Modelling the Rossiter-McLaughlin Effect: Impact of the Convective Centre-to-Limb Variations in the Stellar Photosphere

0.71 MB·

by  H. M. Cegla

Draftversion January12,2016
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
MODELLING THE ROSSITER-MCLAUGHLIN EFFECT: IMPACT OF THE CONVECTIVE
CENTRE-TO-LIMB VARIATIONS IN THE STELLAR PHOTOSPHERE
H. M. Cegla1, M. Oshagh2, C. A. Watson1, P. Figueira2, N. C. Santos2,3, S. Shelyag4
Draft version January 12, 2016
ABSTRACT
ObservationsoftheRossiter-McLaughlin(RM)effectprovideinformationonstar-planetalignments,
6 which can inform planetary migration and evolution theories. Here, we go beyond the classical RM
1 modelling and explorethe impact ofa convectiveblueshift that varies acrossthe stellar disc andnon-
0 Gaussian stellar photospheric profiles. We simulated an aligned hot Jupiter with a 4 d orbit about a
2 Sun-like starandinjected centre-to-limbvelocity(andprofile shape)variationsbasedonradiative3D
magnetohydrodynamic simulations of solar surface convection. The residuals between our modelling
n
a andclassicalRMmodellingweredependentonthe intrinsicprofilewidthandvsini; the amplitude of
J the residuals increased with increasing vsini, and with decreasing intrinsic profile width. For slowly
rotating stars the centre-to-limb convective variation dominated the residuals (with amplitudes of
9
10s of cm s−1 to ∼1 m s−1); however, for faster rotating stars the dominant residual signature was
] due a non-Gaussian intrinsic profile (with amplitudes from 0.5-9 m s−1). When the impact factor
P
was 0, neglecting to account for the convective centre-to-limb variation led to an uncertainty in the
E obliquity of ∼10-20◦, even though the true vsini was known. Additionally, neglecting to properly
. modelanasymmetricintrinsic profilehadagreaterimpactformorerapidlyrotatingstars(e.g. vsini
h
p = 6 km s−1), and caused systematic errors on the order of ∼20◦ in the measured obliquities. Hence,
- neglecting the impact of stellar surface convectionmay bias star-planet alignment measurements and
o consequently also theories on planetary migration and evolution.
r
t Subject headings: Line: profiles–Planetsandsatellites: detection–Sun: granulation–Stars: activity
s – Stars: low-mass – Techniques: radial velocities
a
[

1 1. INTRODUCTION (Pepe et al. 2014), promise precisions of 10 cm s−1 or
v Radial velocity (RV) precision is primarily limited by better by as early as 2017. Such astrophysical phenom-
4 ena affect any high precision RV study. Spectroscopic
instrumentation and our understanding of stellar spec-
5 observations of exoplanets are particularly affected by
tral lines. Consequently, the continued improvement in
0 these phenomena as it can be extremely difficult to dis-
instrumental precision demands an evermore accurate
2 entangle planetary and stellar signals from one another.
treatment of spectral line behaviour. This is clearly ev-
0 This is in addition to the fact that stellar signals can
ident now as current spectrographs, such as HARPS,
1. can routinely offer a precision of ∼ 0.5 m s−1, while as- masqueradeasplanetarysignals(e.g.Queloz et al.2001;
0 trophysical phenomena can distort stellar lines and in- Desidera et al.2004;Hu´elamo et al.2008;Figueira et al.
6 duce spurious velocity shifts ranging from several 10s 2010; Santos et al. 2014; Robertson et al. 2015).
1 of cm s−1 to 100s of m s−1 for solar-type stars (due Furthermore, ignoring certain astrophysical effects
: mayintroduceerrorsinourmeasurementsofstar-planet
v to, for example, variations in gravitationalredshift, stel-
systems,whichcouldultimatelyimpactplanetformation
i lar surface (magneto-)convection, natural oscillations,
X and evolution theories. For example, Shporer & Brown
meridional circulation, spots, plages, and the atten-
(2011) have shown that ignoring stellar surface convec-
r uation of convective blueshift surrounding regions of
a tion in transit observations of the Rossiter-McLaughlin
high magnetic field; Cegla et al. 2012; Saar & Donahue
(RM) effect (Rossiter 1924; McLaughlin 1924; Winn
1997; Schrijver & Zwaan 2000; Dumusque et al. 2011b;
Beckers2007;Dumusque et al.2011a;Boisse et al.2011; 2007) can lead to a deviation in the RVs on the m s−1
Meunier & Lagrange 2013.) level, which the authors postulate will affect the mea-
Additionally, it is clear that the need for an accu- sured spin-orbit alignment angle. Convection on the
rate description of even low-amplitude phenomena will surface of solar-type stars results in a net convective
only intensify as spectrographs, such as ESPRESSO blueshift (CB) of the spectral lines due to the fact that
theuprising(blueshifted)granulesarebrighterandcover
1Astrophysics Research Centre, School of Mathematics & a greater surface area than the downflowing (redshifted)
Physics, Queen’s University Belfast, University Road, Belfast intergranular lanes (for the Sun this value is ∼ -300
BT71NN,UK;[email protected] m s−1; Dravins 1987). Shporer & Brown (2011) pro-
2Instituto de Astrof´ısicae Ciˆencias do Espa¸co, Universidade duced a simple numerical model to illustrate this effect,
doPorto,CAUP,RuadasEstrelas,PT4150-762Porto,Portugal,
wherein they considered the CB to be a constant value
PT
3Departamento de F´ısica e Astronomia, Faculdade de that varied across the stellar disc due to limb darkening
Ciˆencias, Universidade do Porto, Rua Campo Alegre, 4169-007 and projected area. However, they acknowledged that
Porto,Portugal such a model neglected effects from meridional flows,
4Monash Centre for Astrophysics, School of Mathematical
differential rotation, differences in CB for various stel-
Sciences, MonashUniversity,Clayton,Victoria,3800, AU
2
lar lines, as well as the dependence of the local ob- 300
served CB on the centre-to-limb angle, θ (often denoted
as µ = cos(θ)), and hence may underestimate the total 250
error in RM observations.
Indeed,solarobservationsandstate-of-the-art3Dmag-
200
netohydrodynamic(MHD)simulations(coupledwithra-
diative transport)clearlydemonstrate thatthe observed -1m s)
variation in local CB may vary considerably from that y ( 150
predicted by projection effects alone (see Figure 1 – fur- ocit
el
therdiscussedinSection2). Thisdeviationisdue tothe V 100
corrugatednatureofgranulation. Acrossthestellarlimb
different aspects of the granulationare visible to the ob-
50
server, e.g. when granulation is viewed near the stellar
limb the tops of the granules and bottom of the inter-
granular lanes become hidden while the granular walls 0
become visible. Hence, there are variations in the line- 0 20 40 60 80
Centre-to-Limb Angle (°)
of-sight(LOS)velocitiesandfluxthatalterboththeline
Fig. 1.— The average granulation RVs, relative to disc centre,
shape and centroid, and result in RV variations in the over an ∼ 80 minute time-series from the MHD solar simulation
observed local line profiles. presented in Ceglaetal. (2014) as a function of stellar centre-to-
Inthispaper,weusethecentre-to-limbvariationinCB limbangle(reddots). Asolid(black)lineillustratesafourthorder
predicted by a 3D MHD solar simulation, shown in Fig- polynomialfittothedata,andadashed(black)lineillustratesthe
predicted variation inconvective blueshift due solely to projected
ure1,toadvanceupontheanalysisbyShporer & Brown areafortheSun(i.e. aconstant blueshift×cos(θ)).
(2011). Wecreatestellarsurfacemodelsthatincludenot
only stellar rotation and limb darkening, but also the iscoveredintiles withanareaascloseaspossible to the
variation in CB due to granulation corrugation (whilst areaofthesimulationsnapshots;the3Dgridisthenpro-
accountingfortheprojectedareaatagivenµ). Weinject jected onto a 2D plane (as seen by the observer). The
a transiting planet into these stellar models and use the SOAP-T stellar grid, however, is constructed directly in
planetasaprobetoresolvetheCBvariationinsimulated the2Dplane,withatilesizeoptimisedforplanettransit
Sun-as-a-starobservations;thisallowsustoquantifythe analysis. Both codes inject into each tile a line profile
impact of ignoring the CB variation on RM measure- (representative of the stellar photosphere) including the
mentsforSun-likestars. Wealsoindependentlyquantify effects of limb darkening, projected area, and stellar ro-
the erroronthe projectedspin-orbitmisalignmentangle tationalvelocityshifts5. Aplanetarytransitissimulated
using the software tool SOAP-T (Oshagh et al. 2013a) bymaskingthetilesthatcorrespondtotheregionbehind
as well as the Sun-as-a-star model code developed in the planet, and integrating over the stellar disc.
Cegla et al. (2014). The main difference between these two models is that
In Section 2, we describe the two stellar models used the C14 grid is tiled on a 3D surface and projected onto
throughout this paper. We present the RM waveform a 2D plane, whereas the SOAP-T grid originates in the
expectedsolelyfromacentre-to-limbvariationinnetCB 2D plane. This means that the C14 grid has a greater
for a Sun-like star in Section 3. In Sections 4 and 5, number of visible tiles near the stellar limb than it does
we quantify the deviation of the RM curve due to CB near disc centre, whereas the SOAP-T grid has an even
andthecorrespondingimpactontheprojectedspin-orbit number of tiles throughoutthe stellar disc. Hence, some
alignment angle. Finally, we conclude in Section 6. differencesintheRMcurvesbetweenthetwomodelsare
expected since the tiling is slightly different. When we
2. THESTELLARMODELS
examined the residuals between the two stellar models,
Throughout this paper we use two stellar models, as weconcludedthatalthoughthereweredifferencesonthe
each has one particular advantages over the other. In cm s−1 level that such differences were unlikely to affect
the first instance, we create a stellar grid following that the conclusions; see Appendix A for details.
used in Cegla et al. (2014), hereafter C14, while in the In this paper, we only consider the impact of the local
second instance we use the already established software CB, without temporal variations. In the first instance,
toolSOAP-T.OneadvantageoftheC14modelisthatwe wemodelledthe localintrinsiclineprofilesasGaussians.
caninjectasymmetriclineprofilestorepresentthestellar We use a quadratic limb-darkening law, where the coef-
photosphere(asopposedtothestrictlyGaussianprofiles ficients (c = 0.29, c = 0.34) were determined by fit-
1 2
presently accepted by SOAP-T). Another advantage of ting the intensities from the MHD simulations in C14
the C14 model is that, in a forthcoming paper, we can (a quadratic limb-darkening law was chosen to match
include the variability of the ratio between granularand SOAP-T).TheRVsforeachobservationweredetermined
intergranular lanes on the stellar surface (as the gran- by the mean of a Gaussian fit to the disc-integrated
ules evolvethis ratio constantlychangesand contributes line profiles. This technique was chosen as it is the
a disc integrated RV variability on the order of 10s of same procedure used by the HARPS pipeline. Note
cm s−1). On the other hand, the advantage of SOAP- that the HARPS pipeline operates on the CCF (cross-
T is that it is a well-tested numerical model currently correlation function) created by the cross-correlation of
usedintheliterature,andrepresentsatypicalnumerical the observed spectral absorption lines with a weighted
approach to modelling the RM waveform.
The C14 stellar grid was designed to incorporate line 5Forthisworksolidbodyrotationisassumedinordertoisolate
profilesfrom3DMHDsimulations. Assuch,a3Dsphere theimpactfromconvection.
3
◦
template mask, and our disc-integrated profiles serve as weresimulatedin2 steps–thisstep-sizewaslargelyset
a proxy for the CCFs. It is also important to note that by computational constraints (Cegla et al. 2015).
a Gaussian fit only provides the true velocity centroid if TodeterminethevariationinlocalCBasafunctionof
theobservedlineprofiles(andCCFs)aresymmetric(see centre-to-limb angle, the line profiles from all snapshots
Collier Cameron et al.2010,andSection4.1formorede- in the time-sequence (at all stellar limb positions) were
tails). Finally, each model was assigned the same star- cross-correlatedwith one line profile from a single snap-
planet properties; these are summarised in Table 1. In shot at disc centre. The disc centre template profile was
this work we modelled the transit of a 4 d hot Jupiter chosen at random from the simulation time-series to set
arounda Sun-like star with anorbit that is aligned with the zero-point for the cross-correlation, which was ulti-
the stellar spin axis. If not otherwise stated, the orbital mately removed since we are only interested in the rel-
◦
inclinationwas90 (impactfactorb=0);thisinclination ative centre-to-limb variations. The peaks of the CCFs
was chosen so that the planet transited the maximum (from a second order polynomial fit) were used to deter-
centre-to-limb positions across the stellar disc (note we mine the velocity shifts. To minimise the temporal in-
do not suffer a degeneracy between the projected obliq- fluence (i.e. granulation evolution effects), all velocities
uityandthevsini,despiteazeroimpactfactor,because at a given stellar limb position were averaged together
we know the true stellar rotation of our model stars). over the 80 min. time-series6; the results are shown as
red dots in Figure 1. To incorporate the CB variation
in SOAP-T, we fit a fourth order polynomial to these
TABLE 1
points (solid line in Figure 1). For consistency,the same
Starandplanetparametersin the model RMobservations
polynomial was used to introduce the CB velocity shifts
Parameter Star Planet in the C14 grid. Note we opted not to extrapolate the
Period variablea 4d netCBbeyondthe80◦ centre-to-limbangle;thiswasbe-
Mass 1M⊙ 1MJ causethe slopeofthe polynomialfitatthis limbangleis
Radius 1R⊙ 1RJ very steep (predicting an increase of 300m s−1 from 80-
Eccentricity – 0 90◦)andsincewedonotknowifthisistrulyphysicalwe
Inclination 90◦ –
opted for a slight underestimation of the CB variation
ImpactFactor – variableb
as opposed to a potentially large over estimation. All
TpΩeri –– 900◦ tiles with a centre-to-limb angle greater◦than 80◦ were
γ – 0◦ assigned the net CB corresponding to 80 .
1a−St1e0llakrmrost−a1ti.onwasvariedthroughout,correspondingtovsini= 3. RMWAVEFORMFROMCENTRE-TO-LIMBCB
VARIATIONS
bInitially b = 0, but in later sections it was varied to 0.25 and
0.5. If the observed stellar surface velocities are only due
to rotation, then a non-rotating star will have no RV
For each model we produced two sets of 93 observa- anomaly during the planet transit and hence the RM
tions,onewithandonewithouttheCBvariation. These waveformwill be a flat line at zero velocity. However,in
were centred about mid-transit with a cadence of 200 s thepresenceofcentre-to-limbCBvariations,RVanoma-
(this gives close to 1 hr of out-of-transit time on either lies will still be apparent. To investigate the nature of
side of the transit). In the zero CB models, the intrinsic such a signal, we injected the transiting planet into a
lineprofileswereonlyDopplershiftedbytheappropriate system with the position-dependent net CB (shown in
stellar rotational velocity (no other line shifting mecha- Figure 1) for a non-rotating star. Since SOAP-T is not
nisms are included). For models with CB, the intrinsic designedtohandlezerostellarrotation,thistestwasonly
profiles were shifted by both the stellar rotation and the performed using the C14 grid. In this instance, we in-
simulated local CB variation from the solar simulations jected Gaussianline profiles with a FWHM of 5 km s−1;
in C14. this width was chosen as it is similar to the aforemen-
The solar simulations in C14 were created with the tioned6302.5˚AFeIlineprofile(fromthe3DMHDsolar
MURaM code (Vo¨gler et al. 2005), which has a simula- simulations)atdisc centre andtherefore representsa re-
tionboxcorrespondingtoaphysicalsizeof12×12Mm2 alisticFWHMgiventheinjectedCB.ThemeasuredRVs
in the horizontal directions and 1.4 Mm in the vertical for this set of observations is shown in Figure 2 (along-
direction. The initial magnetic field was 200 G, which sideaschematicoftheplanettransit,colour-codedbythe
is only slightly higher than the unsigned average mag- netconvectivevelocitiesrelativetodisccentre). TheRVs
netic field in the ‘quiet’ solar photosphere (i.e. 130 G; near ingress and egress are blueshifted since the planet
Trujillo Bueno et al. 2004). The photospheric plasma obscures the local CBs with the highest redshifts (rela-
parametersfromtheMHDmodelwereusedtosynthesise tive to disc centre) and redshifts near mid-transit where
the6302.5˚AFeIline(withtheSTOPROcode). Atime- theplanetobscuresmoreblueshiftedregionsofthestellar
sequenceof190individualsnapshotswasproduced,with disc. Hence, fromFigure 2 we can see that a localvaria-
a cadence of ∼ 30 s (except near the start of the simula- tionin CBcontributesto the RV anomalyobserveddur-
tionwherethecadencewascloserto15s). Thesequence ing transit, and leads to a non-zero RM waveform even
covers approximately 80 min., corresponding to ∼ 10-20 whennostellarrotationisobserved(theexactshapeand
granularlifetimes. SeeCegla et al.(2013)forfurtherde-
tailsonthesimulationatdisccentre. Tocreatesnapshots 6Notethatshorteraveragingtimescalesintroducescatterabout
the mean values over the entire (80-min.) time series, i.e. scatter
offdisccentre,thehorizontallayersofthesimulationbox
about the red points plotted in Figure 1. For example, 5 minute
were shifted to allow the LOS ray to penetrate the box averages introduce scatter of ∼ ± 50 m s−1 for positions < 60◦
from different angles. Centre-to-limb angles from 0-80◦ and∼±10ms−1 (orless)furthertowardthelimb.
4
C14 Grid: RM Curve with CB & without Rotation; Injected FWHM = 5 km s-1 broadening – and to a lesser extent a number of col-
0.6 -1m s) lisional broadening mechanisms), as well as the instru-
00..24 °Net CB Rel. to 0 (Log 10112...050 omuRfaMelnsstwteabiltllehatCprwBreoreofi−ntleaR.toiMbosnCewroirvtnahastoteeuiqsotCunaBesnn)dwtlfyioti,rnhjwseaycensttedeedxmpwpsliortowrhefioidtluehttFhaCeWvBarHer(Miseii.dtesy-..
We remind the reader that at this stage all models are
-1m s) injected with local Gaussian profiles (though the disc-
ocity ( -0.0 intTehgrearteesdidCuBalmRoMdecluprrvoefislefsorarsetaarssymwimthetaricfi)x.ed intrin-
Vel sic profile FWHM of 5 km s−1 and vsini from 1 - 10
-0.2 km s−1 are shown in Figure 3 for both stellar mod-
els (left: SOAP-T; right: C14 grid). One might ex-
pect the amplitude of these residuals to decrease once
-0.4
the LOS stellar rotation is large enough to dominate
the RVs over the variation in local CB. Interestingly,
-0.6 this is not observed (however, do note that this is
0.48 0.49 0.50 0.51 0.52
Phase the case if the residuals are normalised by the max-
imum amplitude of the RM signal). The amplitude
Fig. 2.—Main: The measured RVs fromatransit injected into
the C14 grid for a non-rotating star (with Gaussian line profiles of these residuals varies from ∼0.1-1 m s−1, depend-
injectedintothediscwithaFWHM=5kms−1). Inset: Schematic ing on vsini, which will be important for, and de-
oftheplanettransitacrossthestellardisc,colour-codedbythelog tectable with, future instruments such as ESPRESSO.7
ofnetconvective velocitiesrelativetodisccentre.
Forthe slowlyrotatingstars,these residualsshowasim-
amplitudeofthiswaveformwilldependontheplanet-to- ilar overall behaviour to that seen in Figure 2. How-
star ratio and the convective properties of the star). ever, as vsini becomes larger than the injected pro-
file FWHM the ingress and egress regions switch from
It is also important to note that the inclusion of the
blueshifted to redshifted. The originfor this unexpected
CBvariationacrossthestellarlimbcausesanasymmetry
behaviour is not clear, but could be related to the er-
in the disc-integrated line profiles. This asymmetry is
rors introduced when fitting a Gaussian function to an
seen even for out-of-transit observations and even if the
asymmetricprofileand/orbecausethelimbcontribution
intrinsicprofilesareGaussian. Moreover,itleadstonon-
(where the net CB is most redshifted) impacts the disc-
zero out-of-transit RVs in the models with CB (that are
integratedprofilemoreoncethevsiniisgreaterthanthe
removed as we are only interested in the relative RVs).
intrinsic broadening (Gray & Toner 1985; Smith et al.
This effect is similar to the ‘C’-shape bisector seen in
1987; Dravins & Nordlund 1990; Bruning & Saar 1990).
stellar observations of cool stars (Gray 2005). In this
Greaterstellarrotationalsoleadstoanincreasedredshift
instance, the asymmetry arises from the combination of
at mid-transit and a decreased redshift in the regions
limb-darkeningandradialCBvariation,i.e. thebrightest
between ingress/egress and mid-transit. Hence, a larger
regionsofthedisc(nearthecentre)willhaveprofileswith
stellar rotation increases the overall amplitude between
a much bluer net CB compared to the darker regions of
the local maxima and minima in this region (which ex-
the disc (near the limb), that will have profiles with a
cludestheingress/egresspoints). Thebehaviourofthese
local CB that is redshifted relative to the value at disc
residuals is similar in both SOAP-T and the C14 grid,
centre. Hence, integrated annuli near disc centre will
thoughtheexactshapeandamplitudeofthecurvesdoes
have a different brightness and net RV shift compared
differ slightly (likely due to the tiling differences). We
to those near the limb, and summing over these annuli
also found a very similar, though opposite, behaviour
createstheasymmetry. Thelevelofasymmetrywillvary
in the residuals when we held the vsini constant (at
based on the FWHM of the injected line profile and the
5 km s−1) and varied the injected line profile FWHM;
stellar rotation. This asymmetry also depends on the
this is because the shape of the disc-integrated profile
shape and amplitude of the centre-to-limb CB, which is
depends heavily on both the rotational broadening and
expected to increase with decreasing magnetic field (as
the width of the intrinsic profiles on the stellar surface.
the convective flows will flow more freely), and on the
Note thatunlike the RMcurvein Figure2 (whichhad
observedstellarlinesandthespectraltype(notevarying
CBvariation,butnostellarrotation),theseresidualsare
these parameters is beyond the scope of this paper).
notsymmetricaboutmid-transit(inagreementwiththat
4. RMCURVESWITHANDWITHOUTCBEFFECTS foundinDravins et al.2015); this isparticularlyevident
in the ingress/egressregions. From a purely mathemati-
4.1. The impact of vsini and intrinstic profile FWHM
calpoint-of-view,theseresidualsshouldbesymmetricas
The observed RVs depend not only on the given star- they are the result of an odd function (stellar rotation
planetsystem(i.e. star/planetmasses,radii,orbitalsep- RVs) being subtracted from a function which is the sum
aration,inclination,and alignment), but also onthe line of an odd and even function (stellar rotation RVs + ra-
broadening inherent to the star as this impacts the ob- dial CB variations). To understand the non-symmetric
servedline profile asymmetries,and hence the measured residuals, it is important to keep in mind that the RVs
line centre. The disc-integrated profile width/shape
depends on the observed stellar rotation (i.e. vsini) 7 WenotethatinthisRVregime,otherphysicaleffects suchas
and the intrinsic profile width (set largely by convec- gravitational microlensing of the transiting planet may also need
tive broadening, i.e. ‘macroturbulence’, and thermal tobetakenintoaccount (Oshaghetal.2013b).
5
SOAP-T: RM Residuals (with CB - without CB); Injected FWHM = 5 km s-1 C14 Grid: RM Residuals (with CB − without CB); Injected FWHM = 5 km s−1
νsin(i) = 1 km s−1 νsin(i) = 1 km s−1
1.0 νsin(i) = 2 km s−1 1.0 νsin(i) = 2 km s−1
νsin(i) = 3 km s−1 νsin(i) = 3 km s−1
νsin(i) = 4 km s−1 νsin(i) = 4 km s−1
νsin(i) = 5 km s−1 νsin(i) = 5 km s−1
νsin(i) = 6 km s−1 νsin(i) = 6 km s−1
νsin(i) = 7 km s−1 νsin(i) = 7 km s−1
νsin(i) = 8 km s−1 νsin(i) = 8 km s−1
νsin(i) = 9 km s−1 νsin(i) = 9 km s−1
νsin(i) = 10 km s−1 νsin(i) = 10 km s−1
0.5 0.5
-1m s) −1m s)
Velocity ( Velocity (
0.0 0.0
-0.5 −0.5
0.48 0.49 0.50 0.51 0.52 0.48 0.49 0.50 0.51 0.52
Phase Phase
Fig.3.—ResidualRMcurvesforbothSOAP-T(left)andtheC14grid(right),wheretheresidualsaredefinedasobservationswithCB
-observations withoutCB.InbothcasestheFWHMoftheinjectedprofileis5kms−1 andthevsiniisvariedfrom1-10kms−1.
are measured by fitting a Gaussian function to the ob- amplitudeoftheresidualsinFigure3isatleastpartially
served disc-integrated line profile. due to the errors introduced by fitting Gaussians to the
FittingaGaussianfunctiontoanasymmetriclinepro- disc-integrated profiles in order to obtain the RV.
filedoesnotprovidethetruevelocitycentroidofthevisi-
ble light. Ifwe areinterestedinrelativevelocitychanges Toillustratethe shapesofthe disc-integratedline pro-
then this offset does not matter, as long as the asym- files on either side of mid-transit, we show profiles at
metry remains the same. For a (model) star with CB ingress divided by those at egress (where the asymme-
andwithoutstellarrotation(seeSection3),theasymme- try in the RM residuals seen in Figure 3 is largest) in
triesinthedisc-integratedlineprofileswillchangeduring the top plot in Figure 4; the two bumps are due to the
transit. However,sincetheCBisanevenfunction,these Doppler shift between the profiles and any difference in
asymmetries will be the same for a given centre-to-limb lineshape. ThesearepresentedonlyfortheC14gridand
position, and will lead to symmetric RVs (for aligned onlyfortheobservationswhichvariedthevsini;analysis
star-planet systems) as the offsets in the true velocity usingSOAP-T(andvaryingthe FWHM)showedsimilar
centroid will also be symmetric. For (model) stars with results. If the above reasoning is correct (and the Gaus-
stellarrotationandwithoutCB,theasymmetrieswillbe sian approximation is responsible for the asymmetry in
mirror images of one another about mid-transit (hence the RM residuals)then the models that include bothro-
thetypicalRMeffect)andwillleadtoRVsthataresym- tation and CB must create profiles that differ in a way
metric about mid-transit8. For stars with both CB and thatisnotasimplemirrorimage. Iftheprofilesaremir-
stellar rotation, the asymmetries are not the same for a rorimagesofoneanother,thenflippingandreversingall
given centre-to-limb angle, nor are they mirror images thefluxvaluesthatcorrespondtothe redshiftedvelocity
of one another. As a result, the offset in absolute veloc- space should result in points that lie exactly on top of
ity as measured by the Gaussian function will vary in a thoseintheblueshiftedvelocityspace. Thistestisshown
complexway. Hence,theRVswillnotrepresentperfectly in the middle andbottom plots in Figure 4 for the mod-
the sum of an odd and even function and therefore the elsexcludingandincludingCB,respectively. Fromthese
RM residuals between the observations with and with- we see that the profiles without CB are in fact mirror
out CB will not be perfectly symmetric (however, note images of one another (as expected from stellar rotation
that the asymmetry in the residuals found here is on alone). We also see that the profiles including CB shifts
the <10 cm s−1 level). This is a fundamental limitation are definitely not mirror images of one another. Hence,
of the Gaussian fit RV technique, which can introduce this allows for the possibility that the errors in the RV
offsets/systematic errors into high precision RV studies measurementdue to the Gaussianfit maydiffer between
(e.g. seeTriaud et al.2009;Collier Cameron et al.2010; theseprofilesandcouldthereforeproduceRVshifts that
Miller et al. 2010, and references therein). Furthermore, are not equal in magnitude.
both Hirano et al. (2010) and Bou´e et al. (2013) have 4.2. The impact of line profile shape/symmetry
shown that the errors introduced by the Gaussian ap-
We also explored the impact of injecting an asymmet-
proximationwillscalewithbothvsiniandintrinsicpro-
ric intrinsic line profile into the stellar disc. In the first
file width. Accordingly, we believe that the increase in
case, we injected one line profile throughout the C14
8 Note that although these RVs will be symmetric about mid- gridrandomly chosenfrom a disc centre snapshot in the
transit, the errrorsintroduced from the Gaussianfit can stillbias solar (MHD) simulation time-series (SOAP-T does not
the analysis. For example, Triaudetal. (2009) proposed that the yet have the ability to accept asymmetric line profiles);
errorsintroducedbytheGaussianapproximationwereresponsible such a profile was chosen as the asymmetries are realis-
forthems−1residualsbetweentheirmeasuredRVsandRMmodel
tic andrepresentativeof thoseproducedby solarsurface
for the transit of HD 189733 b. Additionally, they argued that if
theseerrorswerenottakenintoaccountthemeasuredvsinicould (magneto-)convection. The left plot in Figure 5 shows
beoffbyasmuchas∼300ms−1 forthissystem. the residuals of the RM curve comparing model stars
6
with Gaussianintrinsic line profiles and no CB, to those thereforedifficulttodetectwithcurrentinstrumentation
thatincludeCBandhaveasymmetriclineprofilesfordif- (but not beyond the reach of future spectrographs). For
ferent vsini (1 - 10 km s−1). For very slow rotators(i.e. faster rotators (3 km s−1 ≤ vsini ≤ 10 km s−1) the
vsini≤ 2 kms−1) the differences are≤∼0.5ms−1 and differences can be as large as ∼ 1 - 4 m s−1 in ampli-
tude, and are therefore readily detectable with current
Line Profile Residuals (Ingress/Egress); FWHM = 5 km s−1 spectrographs such as HARPS, HARPS-N, and HIRES
1.003
With CB (note, that for higher vsini the total amplitude of the
Without CB signalwill also be higher, andthe relative impacton the
1.002 RMmodellingmaybelesssignificant). Forthefasterro-
tators,theeffectsoftheCBvariationbecomelessvisible
and the impact of the profile shape dominates. How-
1.001
ever,whatmightbe moresignificantthantheamplitude
of the residuals is the asymmetries present between the
Flux 1.000 RVs on either side of mid-transit, as well as the net ve-
locity at mid-transit (which is non-zero and dependent
νsin(i) = 1 km s−1 onthe vsini). These asymmetries in the RM curve may
0.999 ννssiinn((ii)) == 32 kkmm ss−−11 be incorrectly interpreted as a non-zero spin-orbit mis-
ννssiinn((ii)) == 54 kkmm ss−−11 alignment since the impact factor is usually fixed by the
0.998 ννssiinn((ii)) == 76 kkmm ss−−11 light curve.
ννssiinn((ii)) == 98 kkmm ss−−11 In reality, the intrinsic line shape changes as a func-
νsin(i) = 10 km s−1 tion of centre-to-limb angle. To examine this effect, we
0.997
−20 −10 0 10 20 used the same simulation snapshot as before (to avoid
Velocity (km s−1)
changes from granular evolution), but injected profiles
Line Profile Residuals without CB (Ingress/Egress); FWHM = 5 km s−1 from the snapshot when inclined as close as possible to
1.003
νsin(i) = 1 km s−1 the same centre-to-limb angles as the tiles in the stel-
νsin(i) = 2 km s−1
νsin(i) = 3 km s−1 lar grid. The residuals between the stars that excluded
νsin(i) = 4 km s−1
νsin(i) = 5 km s−1 CB and had intrinsic Gaussian profiles and those stars
νsin(i) = 6 km s−1
1.002 νsin(i) = 7 km s−1 withCBandthelimb-dependentasymmetricprofilesare
νsin(i) = 8 km s−1
νsin(i) = 9 km s−1 showninthe rightofFigure5. Theseresidualsaremuch
νsin(i) = 10 km s−1
larger in amplitude than any of the previous, with RVs
near 10 m s−1 for the fastest rotators. If the observed
Flux 1.001 CCF of the local stellar photosphere varies as much as
theinjectedline profilefromthe radiative3DMHDsim-
ulation,thenthesedifferencesshouldbeeasilydetectable
(note that an observed CCF may experience less centre-
1.000
to-limbvariabilitysinceitiscreatedfromtheinformation
contentofthousandsoflines thathaveavarietyofgran-
Blueshifted Half ulation sensitivity). We note that a high sampling rate
Redshifted Half
at ingress/egresswould be beneficial for such an empiri-
−14 −12 −10 −8 −6 −4 −2 0 calverificationsince these regions experience the largest
Velocity (km s−1)
discrepancies.
Line Profile Residuals with CB (Ingress/Egress); FWHM = 5 km s−1
1.003
νsin(i) = 1 km s−1 5. THEIMPACTOFCENTRE-TO-LIMBCBVARIATIONS
ννssiinn((ii)) == 23 kkmm ss−−11 ONSPIN-ORBITMISALIGNMENTMEASUREMENTS
νsin(i) = 4 km s−1
νsin(i) = 5 km s−1 In the previous section we have shown that ignoring
νsin(i) = 6 km s−1
1.002 νsin(i) = 7 km s−1 the effects of CB and the formation of asymmetric line
ννssiinn((ii)) == 89 kkmm ss−−11 profiles can alter predicted RVs by 10s of cm s−1 to m
νsin(i) = 10 km s−1 s−1. However, the RM effect is primarily studied to de-
termine the alignmentofplanetary systemswith respect
Flux 1.001 to the host star spin axis.
As such, we wish to quantify the impact of the con-
vective centre-to-limb variation on measurements of the
projectedspin-orbitalignmentangle,λ. Todosowesim-
1.000 ulated the aforementioned aligned(λ = 0◦) star-planet
system with a stellar model that included the CB varia-
Blueshifted Half tion to act as our observed data. To fit these simulated
Redshifted Half
observations, we applied models that assumed no con-
−14 −12 −10 −8 −6 −4 −2 0 vective blueshift terms, and intrinsic Gaussian profiles –
Velocity (km s−1)
inline with traditional RM studies. To fit the data, λ
Fig.4.— Top: Residuals from a line profile at ingress divided was allowed to vary ±30◦ in 1◦ intervals; the fits were
bytheequivalentprofileategressforobservationswith(solid)and generated using both the C14 and SOAP-T packages9.
without (dashed) CB forstars with varyingvsini. Middle: Same
astop,butonlyformodelwithoutCBandwheretheredshiftedflux
valueshavebeenflipped,reversedandoverplottedasdashedlines. 9Notewedidfurthertestfitswith10◦ stepsinλfrom40-90◦to
Bottom: Sameasmiddle,butforthemodelwithCBincluded. ensurethefitsdidnotchangeoutsidethechosen±30◦ fitinterval.
7
C14 Grid: RM Residuals with CB (Asym with CB − Gauss without CB) C14 Grid: RM Residuals with CB (Snapshot with CB − Gauss without CB)
10
νsin(i) = 1 km s−1
4 ννssiinn((ii)) == 23 kkmm ss−−11
νsin(i) = 4 km s−1
νsin(i) = 5 km s−1
νsin(i) = 6 km s−1
νsin(i) = 7 km s−1 5
2 ννssiinn((ii)) == 89 kkmm ss−−11
νsin(i) = 10 km s−1
−1m s) −1m s)
Velocity ( 0 Velocity ( 0
νsin(i) = 1 km s−1
νsin(i) = 2 km s−1
−2 νsin(i) = 3 km s−1
−5 νsin(i) = 4 km s−1
νsin(i) = 5 km s−1
νsin(i) = 6 km s−1
νsin(i) = 7 km s−1
νsin(i) = 8 km s−1
−4 νsin(i) = 9 km s−1
νsin(i) = 10 km s−1
−10
0.48 0.49 0.50 0.51 0.52 0.48 0.49 0.50 0.51 0.52
Phase Phase
Fig.5.—Left: ResidualRMcurves formodelobservations wherevsiniwasvariedfrom1-10kms−1. Theresidualswereconstructed
asthedifferencebetweenthemodelstarswherethegridwasinjectedwithGaussianlineprofileswithaFWHMof5kms−1 excludingCB,
and model stars injected with one asymmetric line profile chosen at random from a disc centre snapshot of the radiative 3D MHD solar
simulation including CB variations. Right: Same as Left, but injected with asymmetric profiles chosen from the (same) solar simulation
snapshotwheninclinedfrom0-80◦ onthestellardisc(rather thaninjectingthedisccentreprofileeverywhere).
Since the RM residuals between models with and culation 1000 times for different generations of random
without CB are dependent on the stellar rotation, we noise. The average χ2 of the 1000 generations was then
performed this comparison for both a slow (vsini = usedtocomparethemodelswithCBtothemodelswith-
2 km s−1) and moderately rapidly (vsini = 6 km s−1) out CB, with the best-fit model corresponding to the χ2
rotating star. The RM signal is also dependent on the minimum. Theobliquitiesthatcorrespondtothebest-fit
correctmodelling ofthe intrinsic profile shape, hence we models can be found in Table 2, alongside the reduced
repeated these tests while varying the intrinsic profile in χ2 (shownto illustrate the goodnessof fit betweenmod-
thestellarmodelrepresentingtheobserveddata. Thein- els, hereafter χ2). The error quoted on λ corresponds to
r
jected intrinsic profiles were either Gaussian (matching the 3σ confidenceintervalonthe χ2 minimum(i.e. since
thefitteddata),orasingleasymmetricprofile(fromthe we have 1 free parameter, λ, this interval corresponds
MHD simulationat disc centre), or a range of asymmet- to a ∆χ2 = 9); note that at times an uncertainty of 0◦
ric profiles (from a single MHD snapshot of granulation, arose due to the limitation of our 1◦ step-size in λ – for
◦
inclined from 0-80 on the stellar disc). these systems the fitted λ was allowed to vary in finer
We decided to also test two non-zero impact factors. 0.1◦ steps.
This is because for an impact factor of zero, the sym- Ifthetrueintrinsicprofilecanindeedberepresentedby
metry of the RM signal is unaffected by the spin-orbit a Gaussian function, then our best-fit models indicated
alignment if one assumes the observed RV signal origi- little or no spin-orbit misalignment. This was regardless
natesonlyfromstellarrotationandtheintrinsicprofileis oftheimpactfactorandvsinichosen,witheachscenario
Gaussian. In this scenario, changing the alignment only achieving a χ2 near 1 – though there was one instance
r
alterstheamplitudeoftheRMsignal(similartoachange when comparingwith SOAP-T that the χ2 is closerto 2
r
in stellar rotation rate – note since we know the true (fast rotator when b = 0.25). Additionally, for the C14
stellar rotation we do not suffer the usual degeneracies gridwefoundthe3σ confidenceintervalcorrespondedto
between vsini and the projected obliquity when fitting a variation in λ of ∼10◦ when b = 0, but decreasedto a
a system with b = 0). On the other hand, the shape of variation of only 1-2◦ for non-zero impact factors.
theRMsignalfromtransitswithnon-zeroimpactfactors If the true intrinsic profile is instead represented by a
areinfluencedbythespin-orbitalignment(andhenceare single (i.e. constant across the stellar disc) asymmetric
typically targeted for RM observations). Including the profile, then for the slowly rotating star we can still re-
CB variation (and asymmetric intrinsic profiles) alters coverλ values that indicate spin-orbit alignment. Again
the symmetryofthe RMsignal,regardlessofthe impact the errors on λ were much larger when b = 0 for the
factor. Hence, for a more complete view of the influence C14 comparison, but the fit was worse than when the
ofconvectionon the measurementsof λ we alsoconsider true intrinsic profile was Gaussian. For the fast rotator
impact factors of 0.25 and 0.5. Exploring additional im- with b=0, there were two local minima at λ=±23±1◦
pact factors is beyond the scope of this paper and will for the C14 case and two local minima at λ = −25 and
be pursued in forthcoming publications. +23+12◦ for the SOAP-T case (see bottom right of Fig-
−5
To determine the impact on the measured λ, we per-
ure6forillustrationofthetwolocalminima);thefitwas
formed a χ2 minimisation between the models with CB
also much worse with χ2 = 2.72 and 2.79, respectively.
and those without. Before doing so, we added Gaussian r
noiseatthe 0.5ms−1 level(consistentwithhigh-quality Hence, for this case we could not recover the spin-orbit
alignmentwhenignoringthe CB effects. When b= 0.25
HARPS observations) to the models with CB acting as
and0.5,wewereabletorecoverthespin-orbitalignment,
the observed data. The χ2 calculation was then deter- but then the fits achieved a χ2 = 7.53 and 6.56, respec-
r
mined in a Monte-Carlo fashion by repeating the cal-
8
TABLE 2
Recovered obliquities ofthe alignedmodel RM observationsasdetermined byχ2 minimisation
StellarGrid C14
ImpactFactor b=0.0 b=0.25 b=0.5
vsini
IntrinsicProfileRepresentedbyaGaussian
2kms−1 λ=−5+17◦;χ2=1.19 λ=2+1◦;χ2=1.06 λ=0.3+1.5◦;χ2=1.07
−6 r −2 r −0.9 r
6kms−1 λ=0+8◦;χ2=1.25 λ=0.5±0.6◦;χ2=1.05 λ=0.1+0.3◦;χ2=1.02
−7 r r −0.2 r
IntrinsicProfileRepresented byaSingleAsymmetricProfile
2kms−1 λ=3+10◦;χ2=1.39 λ=2±2◦;χ2=1.22 λ=1±1◦;χ2=1.19
−16 r r r
6kms−1 λ=±23±1◦;χ2=2.72 λ=1.5+0.7◦;χ2=7.53 λ=0.7+0.2◦;χ2=6.56
r −0.3 r −0.4 r
IntrinsicProfileRepresentedbyaRangeofAsymmetricProfiles
2kms−1 λ=3+6 ◦;χ2=1.21 λ=1+2◦;χ2=1.16 λ=1±1◦;χ2=1.37
−12 r −1 r r
6kms−1 λ=±27±1◦;χ2=4.41 λ=0.1+0.6◦;χ2=18.85 λ=0.5+0.2◦;χ2=10.96
r −0.5 r −0.4 r
StellarGrid SOAP-T
IntrinsicProfileRepresentedbyaGaussian
2kms−1 λ=−3+8◦;χ2=1.18 λ=−2+10◦;χ2=1.05 λ=−1+8◦;χ2=1.18
−5 r −12 r −7 r
6kms−1 λ=−6+11◦;χ2=1.05 λ=0+4◦;χ2=1.97 λ=0±3◦;χ2=1.14
−6 r −6 r r
IntrinsicProfileRepresented byaSingleAsymmetricProfile
2kms−1 λ=−11+3◦;χ2=1.27 λ=−1+12◦;χ2=1.32 λ=−4+6◦;χ2=1.03
−2 r −14 r −8 r
6kms−1 λ=−25,+23+12◦;χ2=2.79 λ=−2+10◦;χ2=3.95 λ=−1+6◦;χ2=2.33
−5 r −8 r −5 r
IntrinsicProfileRepresentedbyaRangeofAsymmetricProfiles
2kms−1 λ=−5+8◦;χ2=1.02 λ=5+8◦;χ2=1.02 λ=−2+9◦;χ2=1.06
−7 r −7 r −8 r
6kms−1 λ=±27+10◦;χ2=2.75 λ=0+8◦;χ2=4.31 λ=−1±6◦;χ2=3.63
−5 r −7 r r
tively for the C14 case and 3.95 and 2.33, respectively, we cannot conclude that a degeneracy between CB and
for the SOAP-T case. Note that given the degrees of λ can be completely broken for non-zero impact factors
freedom in this data set, according to a χ2 distribution asthiswouldrequireustoexplorearangeofimpactfac-
there is a < 0.1% probability of achieving a χ2 > 1.8, tors and star-planet systems, as well as allowing for ad-
r
and therefore any fits with such a high χ2 should not be ditional effects such as differential rotation (all of which
r
trusted. is beyondthe scope ofthis paper, but will be pursued in
forthcoming publications).
Finally, we consideredthe case when the true intrinsic Overall,ourresultsprovideevidencethatthe presence
profiles were represented by limb-dependent asymmetric ofavariableCBmayinflateerrorsonλ,atleastforvery
profiles. In this case, the fits respond similarly to the lowimpactfactors. Additionally,both stellargridsshow
previous case with the constant asymmetric profile: the that neglecting to model an asymmetric intrinsic profile
errorsonλwerehigherwhenb=0fortheC14case,and is more important for fast rotatorsand may result in in-
alignmentwasfound forallimpactfactorsfor the slowly correct misalignment measurements and/or very poorly
rotating star and also for the fast star when b 6=0. The fit models (which may cause observers to overestimate
maindifferencebetweenconsideringarangeofasymmet- their errors in an attempt to improve the fit).
ricprofiles,ascomparedtoasingle(constant)asymmet-
ric profile, was that the goodness of fit was significantly 6. SUMMARYANDCONCLUDINGREMARKS
worseforthefastrotatingstarwhencomparingwiththe Throughoutthispaper,wegobeyondtheclassicalRM
C14 grid (with χ2r = 4.41,18.85, and 10.96 for b = 0, modellingbyincludingtheexpectedvariationsacrossthe
0.25, and 0.5, respectively). We note that such poor stellar disc in the both the net convective blueshift and
fits could cause observers to assume they have under- the stellar photospheric profile shape. To study the im-
estimatedtheir errors,evenif they havein factobtained pactofthesevariationsweusedtwodifferentstellarmod-
the trueobliquity. Inturn,this maypromptarenormal- els,SOAP-TandtheSun-as-a-stargridfromCegla et al.
isation of the errors to achieve a best fit χ2r closer to 1; (2014). We simulated the transit of an aligned hot
inwhichcase,someerrorsonλreportedintheliterature Jupiter with a 4 d orbit, and explored a range of (solid
may actually be overestimatedfor faster rotators. body) stellar rotation rates and intrinsic profile widths
Ingeneral,theC14gridproducedmuchlargererroron and shapes. The convective centre-to-limb variation in
λ when b = 0, and also to a lesser extent when the star themodelstarswasbasedoffresultsfroma3DMHDso-
rotatedslower. Thisisbecausetheχ2-λdistributionhas lar surface simulation. The asymmetry/shape of the in-
abroadminimawhenb=0thatnarrowswithhigherim- trinsic profile, representing the stellar photosphere, was
pact factors (and is also slightly narrower for the faster varied by injecting granulation line profiles synthesised
star) – see Figure 6 for examples. Hence, there is a de- fromthe aforementionedMHDsimulation;notethe sim-
generacy between the minimum χ2 and the recoveredλ, ulated line profiles were taken from only one position in
atleastfor verylow impactfactors. This indicates apo- timeaswewantedtoisolatethecentre-to-limbvariations
tential degeneracy between recoverable obliquities and from any temporal variability (i.e. granular evolution).
the CB variation. However, the narrowing of the χ2-λ We also quantified the impact of these convective effects
distribution for higher impact factors indicates that this on the measured obliquity of this planetary system.
potentialdegeneracymayweakenwhenb6=0. Notethat To quantify the impact on obliquity, we examined
9
FWHM = 5 km s−1, νsin(i) = 2 km s−1; b = 0.0 FWHM = 5 km s−1, νsin(i) = 2 km s−1; b = 0.25
200 200
180 180
160 160
χ2 140 χ2 140
120 120
100 100
χ2min at −5° χ2min at 2°
80 80
−30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30
λ (°) λ (°)
FWHM = 5 km s−1, νsin(i) = 2 km s−1; b = 0.5 FWHM = 5 km s−1, νsin(i) = 6 km s−1; b = 0.0
200
800
180
160
600
χ2 140 χ2
400
120
100 200
χ2min at 1° χ2min at ± 23°
80
−30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30
λ (°) λ (°)
Fig.6.—χ2 maps forfourdifferentsystems,usingtheC14grid. Thesolidvertical linesindicatetheχ2 minima,thehorizontal dashed
anddot-dashed lines representthe ∆χ2=9regions, andthe vertical dashed redlines indicatethe correspondingλlimitsthat fallwithin
∆χ2=9(andthereforeindicatea3σconfidenceintervalontheminimumχ2). TopandBottomLeft: illustratethedecreaseindegeneracy
between χ2 and λ at increasing impact factor, in clockwise order (examples illustrated only for the Gaussian intrinsic profile scenario).
Bottom Right: illustratesadoubleχ2 minimumfound(exampleisforthesingleasymmetricintrinsicprofilescenario).
the best-fit (as determinedby χ2 minimisation) between • The shape and amplitude of the residuals between
models without convection (but with a variety of obliq- RM curves with and without a centre-to-limb con-
uities) and models with convective centre-to-limb varia- vectivevariationdependonthestar’svsiniandin-
tions (and a variety of true intrinsic profile shapes, i.e. trinsic profile FWHM. The amplitude of the resid-
Gaussian, constant asymmetric, range of asymmetries). uals increase with increasing vsini, and decreas-
These tests were carried out for both a fast (vsini = 6 ing FWHM. We believe this unexpected behaviour
km s−1) and slowly (vsini = 2 km s−1) rotating star, could be related to two phenomena. First, fitting
and for systems with impact factors of 0, 0.25, and 0.5. a Gaussian to an asymmetric profile produces off-
The findings of our study are summarised below: sets from the true velocity centroid, and these off-
sets/errors increase with increasing vsini and de-
• The presence of a centre-to-limb variation in the creasing FWHM. Second, it may be caused by the
net convective blueshift produces an asymmetric increased contribution from the limb to the disc-
disc-integrated profile, even if the local intrinsic integratedprofileatgreatervsini(Bruning & Saar
line profiles are Gaussian. This is because limb 1990; Smith et al. 1987, and references therein),
darkening creates an uneven weighting across the where the net CB is most redshifted.
radially symmetric centre-to-limb velocity shifts
• Whenthevsiniofthe starislessthentheFWHM
(e.g. anannuliatdisccentrehasadifferentbright-
of the intrinsic profile, the residuals between a
nessandnetRVshiftthananannulinearthelimb).
model star with and without a centre-to-limb con-
• The RVs measured during transit should be the vective variation results in a blueshift at ingress
sum of an odd (stellar rotation) and even (convec- and egress (where the obscured convective veloci-
tive variation) function. However, this is not re- ties are most redshifted) and a redshift near mid-
flectedinthevelocitycentroiddeterminedfromthe transit (where the convective velocities are most
mean of a Gaussian fit because the profiles on the blueshifted). However, if the vsini of the star is
blueshiftedhemispherehaveadifferentasymmetry greater than the FWHM of the intrinsic profile,
to those on the redshifted hemisphere (due to the then the ingress and egressare also redshifted; the
interplay of the rotation and convection). Hence, reason for this behaviour is not clear, but it may
the residuals between models with and without also be related to the RV fitting procedure and/or
convection are slightly asymmetric. the increased contribution from the net CB at the
10
limb once the vsini is greater than the intrinsic spectrographs if the vsini is greater than ∼ 3 km s−1
broadening (Gray & Toner 1985). and if the shape of the intrinsic profile/CCF is non-
Gaussian. Herein, we have shown that these residuals
• The amplitude of the residuals between stars with increase with increasing vsini and decreasing intrinsic
and without centre-to-limb convective variations profile FWHM. Furthermore, these effects may even be
also depends on the correct modelling of the in- able to explain some of the correlatedresiduals reported
trinsic line profile shapes. For slow rotators, in the literature between observed transits and previous
vsini ≤ 2 km s−1, the impact of the convective RMmodels(e.g. those foundforHD 189733;Winnet al.
blueshift contribution can be seen in the residuals, 2006;Triaud et al.2009–note,thisisinagreementwith
and dominates over the intrinsic profile modelling, the hypothesis putforwardby Czesla et al.2015intheir
with amplitudes < 0.5 m s−1. While these effects study of the centre-to-limb intensity variations).
may be negligible now, this is unlikely to be the Inforthcomingpublications,weaimtosearchforthese
caseoncespectrographsreach10cms−1 precision. effects observationally and also to predict them for a
Forfasterrotators,3kms−1 ≤vsini≤10kms−1, variety of star-planet systems (e.g. with varying obliq-
anincorrectmodellingoftheintrinsicprofileshape uity,planetmass/radius/separation,impactfactors,stel-
dominates the residuals. If the true intrinsic pro- lar rotation, and spectral type/magnetic field strength).
filecanberepresentedbyoneconstant,asymmetric Of particular importance is to quantify the convective
profile(butisincorrectlymodelledbyaGaussian), contribution to the observed RM signal for small plan-
theresidualsrangedfrom∼1to4ms−1,butifthe ets,asitmaycompletelydominateoverthe contribution
asymmetries changed across the stellar disc then fromstellarrotation(especiallyforslowrotators),andto
the residuals ranged from ∼ 0.5 to 9 m s−1 (with accountfor temporalvariationsfromgranularevolution.
greaterresidualsforgreatervsini). Theexactam- As instrumental precisionincreasesit is ever more im-
plitude of the residuals will depend on the convec- portant to correctly account for the contribution from
tivepropertiesofthestarandthelevelofasymme- the stellar surface in the observed RVs of high precision
try of the observed intrinsic line profile/CCF, and transitmeasurements. Our results indicate thatneglect-
therefore may be greater or less than found here. ingtodosaymayhamperand/orbiasourinterpretation
ofplanetary evolutionand migration. Fortunately,some
• For a hotJupiter with a 4d orbitabouta Sun-like
of the residuals from failing to account for convection in
star, neglecting to account for the centre-to-limb
theobservedRMwaveformshouldbereadilydetectable,
variation in convective blueshift led to an uncer-
andthereforemayhelpconfirmtheproperwaytoinclude
tainty in the obliquity of ∼10-20◦ for aligned sys-
the convective effects in future RM modelling.
tems with an impact factor of 0. We believe this
is due to a potential degeneracy between the pro-
jected obliquity, λ, and the convective blueshift.
ACKNOWLEDGMENTS
The uncertainty on the obliquity may decrease for
non-zero impact factors, down to 1-3◦. However, We thank the anonymous referee for their consid-
we cannot claim that such a degeneracy is com- ered report, that led to a much clearer and more con-
pletely broken as this was not found with both cise manuscript, and provided important insight into
stellar grids; and also because we have only tested the behaviour of the residuals. The authors would
one aligned system under the assumption of solid also like to thank E. de Mooij for useful discussions
body rotation (ignoring granular evolution and that improved computational speed. HMC and CAW
other contributions to the observedRVs). We also gratefully acknowledge support from the Leverhulme
found that neglecting to properly model an asym- Trust (grant RPG-249). CAW also acknowledges sup-
metric intrinsic profile may result in incorrectmis- port from STFC grant ST/L000709/1. MO acknowl-
alignment measurements for fast rotators (off by edges support by the Centro de Astrof´ısca da Univer-
∼ 20◦ from the true projected obliquity). Addi- sidade do Porto through grant CAUP-15/2014-BDP.
tionally, incorrectly modelling the intrinsic profile This work was supported by Funda¸ca˜o para a Ciˆencia
shapealsoproducedworsemodelfits,especiallyfor e a Tecnologia (FCT) through the research grants
thefasterrotatingstarswhichhad‘best-fit’models UID/FIS/04434/2013 and PTDC/FIS-AST/1526/2014.
with extremely unlikely probabilities (<0.1%). PF and NCS also acknowledge the support from
FCT through Investigador FCT contracts of reference
Inthispaper,wehavefoundthattheconvectivecentre- IF/01037/2013 and IF/00169/2012, respectively, and
to-limb variations in the stellar photosphere of a Sun- POPH/FSE (EC) by FEDER funding through the pro-
likestarhavethepotentialtosignificantlyaffecttheRM gram “Programa Operacional de Factores de Competi-
waveform,evenforthetransitofahotJupiter. Notonly tividade - COMPETE”. PF further acknowledges sup-
canthese variationslead to residualson the m s−1 level, port from Funda¸ca˜o para a Ciˆencia e a Tecnologia
but if unaccounted for can also lead to both incorrect (FCT) in the form of an exploratory project of refer-
(projected) obliquity measurements and incorrect error ence IF/01037/2013CP1191/CT0001. SS is the recipi-
estimations on the (projected) obliquity. entofanAustralianResearchCouncilsFutureFellowship
Theresidualspredictedbetweenobserveddataandtra- (projectnumberFT120100057). Thisresearchhasmade
ditionalRMmodels (that ignorethe centre-to-limbvari- use of NASA’s Astrophysics Data System Bibliographic
ation in convection) should be measurable by current Services.
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Beckers,J.M.2007,AstronomischeNachrichten, 328,1084 Boisse,I.,Bouchy,F.,H´ebrard,G.,etal.2011, A&A,528,A4+