan-
etsprovideopportunitiesfortestingphysicallawsofnature,
especially fundamental theories of gravity,in theplaces be-
⋆ E-mail:[email protected]
yondthe Solar system.
1 http://exoplanet.eu/catalog/
2 As pointed out by Kipping (2011), the so-called “mid-transit But, what is the necessity of performing these tests in
time” in the exoplanet literature is highly ambiguous. Follow-
exoplanetary systems? After all, modified and alternative
ingtheterminologyinKipping(2011),wewilluse“transitmini-
relativistic theories of gravity have been tested in the So-
mum”and“timeoftransitminimum”inthispaper.Because,for
lar system with very high precision (see Will 1993, 2006;
a limb-darkened star, the transit minimum occurs when the ap-
Turyshev 2008, for reviews), whereas tests in exoplanetary
parent sky-projected separation between the exoplanet and the
star reaches a minimum, which has a completely unambiguous systems in current stage are expected tobe much worse for
definition. So “transit timing variations” also refers to “changes lack of high-accuracy observations. However, some obser-
oftimesoftransitminimum”. vationsindicate thefundamental constants of naturemight
(cid:13)c 2002RAS
2 Y. Xie & X.-M. Deng
haveatemporalandspatialvariation(e.g.Webb et al.1999; beavailable in thefuture. It will also provideopportunities
Murphyet al. 2001a,b,c,d; Webb et al. 2001; Murphy et al. to test the fundamental theories of gravity in quite a large
2003,2007,2008;Webbet al.2011),whichimplysomefun- number of different and unique locations beyond the Solar
damental laws of nature may also have variations in such system. This will make transiting exoplanets very similar
a manner although it ought be very small. Furthermore, to binary pulsars in testing physical laws describing grav-
scalar-tensor theories of gravitation, as alternatives of gen- ity (e.g. Bell et al. 1996; Damour & Esposito-Far`ese 1996;
eral relativity (GR), also theoretically imply these subtle Kramer et al. 2006; Iorio 2007b; Deng et al. 2009; Li 2010;
variations may exist even in the scale of planetary systems Deng 2011; Li 2011; DeLaurentis et al. 2012; Ragos et al.
due to the possible couplings between matter and scalar 2013; Xie 2013).
fields (see Fujii & Maeda 2007, for a review). In order to Hence,wewill investigatethe(im)possibility of detect-
(dis)prove it empirically, we need to go to different times ingfifth-force-likeYukawa-typeeffectsonthesecularTTVs
andplaces.Exoplanetarysystemscanserveastestbedsout- as an example of trying to test new physics in exoplanets.
side the Solar system for conducting those tests owning to Afteranalysingtheirobservabilityinexoplanetarysystems,
theiruniquelocations.Therefore,inthiswork,wewillfocus we find that the most optimistic values of this type secular
on testing theinverse-squarelaw (ISL) of gravity. TTVsareatthelevelof 0.1secondsperyear.Thosevalues
∼
A way to parameterize deviation from the ISL of unfortunately appear only in rarely uniquecases and, most
gravity, which might be caused by new physics beyond importantly, they are still at least two orders of magnitude
the standard model of particles and GR, is the fifth- below the current capabilities of observations. Such devia-
force-like Yukawa-type correction to the Newtonian grav- tion from theISL of gravity is likely too small to detect for
itational force, which has been intensively studied (e.g. theforeseeablefuture.Meanwhile,systematicuncertainties,
Fischbach et al. 1986, 1992; ?; Adelberger et al. 2003; Iorio suchasthepresenceofadditionalandunknownplanets,will
2007a; Lucchesi& Peron 2010; Haranas & Ragos 2011; likelybeexceptionallydifficulttoremovefromasignalthat
Haranas et al.2011;Lucchesi2011;Iorio2012b;Deng& Xie should be seen.
2013).The gravitational potential with this correction is The rest of the paper is organized as follows. Section
2 is devoted to describing TTVs under the Yukawa-type
V =VN(r)+VYK(r), (1) correction. In section 3, we present an analysis about its
wheretheNewtonianpotentialandYukawa-typecorrection observability in the secular TTVs. Finally, in section 4, we
are, respectively, summarize our results.
VN(r) = GM1M2, (2)
r
2 TTVS CAUSED BY YUKAWA-TYPE
VYK(r) = GM1M2αexp r . (3) CORRECTION
r (cid:18)− λ(cid:19)
To describe the dynamics of a transiting exoplanetary sys-
Here G is the gravitational constant, M (i = 1,2) is the
i tem, understand its transit light curve and represent the
massoftheithbodyandr isthedistancebetweenthem.α
observables, we adopt the coordinate systems defined and
is a dimensionless strength parameter of thecorrection and
applied in Kipping (2011). The plane of Xˆ-Yˆ is defined as
λ is a length scale for it (see Fischbach & Talmadge 1999,
the plane of the sky, the host star is at the origin O and
for a review of constraints on α and λ). Under this param-
the observer is located at (X,Y,Z)=(0,0,+ ). Then , in
eterization, if the inverse-square law of gravity is violated, theXˆ-Yˆ-Zˆ system,theinclination ofatransit∞ingexoplanet
an extraforce will exert onexoplanetsand causeadditional
i is close to 90◦. Hereafter, we take widely used notations
secular TTVs.
in celestial mechanics: a is the semi-major axis, e is the
Among mainly known sources of secular TTVs,
eccentricity, i is the inclination, Ω is the longitude of the
the general relativistic periastron advance (GRPA) con-
ascending node, ω is the argument of periastron, M is the
tributes. Its observability in exoplanets has been in-
mean anomaly and f is the true anomaly. The normalized
vestigated in several works (e.g. Miralda-Escud´e 2002;
apparent(sky-projected)separationbetweentheplanetand
Adams& Laughlin 2006a,b,c; Iorio 2006; Heyl& Gladman
thestar is defined as (Kipping 2011)
2007; Jord´an & Bakos 2008; Pa´l & Kocsis 2008; Li 2010;
Iorio 2011a,b; Li 2012; Zhao & Xie 2013). It is found that 1
S X2+Y2
GRPAcanbedetectableontimescalesoflessthanabout10 ≡ R∗p
years with current observational capabilities by observing a
= ̺(f) 1 sin2(ω+f)sin2i, (4)
thetimes of transits in exoplanets (Jord´an & Bakos 2008). R∗ q −
This means that, like the well-known phenomena in
where ̺(f) (1 e2)/(1+ecosf) and R is the radius
the Solar system, such as the anomaly in the perihelion ≡ − ∗
of the star. For mathematical convenience in the following
shift of Mercury (Nobili & Will 1986) that gave a hint at
parts of this paper, we will also use theexpression for S2:
new physics about GR and the dynamics of planets which
could be used to test fundamental laws of physics (e.g. a2
S2 = ̺2(f)[1 sin2(ω+f)sin2i]. (5)
Iorio 2005a,b,c,d; Iorio & Giudice 2006; Ruggiero & Iorio R2 −
∗
2007; Folkner 2010; Pitjeva 2010; Fienga et al. 2011; Iorio
2010; Iorio et al. 2011; Iorio 2011c, 2012a,c; Pitjeva 2012;
2.1 Transit minima
Pitjev & Pitjeva 2013; Pitjeva & Pitjev 2013; Xie& Deng
2013),observingofsecularTTVscanalsoserveasatestbed For a Kelperian transiting exoplanet, the instants of tran-
with the help of high-precision measurements which might sit minima (and maxima) occur when dS/dt = 0 (Kipping
(cid:13)c 2002RAS,MNRAS000,1–??
Impossibility of testing ISL by TTVs 3
2011),which leads to Ifatwo-bodyproblemwiththefifth-force-likeYukawa-type
correction is considered, we can find that
dS dSdf
= =0. (6) dηT h+e2 dω
dt df dt 1 = cos2i , (13)
(cid:28) dt (cid:29) −(1+h)2 (cid:28)dt(cid:29)
It is worth mentioning that condition defined by equation dη2T = k2+h+e2 cos4i dω , (14)
(6)isapuregeometriccriterionwithoutanyambiguity.Be- (cid:28) dt (cid:29) − (1+h)3 (cid:28)dt(cid:29)
causedf/dt=0forplanetaryorbitalmotions,thecondition dηT 1
dS/dt=0is6equivalenttodS/df =0.Aneasierwaytohan- 3 = (1+h)−5(2h3k2+2hk4+h2k2 2k4 2h3
(cid:28) dt (cid:29) 2 − −
dlethemathematicsistomakeuseofdS2/df =0,andsuch
dω
a condition can be proven to be equivalent to dS/df = 0 7hk2 4h2 6k2 2h)cos6i . (15)
− − − − (cid:28)dt(cid:29)
(Kipping 2011). To obtain the true anomaly at the transit
minima fT (the subscript “T” denotes transit), one needs Here,thesecularvariationsofa,e,iandΩarezeroandonly
to solve a quartic equation involving cosf [see eq. (4.5) in dω/dt contributesin theaboveones [see eqs. (18)–(22) in
h i
Kipping(2011)].Althoughsolvingitismathematicallypos- Deng& Xie(2013)]because
sible,thesolutionsareprettylengthyandimpractical.Treat-
ing cos2i as a small quantitywhich is very close tozero for dω =αna√1−e2 exp a I1 ae , (16)
transiting exoplanets and using the Newton-Raphson iter- (cid:28)dt(cid:29) eλ (cid:18)− λ(cid:19) (cid:18) λ (cid:19)
ation method, Kipping shows the series expansion solution
where I1(z) = dI0(z)/dz and I0(z) is the modified Bessel
for fT can bewritten as (Kipping2011) function of thefirst kind(Arfken & Weber2005).
However,thesecularvariationoff isnotanobservable
T
n
π practicallysothat,forrealisticmeasurements,itneedstobe
f = ω ηT, (7)
T (cid:20)2 − (cid:21)− j convertedtothesecularvariationoftimeoftransitminimum
Xj=1
t , i.e. secular TTV,
T
where, by h esinω and k ecosω, dt dt df 1 df
≡ ≡ T = T T = ̺2 T , (17)
(cid:28) dt (cid:29) (cid:28)dfT dt (cid:29) n√1 e2 T(cid:28) dt (cid:29)
k −
ηT = (cos2i)1, (8)
1 (cid:18)1+h(cid:19) where̺T ̺(fT)andn=2π/P.Bysubstitutingequations
≡
(7), (12) and (16) into above one, we can have
k 1
ηT = (cos2i)2, (9)
2 (cid:18)1+h(cid:19)(cid:18)1+h(cid:19) dtT = 1 ̺2 dω + (cos2i)
k 6(1+h)+k2(1 2h) (cid:28) dt (cid:29) −n√1 e2 T(cid:28)dt(cid:29) O
η3T = (cid:18)1+h(cid:19)(cid:20) 6(1+h)3− (cid:21)(cos2i)3. (10) = −αe λa̺2T−exp(cid:18)− λa(cid:19)I1(cid:18)aλe(cid:19)+O(cos2i).(18)
It is demonstrated (Kipping 2011) that using a first-order
Foratimeduration∆t,theYukawa-typesecularTTV ∆t
expansion solution can reduce the error to less than a mil- T
is
lisecond for a highly eccentric planet with a short period.
SfoouluntdioinnsKeixpppainngde(d20t1o1)h.igher orders (up to n = 6) can be ∆∆ttT =−αeξ̺2Texp(−ξ)I1(ξe)+O(cos2i), (19)
where
̺ =(1 e2)/(1+esinω)+ (cos2i), (20)
T
− O
and
2.2 Yukawa-type secular TTVs
ξ a/λ. (21)
≡
With the same approach used in Iorio (2011a) to work out
Itisworthmentioningthatalthoughequation(19)doesnot
long-termtimevariationsofsomeobservablesfortransiting
divergewhen e=0 because
exoplanets, for a given observable Γ that is a function of
Keplerian orbital elements, i.e. Γ = Γ( σ ), where σ = 1 ξ
{ } { } lim I1(ξe)= , (22)
a,e,i,Ω,ω,M , if perturbations on the Keperlian orbital e→0e 2
{ }
motion are taken into account, we can calculate its secular it is not suitable for the case that e is extremely close to
variation by averaging: 0 because it makes ω ill-defined. In order to avoid this, one
needstoreformulateitintermsofsingularity-freeorbitalel-
dΓ 1 P dΓ 1 P ∂Γdκ ements (Danby 1962). While the orbit may precess rapidly
= dt= dt, (11)
(cid:28)dt(cid:29) P Z dt P Z ∂κ dt nearzeroeccentricity,itseffectwouldsimplybeachangeto
0 0 κX∈{σ}
the measured orbital period (yielding an orbital frequency
where P is theKeplerian period of theorbit. Applyingthis of 2π/Pmeasured = 2π/Ptrue +ω˙). The TTV signal in this
caseisthedeviation from aconstant intervalbetween tran-
approach to f ,we can obtain its secular changes as
T sits. Thus, the changing angular velocity of a planet on an
elliptical orbit is an essential aspect of thesignal. However,
df dω n dηT
T = j . (12) due to the requirement of an elliptical orbit, short-period
(cid:28) dt (cid:29) −(cid:28)dt(cid:29)−Xj=1(cid:28) dt (cid:29) planets(withperiodsofonlyafewdays)arenotlikelytobe
(cid:13)c 2002RAS,MNRAS000,1–??
4 Y. Xie & X.-M. Deng
good candidates because of the circularization of the orbit e = 0.01; (b) e = 0.1; (c) e = 0.3; and (d) e = 0.6. These
from tides. It can also befound that,from equation (20), sub-cases share identical logarithmic color bars in the unit
of second per year (s yr−1) and are all generated by taking
(1 e)2+ (cos2i)6̺2 6(1+e)2+ (cos2i), (23)
− O T O ω=270◦ whichmakes̺T maximum.Itcanbecheckedthat
which makes ∆t /∆t not sensitive to ω according to equa- theirpatternsbarelychangefordifferentvaluesofωaccord-
T
tion(19).And,sinceξ isadimensionlessparameterasara- ing to equation (23). They tell us that the most optimistic
tioof thesemi-major axisaand thelength scale ofYukawa valuesofYukawa-typesecularTTVsareatthelevelof 0.1
∼
correctionλ,itsuggeststhatitwillbeverydifficulttodeter- secondsperyear.Unfortunately,thosevaluesappearonlyin
mine λ by observations on TTVs alone. In ∆t /∆t, α and rarely uniquecases (very small regions surrounding ξ 2.2
e play more important roles (see nextsection fTor details). and α = 10−8 in each sub-figures) and, most importa≈ntly,
Equation(19)describeschangesoftimesoftransitmin- theyarestillatleasttwoordersofmagnitudebelowthecur-
imum for a time duration which may be much longer than rent capabilities of observations. Therefore, such deviation
the Keplerian period of the orbit. In practice, an impor- from the ISL of gravity is likely too small to detect for the
tant directly measurable quantity is the temporal interval foreseeable future.
PT between successive transits. It will change as well be- There are some other effects that will make the
causeofω˙.Thisvariationgivestheobserveddeviationsfrom detection more complicated. For example, GR, a stellar
alinear ephemeris. Up tothefirst orderof e,thederivative quadrupole moment, tidal deformations and an ad-
of the transit period PT is given by (Miralda-Escud´e 2002; ditional planet (perturber) (e.g Miralda-Escud´e 2002;
Heyl& Gladman 2007; Jord´an & Bakos 2008) Holman & Murray2005;Agol et al.2005;Heyl& Gladman
2007; Jord´an & Bakos 2008; Iorio 2011a, 2012d;
ω˙ 2
P˙ =4πe sin(M ) (24) Nesvorny´ et al. 2012; Iorio 2013) can also cause secu-
T (cid:18)n(cid:19) T
larperiastron advancesrespectively: ω˙ , ω˙ , ω˙
h iGR h iquad h itide
so that the contribution caused by Yukawa-type correction and ω˙ [seeeqs.(1),(3),(5)and(7)inJord´an & Bakos
h ipert
is (2008) for their expressions]. It is found (Jord´an & Bakos
P˙ =4πα2ξ2e−1(1 e2)exp( 2ξ)I2(ξe)sin(M ), (25) 2008) that ω˙quad is usually much less than ω˙GR; ω˙tide
T − − 1 T and ω˙pert are of comparable magnitude to ω˙GR. The
where M is themean anomaly at transit (Miralda-Escud´e GR periastron advance can result in secular TTVs as
T
2002)andisrelatedtothetrueanomalyattransitf ofthe (Jord´an & Bakos 2008; Iorio 2011a; Zhao & Xie 2013)
T
first order of e by M =f 2esinf .
T T T
−
∆t √1 e2 GM
T = 3 − ∗ + (cos2i)
3 OBSERVABILITY OF YUKAWA-TYPE ∆t (cid:12)(cid:12)GR − (1+h)2 c2a O
SECULAR TTVS (cid:12)(cid:12) 47.7s M 2/3 P −2/3
∗ (1 2h)
This section will be dedicated to an important is- ≈ −(cid:18) 1yr (cid:19)(cid:18)M⊙(cid:19) (cid:18)1day(cid:19) −
sue: the observability of these Yukawa-type effects. + (cos2i,e2), (27)
O
Based on the literature about Yukawa-type correction
(e.g. Fischbach & Talmadge 1999; Adelberger et al. 2003;
Denget al. 2009; Lucchesi & Peron 2010; Deng 2011; wherecisthespeedoflight andM isthemassofthehost
∗
Lucchesi 2011; Iorio 2012b) and the catalog of confirmed star. It means, in a transiting exoplanet system with Sun-
transiting exoplanets 3, we will focus on the domain of the likemass,periodof 1dayandrelativelysmalleccentricity
parameters in equation (19) as of 0.1, the TTVs∼caused by GR can reach 40 seconds
∼ ∼
= (ξ,α,e,ω) 10−2 6ξ6101, 10−12 6α610−8, inayear.TocomparethemagnitudesofTTVstriggeredby
D { | GR and Yukawacorrection, we can calculate theratio as
0.016e60.6, 06ω62π . (26)
}
Intheconstructionofthisspaceofparameters,wetakecon-
firmed transiting exoplanets as samples of orbital configu- η (∆tT/∆t)GR = hω˙iGR
rations. We also consider constraints on Yukawa correction GR/YK ≡ (∆tT/∆t)YK hω˙iYK
in the Solar system. For instance, one of the upper bounds = 3κ1/3c−2µ2∗/3P−2/3f1gYK, (28)
ofYukawacorrection isgivenbyIorio(2007a)usingtheSo-
larsystemplanetsorbitalmotion withEPM2004 ephemeris
(Pitjeva 2005): α∼10−9 and λ∼0.18 au. A crucial differ- where κ 4π2, µ∗ GM∗ and
ence between the case of the Solar system and the one of ≡ ≡
exoplanetary system is that we know major objects in the
Solarsystemverywellbutmostexoplanetswithlowmasses f1 = e(1 e2)−3/2, (29)
(.M⊕)currentlyremainunknown.Inordertoincludemore −
exp(ξ)
possible and potential cases, even some of which have not gYK = . (30)
been observed yet,we enlarge thedomain as well. αξI1(ξe)
D
Figure1showscolor-indexed∆t /∆tinfourcases:(a)
T
Similarly, we also can have three other ratios involving the
3 http://exoplanet.eu/catalog/ stellarquadrupolemoment,tidaldeformations andtheper-
(cid:13)c 2002RAS,MNRAS000,1–??
Impossibility of testing ISL by TTVs 5
turberas withlowmassesarecurrentlydifficulttodetectandremain
unknowninmostcases.Theresultinguncertaintieswillruin
ηquad/YK ≡ ((∆∆ttTT//∆∆tt))qYuKad aannydtehffeoyrtwsitllolitkeesltyIbSeLeoxfcegprativointyalluysidnigffiecxuoltptlaonreetmaroyveTfTroVms
3
≈ 2κ2/3J2∗µ−∗2/3R∗2P−4/3f2gYK, (31) asignalthatshouldbeseen.Separating,discriminatingand
extractingvariouscontributionsinTTVsforfuturepositive
η (∆tT/∆t)tide detectionofpossibledeviationfromISLrequiretremendous
tide/YK ≡ (∆tT/∆t)YK advances of techniques for observations and sophisticated
≈ 15κ5/3µ−∗5/3P−10/3k2,pTRp5MM∗pf3gYK, (32) methodsof data analysis.
η (∆tT/∆t)pert
pert/YK ≡ (∆tT/∆t)YK
≈ 43MM∗2PP222f2gYK, (33) 4 CONCLUSIONS AND DISCUSSION
In the context of potential and considerable increase of the
where
number of transiting exoplanets and the precision of mea-
f2 = e(1 e2)−1/2, (34) sured times of transit minima by ground-based and space-
− borneobservatoriesusedforstudyingexoplanettransitsnow
3 1
f3 = e(1 e2)−11/2 1+ e2+ e4 , (35) andin thefuture,we studythepossibility of testing funda-
− (cid:20) 2 8 (cid:21)
mental laws of nature in these system via TTVs. Focusing
= 1+ k2,s R∗ 5 Mp 2. (36) onpresumableviolationsoftheISLofgravitywhicharepa-
T k2,p(cid:18)Rp(cid:19) (cid:18)M∗(cid:19) rameterized by the fifth-force-like Yukawa-type correction
to the Newtonian gravitational force, we investigate their
Here,J∗ isthestellar quadrupolemoment,R istheradius
2 ∗ effectsonsecularTTVsandanalyzetheirobservability.Itis
of the host star; M is the mass of the planet, R is the
p p foundthat themost optimistic valuesof Yukawa-typesecu-
radius of the planet; k2,s and k2,p are the apsidal motion lar TTVs are at the level of 0.1 seconds per year. Those
constantsforthestarandplanetrespectively,whichdepend ∼
valuesunfortunatelyappearonlyinrarelyuniquecasesand,
on the mass concentration of the tidally deformed bodies
most importantly, they are still at least two orders of mag-
(Sterne 1939); M2 is the mass of the second planet and P2 nitude below the current capabilities of observations. Such
is the Keplerian period of its orbit. In the calculations of
deviationfromtheISLofgravityislikelytoosmalltodetect
theseratios[equations(28)and(31)–(33)],equation(16)in
for theforeseeable future.
this work and eqs. (1), (3), (5) and (7) in Jord´an & Bakos
Moreover, exoplanetary systems are full of complexity
(2008) are used.
so that many sources can trigger secular TTVs, such as
In order to estimate these ratios, we fix ξ = 2.2 and
GR, a stellar quadrupole moment, tidal deformations and
α=10−8 because they generate the most optimistic values
a perturber. After calculating the ratios between TTVs by
ofYukawa-typesecularTTVs,whichareatthelevelof 0.1
∼ Yukawacorrectionandthosecausedbythesefoureffects,we
seconds per year (see figure 1). After evaluating them, we
findthesignals ofYukawacorrection aremuchweakerthan
findsecularTTVscausedbyGR,thestellarquadrupolemo-
others. The uncertainties of perturbers with low masses,
ment,tidal deformations and a perturberare usually larger
which are usually unknownfor now, will ruin any efforts to
than the one due to the Yukawa correction by several or-
test ISL of gravity using exoplanetary TTVs and they will
ders of magnitude(see figure 2). Figure 2(a) shows η
GR/YK likelybeexceptionallydifficulttoremovefromasignalthat
with respect to P, where m M /M . It indicates, for an
≡ ∗ ⊙ should be seen. Separating, discriminating and extracting
exoplanet with P 10 days, the TTVs caused by GR are
about102 timesgre∼aterthanthosetriggeredbypossiblede- variouscontributionsinTTVsforpositivedetectionrequire
tremendousadvancesoftechniquesforobservations andso-
viationfromISLofgravity.ThecomparisonofTTVsbythe
phisticated methodsof data analysis.
stellar quadrupole moment and Yukawa correction is given
in figure 2(b) in which j J∗ 107, r R /R and e is
≡ 2 × ≡ ∗ ⊙
fixedas0.3.Itsuggestseffectsfromthequadrupolemoment
will suppress the Yukawa correction on a close-in planet.
Some curves of η are presented in figure 2(c) where
tide/YK ACKNOWLEDGMENTS
we assume a Jupiter-like giant planet with k2,p =0.25 by a
polytrope of index n 1 (Hubbard 1984) and a host star Weacknowledgeveryusefulandhelpfulcommentsandsug-
≈
with k2,s = 0.01 (Claret & Gimenez 1992). Like the trends gestions from our anonymous referee. The work of YX is
in figure 2(b), the TTVs by tidal deformations are much supported by the National Natural Science Foundation of
larger than those by Yukawa correction for hot-Jupiters. ChinaGrantNo.11103010, theFundamentalResearchPro-
In figure 2(d), these curves show the relative strength of gram of Jiangsu Province of China Grant No. BK2011553
TTVsbyaperturberandthosebyYukawacorrection,where andtheResearch Fundfor theDoctoral Program of Higher
m2 M2/M⊕. The influence of a perturber can reach the Education of China Grant No. 20110091120003. The work
≡
level of several orders of magnitude greater than theeffects of XMD is funded by the Natural Science Foundation of
ofYukawacorrection.Foraperturberwith M ,itscontri- ChinaunderGrantNo.11103085 andtheFundamentalRe-
⊕
∼
butioninTTVscanbeabout10timeslargerthanYukawa- search Program of Jiangsu Province of China Grant No.
correction’s when P :P2 1:2. However, such perturbers BK20131461.
≈
(cid:13)c 2002RAS,MNRAS000,1–??
6 Y. Xie & X.-M. Deng
(a) e=0.01, ω=270° m=0.6, e=0.1
4 m=0.6, e=0.5
-8 0
) m=1.0, e=0.1
1
-r K m=1.0, e=0.5
-9 -2 s y R/Y 3
αg 10 -10 -4 ∆/t) (T ηg 10G 2
lo -6 ∆(t lo
-11 0
1 1 (a)
-8 og
-12 l
-2 -1 0 1 2
-2 -1.5 -1 -0.5 0 0.5 1
log ξ log10 P (d)
10
(b) e=0.1, ω=270° j=2.0, m=0.6, r=0.6
j=2.0, m=1.0, r=1.0
4
-8 0 j=10, m=0.6, r=0.6
αg 10 -1-90 --42 -1∆/t) (s yr)T ηog 10quad/YK 02 j=10, m=1.0, r=1.0
lo -6 ∆(t l
-11 0 (b)
1 -2
-8 og
-12 l -2 -1 0 1 2
-2 -1.5 -1 -0.5 0 0.5 1 log P (d)
10
log ξ
10
m=0.6, r=0.6, e=0.1
(c) e=0.3, ω=270° 10 m=0.6, r=0.6, e=0.5
m=1.0, r=1.0, e=0.1
-8 0 K m=1.0, r=1.0, e=0.5
) Y
-1r de/ 6
αg 10 -1-90 --42 ∆/t) (s yT ηlog 10ti 2
lo -11 -6 ∆ (t0 -2 (c)
1
-8 og -2 -1 0 1 2
-12 l
log P (d)
-2 -1.5 -1 -0.5 0 0.5 1 10
log ξ
10
5
(d) e=0.6, ω=270°
-8 0 1) ert/YK 3
-9 -2 -s yr η 10p
α log 10 --1110 --64 ∆∆ (t/t) (0T log - 11 (d) mmm2m=22==21=1401002
-8 og1 0.2 0.4 0.6 0.8
-12 l P/P2
-2 -1.5 -1 -0.5 0 0.5 1
log10 ξ Figure 2. The curves of ηGR/YK, ηquad/YK, ηtide/YK and
ηpert/YK areshowninthe panels of (a), (b), (c) and(d) respec-
tively. We fix ξ = 2.2 and α = 10−8 for all of them. In panel
Figure 1. Color-indexed ∆tT/∆t in four cases: (a) e = 0.01; (a),m≡M∗/M⊙;in(b),wetakee=0.3andhavedefinitionsas
(b) e=0.1; (c) e= 0.3; and (d) e= 0.6. These sub-cases share j = J2∗×107 and r ≡ R∗/R⊙; in (c), we assume a Jupiter-like
identical logarithmiccolor bars inthe unitof second per year (s giant planet with k2,p = 0.25 and a host star with k2,s = 0.01;
yr−1) and are all generated by taking ω =270◦. Their patterns andin(d),wedefinethemassofaperturberasm2≡M2/M⊕.
barelychangefordifferentvaluesofωaccordingtoequation(23).
(cid:13)c 2002RAS,MNRAS000,1–??
Impossibility of testing ISL by TTVs 7
REFERENCES Iorio, L. 2011b, Ap&SS,331, 485
Iorio, L. 2011c, J. Cosmol. Astropart. Phys., 5, 019
Adams, F. C., & Laughlin, G. 2006a, ApJ,649, 992
Iorio, L. 2012a, Sol. Phys., 281, 815
Adams, F. C., & Laughlin, G. 2006b, ApJ, 649, 1004
Iorio, L. 2012b, Journal of High Energy Physics, 5, 73
Adams,F.C.,&Laughlin,G.2006c,InternationalJournal
Iorio, L. 2012c, J. Cosmol. Astropart. Phys., 7, 001
of Modern Physics D, 15, 2133
Iorio, L. 2012d, Celestial Mechanics and Dynamical As-
Adelberger, E. G., Heckel, B. R., & Nelson, A. E. 2003,
tronomy,112, 117
AnnualReview of Nuclear and Particle Science, 53, 77
Iorio,L.2013,Celestial MechanicsandDynamicalAstron-
Agol, E., Steffen, J., Sari, R., & Clarkson, W. 2005, MN-
omy,116, 357
RAS,359, 567
Iorio,L.,&Giudice,G.2006,J.Cosmol.Astropart.Phys.,
Arfken,G.B.,&Weber,H.J.2005,MathematicalMethods
8, 007
for Physicists (6th Ed.) (Amsterdam; Boston: Elsevier)
Iorio,L.,Lichtenegger,H.I.M.,Ruggiero,M.L.,&Corda,
Bell, J. F., Camilo, F., &Damour, T. 1996, ApJ, 464, 857
C. 2011, Ap&SS,331, 351
Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science,
Jord´an, A., & Bakos, G. A´.2008, ApJ, 685, 543
327, 977
Kipping, D. M. 2011, The Transits of Extrasolar Plan-
Claret, A.,& Gimenez, A. 1992, A&AS,96, 255
etswithMoons,Ph.D.thesis,UniversityCollegeLondon,
Damour,T.,&Esposito-Far`ese,G.1996,Phys.Rev.D,53,
2011
5541
Koch, D. G., Borucki, W. J., Basri, G., et al. 2010, ApJ,
Danby,J.1962,FundamentalsofCelestialMechanics(New
713, L79
York:Macmillan)
Kramer, M., Stairs, I. H., Manchester, R. N., et al. 2006,
De Laurentis, M., De Rosa, R., Garufi, F., & Milano, L.
Science, 314, 97
2012, MNRAS,424, 2371
Li, L.-S. 2010, Ap&SS,327, 59
Deng, X.-M. 2011, Science in China G: Physics and As-
Li, L.-S. 2011, Ap&SS,334, 125
tronomy,54, 2071
Li, L.-S. 2012, Ap&SS,341, 323
Deng, X.-M., & Xie, Y.2013, MNRAS,431, 3236
Deng,X.-M.,Xie,Y.,&Huang,T.-Y.2009, Phys.Rev.D, Lissauer, J. J., Fabrycky, D. C., Ford, E. B., et al. 2011,
79, 044014 Nature,470, 53
Fabrycky, D. C., Ford, E. B., Steffen, J. H., et al. 2012, Lucchesi,D.M.2011,AdvancesinSpaceResearch,47,1232
ApJ,750, 114 Lucchesi, D. M., & Peron, R. 2010, Physical Review Let-
Fienga, A., Laskar, J., Kuchynka,P., et al. 2011, Celestial ters, 105, 231103
Mechanics and Dynamical Astronomy,111, 363 Mazeh, T., Nachmani, G., Holczer, T., et al. 2013, ApJS,
Fischbach,E.,Gillies,G.T.,Krause,D.E.,Schwan,J.G., 208, 16
& Talmadge, C. 1992, Metrologia, 29, 213 Miralda-Escud´e, J. 2002, ApJ, 564, 1019
Fischbach, E., Sudarsky, D., Szafer, A., Talmadge, C., & Murphy, M. T., Tzanavaris, P., Webb, J. K., & Lovis, C.
Aronson, S.H. 1986, Physical ReviewLetters, 56, 3 2007, MNRAS,378, 221
Fischbach, E., & Talmadge, C. L. 1999, The Search for Murphy, M. T., Webb, J. K., & Flambaum, V. V. 2003,
Non-Newtonian Gravity (New York:Springer-Verlag) MNRAS,345, 609
Folkner,W.M.2010,inIAUSymposium,IAUSymposium, Murphy, M. T., Webb, J. K., & Flambaum, V. V. 2008,
vol. 261, edited by S. A. Klioner, P. K. Seidelmann, & MNRAS,384, 1053
M. H. Soffel, 155–158 Murphy,M.T.,Webb,J.K.,Flambaum,V.V.,Churchill,
Ford, E. B., Fabrycky, D. C., Steffen, J. H., et al. 2012, C. W., & Prochaska, J. X.2001a, MNRAS,327, 1223
ApJ,750, 113 Murphy,M.T.,Webb,J.K.,Flambaum,V.V.,Prochaska,
Fujii, Y., & Maeda, K.-I. 2007, The Scalar-Tensor The- J. X., & Wolfe, A.M. 2001b, MNRAS,327, 1237
ory of Gravitation (Cambridge, UK: Cambridge Univer- Murphy, M. T., Webb, J. K., Flambaum, V. V., et al.
sity Press) 2001c, MNRAS,327, 1244
Haranas, I., & Ragos, O. 2011, Ap&SS,331, 115 Murphy, M. T., Webb, J. K., Flambaum, V. V., et al.
Haranas, I.,Ragos, O.,&Mioc, V.2011, Ap&SS,332,107 2001d, MNRAS,327, 1208
Heyl, J. S., & Gladman, B. J. 2007, MNRAS,377, 1511 Nesvorny´, D., Kipping, D., Terrell, D., et al. 2013,
Holman,M.J.,Fabrycky,D.C.,Ragozzine,D.,etal.2010, 1304.4283
Science, 330, 51 Nesvorny´,D.,Kipping,D.M.,Buchhave,L.A.,etal.2012,
Holman, M. J., & Murray, N.W. 2005, Science, 307, 1288 Science, 336, 1133
Hubbard,W.B.1984,PlanetaryInteriors(NewYork:Van Nobili, A.M., & Will, C. M. 1986, Nature, 320, 39
Nostrand Reinhold Co.) Pa´l, A., & Kocsis, B. 2008, MNRAS,389, 191
Iorio, L. 2005a, A&A,431, 385 Pitjev, N. P., & Pitjeva, E. V. 2013, Astronomy Letters,
Iorio, L. 2005b, A&A,433, 385 39, 141
Iorio, L. 2005c, J. Cosmol. Astropart. Phys., 7, 008 Pitjeva, E. 2012, in IAU Joint Discussion, IAU Joint Dis-
Iorio, L. 2005d, J. Cosmol. Astropart. Phys.,9, 006 cussion, vol. 7
Iorio, L. 2006, NewAstron., 11, 490 Pitjeva, E. V.2005, Solar System Research, 39, 176
Iorio, L. 2007a, Journal of High Energy Physics, 10, 041 Pitjeva, E. V. 2010, in IAU Symposium, IAU Symposium,
Iorio, L. 2007b, Ap&SS,312, 331 vol. 261, edited by S. A. Klioner, P. K. Seidelmann, &
Iorio, L. 2010, J. Cosmol. Astropart. Phys., 6, 004 M. H.Soffel, 170–178
Iorio, L. 2011a, MNRAS,411, 167 Pitjeva, E. V., & Pitjev, N.P. 2013, MNRAS,432, 3431
(cid:13)c 2002RAS,MNRAS000,1–??
8 Y. Xie & X.-M. Deng
Ragos, O.,Haranas,I.,&Gkigkitzis,I.2013,Ap&SS,345,
67
Ruggiero, M. L., & Iorio, L. 2007, J. Cosmol. Astropart.
Phys., 1, 010
Steffen, J. H., Fabrycky, D. C., Ford, E. B., et al. 2012,
MNRAS,421, 2342
Sterne, T. E. 1939, MNRAS,99, 451
Turyshev,S.G. 2008, AnnualReview of Nuclear and Par-
ticle Science, 58, 207
Webb,J.K.,Flambaum,V.V.,Churchill,C.W.,Drinkwa-
ter,M.J.,&Barrow,J.D.1999,PhysicalReviewLetters,
82, 884
Webb,J.K.,King,J.A.,Murphy,M.T.,etal.2011,Phys-
ical ReviewLetters, 107, 191101
Webb,J.K.,Murphy,M.T.,Flambaum,V.V.,etal.2001,
Physical Review Letters, 87, 091301
Will,C.M.1993,TheoryandExperimentinGravitational
Physics (Cambridge, UK: Cambridge University Press)
Will, C. M. 2006, Living Reviews in Relativity,9, 3
Xie,Y.2013, ResearchinAstron.Astrophys.(RAA),13,1
Xie, Y., & Deng,X.-M. 2013, MNRAS,433, 3584
Zhao, S.-S., & Xie, Y. 2013, Research in Astron. Astro-
phys.(RAA),13, 1231
(cid:13)c 2002RAS,MNRAS000,1–??