Planetary detection limits taking into account stellar noise. II. Effect of stellar spot groups on radial-velocities

1.9 MB·English

by  X. Dumusque

Astronomy&Astrophysicsmanuscriptno.planetary˙detection˙limits˙2˙arxiv (cid:13)c ESO2011
January6,2011
Planetary detection limits taking into account stellar noise
II. Effect of stellar spot groups on radial-velocities
X.Dumusque1,2,N.C.Santos1,3,S.Udry2,C.Lovis2,andX.Bonfils4
1 CentrodeAstrof´ısica,UniversidadedoPorto,RuadasEstrelas,4150-762Porto,Portugal
2 ObservatoiredeGene`ve,Universite´deGenve,51ch.desMaillettes,1290Sauverny,Switzerland
3 DepartamentodeF´ısicaeAstronomia,FaculdadedeCieˆncias,UniversidadedoPorto,Portugal
4 Universite´J.Fourier(Grenoble1)/CNRS,LaboratoiredAstrophysiquedeGrenoble(LAOG,UMR5571),France
1 ReceivedXXX;acceptedXXX
1
0 ABSTRACT
2
Context.Thedetectionofsmallmassplanetswiththeradial-velocitytechniqueisnowconfrontedwiththeinterferenceofstellarnoise.
n
HARPScannowreachaprecisionbelowthemeter-per-second,whichcorrespondstotheamplitudesofdifferentstellarperturbations,
a
suchasoscillation,granulation,andactivity.
J
Aims.Solarspotgroupsinducedbyactivityproducearadial-velocitynoiseofafewmeter-per-second.Theaimofthispaperisto
5
simulatethisactivityandcalculatedetectionlimitsaccordingtodifferentobservationalstrategies.

Methods.BasedonSunobservations,wereproducetheevolutionofspotgroupsonthesurfaceofarotatingstar.Wethencalculate
] theradial-velocityeffectinducedbythesespotgroupsasafunctionoftime.Takingintoaccountoscillation,granulation,activity,and
P
aHARPSinstrumentalerrorof80cms−1,wesimulatetheeffectofdifferentobservationalstrategiesinordertoefficientlyreduceall
E
sourcesofnoise.
.
h Results.Applyingthreemeasurementspernightof10minuteseverythreedays,10nightsamonthseemsthebesttestedstrategy.
p Depending on thelevel of activityconsidered, fromlogR′ = −5to−4.75,thisstrategywouldallow ustofindplanets of2.5to
HK
- 3.5M⊕ inthehabitable zoneof aK1V dwarf.Using Bern’smodel ofplanetary formation, weestimatethat for thesamerangeof
o activity level, 15 to 35% of the planets between 1 and 5M and with a period between 100 and 200 days should be found with
⊕
r HARPS.AcomparisonbetweentheperformanceofHARPSandESPRESSOisalsoemphasizedbyoursimulations.Usingthesame
t
s optimizedstrategy,ESPRESSOcouldfind1.3M⊕ planetsinthehabitablezoneofKdwarfs.Inaddition,80%ofplanetswithmass
a between1and5M andwithaperiodbetween100and200dayscouldbedetected.
⊕
[
Keywords. stars:individual:αCenB–stars:planetarysystems–stars:activity–Sun:activity–Sun:sunspots–techniques:radial-
1 velocities
v
4
5
1. Introduction turbation produced by three different physical phenomena: os-
9
cillations,granulation,andmagneticactivity.
0
. We have now discovered more than 400 exoplanets with the Oscillations of solar type stars, which can be seen as a di-
1 radial-velocity(RV) technique1. The majorityof them are very latation and contraction of external envelopes over timescales
0
massive and on very short period orbits, something that was of a few minutes (5 minutes for the Sun; Schrijver&Zwann
1
quite unexpected from the theories of giant-planet formation. 2000; Broomhalletal. 2009), is caused by pressure waves
1
However, since a few years, giant planets much more similar (p-modes) that propagate at the surface of solar type stars.
:
v to the solar system giants (e.g. Wrightetal. 2008) as well as The individual amplitudes of p-modes are typically from a
Xi super-Earthplanetshavebeendetected(massfrom2to10M⊕) few to tens of cms−1, but the interference of tens of modes
(e.g. Mayoretal. 2009b,a; Udryetal. 2007). This has become with close frequencies introduce RV variations of several
ar possiblethankstothreeimportantimprovementsoftheRVtech- cms−1, depending on the star’s spectral type and evolution-
nique. First, the precision of spectrographsimprovedconsider- ary stage (Bedding&Kjeldsen 2007; Bouchy&Carrier 2003;
ably, reaching now a 100cms−1 precision level (typically 100 Bedding&Kjeldsen 2003; Schrijver&Zwann 2000). The am-
cms−1 in 1 minute for a V = 7.5 K0 dwarf, using the HARPS plitude and period of the oscillation modes increase with mass
spectrograph on the ESO 3.6 meter telescope, see Pepeetal. alongthemainsequence.Theoryandobservationsshowthatthe
2005).Asecondimprovementhasbeenachievedthroughanap- frequenciesofthep-modesrisewiththesquarerootofthemean
propriateobservationalstrategy,whichallowsustoaverageout densityof the star andthattheir amplitudesare proportionalto
perturbationscausedbystellar oscillations(Santosetal. 2004). theratiooftheluminosityoverthemass(Christensen-Dalsgaard
Finally, several years of follow-up helped us find long-period 2004;O’Tooleetal.2008).
planetsaswellasverysmallmassones.Atthe100cms−1 level The convection in external layers of solar type stars drives
ofaccuracy,westarttobeconfrontedwithnoisescausedbythe differentphenomenaof granulation(granulation,mesogranula-
starsthemselves.Stellarnoiseistheresultofthreetypesofper- tion, and supergranulation),which also affect the RV measure-
ments on time scales going from several minutes to several
hours. In order of timescale and size of convectivepattern, we
1 seeTheExtrasolarPlanetsEncyclopaedia,http://exoplanet.eu firsthavegranulation,whosetypicaltimescaleisshorterthan25
2 Dumusqueetal.:Activitynoiseandplanetarydetection
minutes(Titleetal. 1989; DelMoro2004). Thencomesmeso- 2. Simulationofdifferentactivitylevels
granulationand finally supergranulation,with timescales of up
FordecadessurveyshavestudiedtheactivitylevelofFGKstars
to 33 hours (DelMoroetal. 2004). When integrated over the
in thesolar neighborhood(e.g. Baliunasetal. 1995;Halletal.
entire stellar disk, all these convection perturbations present a
2007).Oneofthemainresultsofthesesurveysisthebi-modality
noiselevelontheorderofthemeter-per-second.
oftheactivityleveldistribution(e.g. Vaughan&Preston1980;
Henryetal. 1996). About 70% of the stars, even those with a
On longer timescales, similar to the star rotational period,
clear Sun-like activity cycle, seem to have an activity levelbe-
the presence of activity related spots and plages perturbs pre-
lowlog(R′ )=−4.75(seeNoyesetal.1984,formoreinforma-
cise RV measurements. Spots and plages on the surface of a HK
tion about the chromosphericemission ratio, R′ ). The others
star will break the flux balance between the red-shifted and HK
have a higher activity level, and no crossing between the low-
the blue-shifted halves of the star. As the star rotates, a spot
activityandhigh-activityregionseemstooccur.Earlyresultsof
group, or a plage, moves across the stellar disk and produces
the Kepler mission arrive at the conclusion that there may be
anapparentDopplershift(Saar&Donahue1997;Quelozetal.
moreactivestarsthanexpected(seeBasrietal.2010).Ofmore
2001; Hue´lamoetal. 2008; Lagrangeetal. 2010). This effect
than100000starswithasurfacegravity,logg,higherthan4and
can be hard to distinguish from the signal caused by the pres-
aneffectivetemperaturebetween3200and19000K,50%seem
ence ofa planet.Forthe Sun at maximumactivityof cycle 23,
to be more active than the active Sun. Nevertheless, very low-
Meunieretal. (2010) find a noise related to spot groups and
plages of 42cms−1. Because the temperature of spot groups activitystarshavebeendiscoveredinthissample.
Inthepresentpaper,wesimulatetheactivityofstarsbelow
and plages are different from the mean stellar surface, and be-
log(R′ ) = −4.75. The goal is to reproduce as realistically as
cause active regions contain both, the noise induced by them HK
possible the appearance of stellar spot groups on the surface
will usually be compensated, but not entirely, because the sur-
of these stars. Because the Sun is the only star with an activ-
face ratio between spots and plages varies(e.g. Chapmanetal.
itylevelbetween−5and−4.75forwhichwecanresolvestellar
2001). According to Meunieretal. (2010), the major perturba-
spotgroups,wewilluseitasaproxyforlow-activitystars.Stars
tiveeffectofactivityonRVs isnottheoneinduceddirectlyby
with a log(R′ ) higher than -4.75 will not be simulated. First
spot groups and plages, but the one caused by the inhibition HK
of all because of a lack of information about the activity phe-
of convection in active regions (e.g. Dravins 1982; Livingston
nomenonofthesestars,andsecondly,becausetheactivitynoise
1982;Brandt&Solanki1990;Gray1992).Thiseffectleadstoa
willprobablybetooimportanttofindverysmallmassplanetsin
blueshift distortion of spectrum lines, which results in a noise
varying from 40cms−1 at minimum activity to 140cms−1 at habitableregionswiththeRVtechnique.
maximum.
2.1.ActivityoftheSun
TheEarthproducesaradial-velocityperturbationof9cms−1
on the Sun, which would be completely masked by the stellar The Sun has a 22-year magnetic cycle as well as a 11-year
noise. If we manage to understand the structure of these dif- sunspot cycle. During these 11 years, the activity level varies
ferent kinds of noise, we could use appropriate observational betweenlog(R′HK) = −5 atminimumandlog(R′HK) = −4.75at
strategies to reduce the stellar noise contribution as much as maximum.Atminimumthereisnospotgrouponthesurfaceof
possible, and thus find very small mass planets far from their theSun.Whentheactivitylevelrises,severalspotgroupsstartto
host star. This investigation of optimized observational strate- appearatalatitudeofabout±30◦.Beforereachingthemaximum
giesisessentialforfutureandmoreaccurateinstruments,such activity level,the numberofspot groupswill increase progres-
asESPRESSO@VLT(http://espresso.astro.up.pt/,precisionex- sively, whereas the latitude of spot groupswill decrease (±15◦
pected: 10cms−1) or CODEX@E-ELT (e.g. Pasquinietal. atmaximumactivity).Afterthismaximumphase,themigration
2008, precision expected: 2cms−1), in order to detect Earth ofspotgroupstowardtheequatorcontinuesandthe numberof
twins(Earth-massplanetsinhabitableregions). groups decreases progressively until they all disappear. At the
endofa cycle,the activitylevelreturnsto log(R′ ) = −5,and
HK
anothersunspotcycleof11-yearbeginsagain.
The two first types of noise, which are caused by oscilla-
tionsandgranulationphenomena,havebeendiscussedinapre- Because the latitude of the spot groups changes during the
viouspaper(Dumusqueetal. 2011, hereafterPaper I).Starting sunspotcycle,itisimportanttorecallthattheSunhasadiffer-
fromHARPSasteroseismologymeasurements,wecharacterized entialrotationinlatitude.Therefore,spotgroupsrotatefasterat
these two kinds of noise using the power spectrum representa- theequatorthanatthepoles,accordingtotheequation
tion. Then we derived an optimized observational strategy, re-
ω= A+Bsin2θ, (1)
ducingatbestthestellarnoisecontributionwhilekeepingarea-
sonabletotalobservationaltime.
whereωistheangularspeedindegree/day,θ thelatitude,and
AandBaretwoconstantsequalto14.476±0.006and−2.875±
Inthepresentpaper,wecontinuethisfirststudyandinclude
0.058(seeHoward1996).
the noise coming from magnetic activity spot groups. Starting
During the 11-year sunspot cycle or an even longer time,
from observationsof the Sun, which is the only star where we
the spot groups do not appear randomly in longitude, but
can resolve spot groups, we simulate the appearance of spot
15deg around preferred active longitudes (e.g. Ivanov 2007;
groupsonitssurfaceandcalculatetheRVcontribution.Adding
Berdyugina&Usoskin 2003). These preferred longitudes turn
thenoisecomingfrommagneticactivitytotheresultsobtained
with the Sun’s differential rotation and are observed for large
in Paper I, we develop an observational strategy that simulta-
spotgroups2.ObservationoftheSunoverseveralsunspotcycle
neously reduces the three kinds of stellar noise. To finish, we
hasshownthat
calculate the correspondingdetection limits, as well as the ex-
pectednumberofplanetsthatcouldbefoundbycomparingour 2 Largespotgroupsrepresent11%ofthetotalnumberofspotgroups
resultswithBern’smodelofplanetaryformation. observed,correspondingto67%ofthetotalspotgrouparea.
Dumusqueetal.:Activitynoiseandplanetarydetection 3
– there are normally two active longitudes in the northern
hemisphereandtwointhesouthernone,
– thetwoactivelongitudesareshiftedby180degineachhemi-
sphere,
– theactivelongitudesinthesouthareshiftedby90deg inre-
gardtothoseinthenorth,
– sometimes, each hemisphere can have a third active longi-
tude,whichisshiftedby90degwithregardtothetwonormal
ones.
Whensunspotsappear,theirdiameterisverysmall,approxi-
mately0.003R .Themajorityofthemwilldisappearinacouple
⊙
ofhours,butafewwillevolveandbecomebiggerspots,which
will gather together as spot groups. These can last for several
daysto a few months.Accordingto Howard&Murdin(2000),
thelifetimedistributionofspotgroupsis
– 50%ofthespotgroupshavealifetimebetween1hourand2
days,
– 40%ofthe spotgroupshavea lifetimebetween2 daysand
11days,
– 10%ofthespotgroupshavealifetimebetween11daysand
2months.
– FortheSunatmaximumactivity(log(R′ ) = −4.75),there
HK Fig.1. Upper Panel: Monthly averages of the sunspot
isonegroupspotpersunspotcyclethatcanlastfrom2to6
numbers. We clearly see the 11-year sunspot cycle (from
months.
http://solarscience.msfc.nasa.gov/SunspotCycle.shtml). Lower
Duringitslifetime,aspotgroupwillrapidlyincreaseinareaun- Panel: VariationoftheS-indexasafunctionoftimefortheSun
til the maximum size is reached, after which the decrease will (fromLockwoodetal.2007).
follow in a slower way (e.g Howard&Murdin 2000). The in-
creaseanddecreasetimesarenotpreciselydefined.Foroursim-
ulation,wewillsetthesetimesto1/3and2/3ofthetotallifetime
ofaspotgroup. – log(R′ ) = −5:theSunisatminimumactivityandthereis
HK
The maximum spot group filling factor3, f , is linked nospotgrouponthesolarsurface.Therefore,nonoisefrom
s,max
to the lifetime of the spot group, T, by the equation thespotgroupsperturbstheRVs(caseofPaperI).
(Howard&Murdin2000) – log(R′ ) = −4.75: the Sun is at maximum activity. The
HK
numberofspotsin thevisible hemisphereis approximately
fs,max =10−5T, (2) equal to 150 and they are located at ±15◦ of latitude (e.g
Howard&Murdin2000).
where f isexpressedinhemisphereandT indays.
s,max – log(R′ ) = −4.9:theSunisatmediumactivity,whichcor-
Once the lifetime and the maximal size of spot groups are HK
respondstoatotalnumberof75spotsat±22.5◦ oflatitude
defined,we needa probabilisticprocessto mimicthe observed
inthevisiblehemisphere.
appearance of spot groups on the solar surface. According to
Hoyt&Schatten (1998), the appearance of spot groups can be
Theactivitylevelisgivenbyadefinednumberofspots,but
described by a Poisson law. We can therefore define the prob-
the properties defined above, such as the lifetime distribution,
ability, P, to have a certain number of spot groups, N, by the
the maximum filling factor (Eq. 2), and the appearance statis-
formula
tical law (Eq. 3), are given for spot groups. In order to use all
e−λτ(λτ)k these properties, we need to know the number of spot groups
P[(N(t+τ)−N(t))=k]= k=0,1,···, (3)
k! of a definednumberofspots. Thiscan be doneusing the work
of Hathaway&Choudhary (2008), which gives the number of
wheretisthetime,τthetimestepandλtheaverageappearance
sunspots per spot group as a function of the spot group maxi-
ofspotgroupsperunitoftime.
mumfillingfactor(seeFig.2).
The upper panel in Fig. 1 represents the continuous
daily observation of sunspots (Marshall Space Flight Centre,
http://solarscience.msfc.nasa.gov/SunspotCycle.shtml).Clearly, 2.2.Activitysimulation
the number of spots varies with the 11-year sunspot cycle of
InordertocheckourabilitytodetectEarth-likeplanetsthrough
the Sun. Comparing the two graphs in Fig. 1, we notice that
activityeffects,oursimulationswilltrytomimictheSunactivity
the number of spots is correlated with the S-index and thus
with the log(R′ ) (because S and log(R′ ) are proportional, phenomenaasmuchaspossible.
see NoyesetalH.K(1984)). In conclusion, noHKspot correspondsto Forthesimulation,westartwiththefollowinginitialcondi-
log(R′ )=−5,andthemaximumnumberofspotscorresponds tions:
HK
tolog(R′ ) = −4.75.Accordingtothisargument,wedecideto
HK – Wesetthenumberofspots,N ,foragivenactivitylevel.
simulatethreedifferentlevelsofactivity: s
– Then,accordingtothelifetimedistributiondescribedinSect.
3 Thefillingfactorisjusttheratiooftheareaofthespotgroupover 2.1, we createa first spotgroupandcalculateits maximum
theareaofthevisiblehemisphere. fillingfactorusingitslifetime(Eq.2).
4 Dumusqueetal.:Activitynoiseandplanetarydetection
Fig.2.Numberofsunspotsperspotgroup(rightscale)asafunc- Fig.3.RVsproducedbyaspotgroupat15degoflatitude,witha
tionofthespotgroupmaximalareainmillionthofhemisphere fillingfactorof0.001.Thesimulationismadeusingtheprogram
(source:Hathaway&Choudhary2008). SOAP.
– This maximum filling factor gives us, using Fig. 2, the
number of spots present in this spot group, N . If
s,group
N < N ,wecreateasecondspotgroupandsoon,until
s,group s
PN = N .
s,group s
– Each spot groupis puton the latitude correspondingto the
activitylevelandonthedifferentactivelongitudes,withno
possible overlap. The longitudes of the spot groups are se-
lectedusingaGaussiandistribution4centeredonactivelon-
gitudesandwithadispersionof15deg.
Oncetheinitialconditionsarecreated,westartthetimeevo-
lutionofthesystem,usinga rotationalperiodofthe Sunatthe
equatorof26days.Thetimestepofthesimulationissettotwo
hours.Allspotgroupsbeginwithanullareaandincreasetheir
sizelinearlyuntiltheyreachthemaximumafter1/3oftheirlife-
time(seeEq.2).Thedecreasingphaseislinearlyduringthere-
Fig.4.Semi-amplitudeoftheRVsignal,K inducedbyaspot
maining2/3oftheirlifetime.Thenumberofnewspotgroupsat RV
groupasafunctionofthespotgroupfillingfactor.Theseresults
eachtimestepisgivenbytheappearancestatisticallaw(seeEq.
havebeenobtainedwiththeprogramSOAP.Wecanseeinblack
3). theresultforspotgroupsatalatitudeof15deg,andingray(red),
At the beginning of the simulation, the value for the aver- of22.5deg.
ageappearanceofspotgroups,λ,isunknown.Tofindtheright
value,webeginthesimulationwithanarbitrarynumberandcal-
culatethetotalfillingfactorforeachtimestep.UsingFig.2,this
fillingfactorgiveusthetotalnumberofspots.Ifthemeantotal
numberofspotsinnotequaltoN inseveralyears,werestartthe To model the radial-velocityvariationsimposed by stellar spot
s
simulationwithanothervalueforλ.Finally,afterseveraltests,λ groups,weusedtheprogramSOAP(Bonfils&Santos,inprep.).
issetto2.4and4.8foranactivityleveloflog(R′ )=−4.9and SOAPcomputestherotationalbroadeningbysamplingthestel-
log(R′ )=−4.75,respectively. HK lar disk on a grid. Each grid cell is assigned a Gaussian func-
FoHrKeachlevelofactivity,wegenerate100simulationswith tionthatrepresentsthetypicalemergingspectralline(or,equiv-
anequalprobabilityofhavingtwoorthreeactivelongitudesper alently, the spectrum’s Cross Correlated Function (CCF)). All
hemisphere. cellsare Doppler-shiftedaccordingto theirprojectedvelocities
towardtheobserver’sline-of-sight,andaveragedwithaweight
following a linear limb-darkening law (α = 0.6). When spot
2.3.Radial-velocitiesinducedbyactivity-relatedspotgroups groupsare added to the stellar surface, SOAP computeswhich
grid cells they occupy and change the weight for those given
Whenspotgroupsare presenton a rotatingstar,theybreakthe
cells. The weight can be set to zero for a dark spotgroup,to a
fluxbalancebetweenthered-shiftedandtheblue-shiftedhalves
fractionofthestellarbrightnessifweassumeagivenbrilliance
of the star. As the star rotates, a spot group moves across the
forthespotgroup,ortomorethanthestellarbrightnesstosimu-
stellardiskandproducesapparentDopplershifts.Thismodula-
lateplages.Finally,foragivenspotgroupconfiguration,SOAP
tionisoftenhard(sometimesimpossible)todistinguishfromthe
deliversthe modified spectral line, the flux, the radial-velocity,
Dopplermodulationcausedbythegravitationalpullofaplanet.
and the bisector span as a functionof the star’s rotating phase.
4 Because 68% of the values generated by a Gaussian distribution NotethatSOAPmerelycalculatesthefluxeffectinducedbyspot
willfallintherangeµ±σ,whereµisthemeanandσthedispersion, groups.Itdoesnotmodelanytypeofgranulation,therefore,the
thistechniqueallowsustoput68%ofthetotalspotgroupsareaat15◦ inhibition of convection(granulation)in active regionssuch as
aroundtheactivelongitudes(67%observed,seeSect.2.1). spotgroupsisnottookintoaccount(Meunieretal.2010).
Dumusqueetal.:Activitynoiseandplanetarydetection 5
Fig.5. Simulations for an equatorial rotational period of 26 days and an activity level log(R′ ) = −4.9 (left panel) and for
HK
log(R′ ) = −4.75 (right panel). The horizontal line on the top and bottom graphs are the RV mean and the spot number mean
HK
overthe4yearsofthesimulation,respectively.Justthe400firstdaysareshownhereforclearness.
Normallyaspotgroupisconstitutedofseveralsmallspots. of48cms−1.Notethatthisvariationisvalidonlyforspotgroups
For the simulation we considered each spot group as a unique andwasderivedwithouttakingintoaccountplagesorinhibition
bigspottofacilitatestheoperation.WeusedtheprogramSOAP oftheconvectioninactiveregions.Inourcase,after100simu-
to calculate the RV effect induced by a spot group presenting lationsofahigh-activitylevel,wearriveatavalueof63cms−1.
a filling factor, f of 0.001 and a latitude of 15deg (Fig. 3) or Ourestimateis30%higherthattheonefoundbyMeunieretal.
s1
22.5deg.Thefirstlatitudecorrespondstoalog(R′ )of-4.75and (2010),whichisbasedonrealpositionandsizeofspotgroups.
HK
thesecondtoalog(R′ )of-4.9.TogettheRVeffectforanyspot This can be explained because Meunieretal. (2010) used ob-
HK
groupfillingfactor, f ,wefirstcalculatetheRVsemi-amplitude servations of the solar cycle 23. This was a fairly quiet cycle,
s2
withSOAP,K ,whichisinducedbydifferentspotgroupsizes. presentingat maximumno morethan 100 sunspotsin the visi-
RV
Then we fitted a power law model to these data (Fig. 4) and blehemisphere(seeFig. 1). Inourcase, themaximumnumber
foundthat ofsunspotswasbasedoncycles21and22,whichweremorein-
tense(150sunspotsinthevisiblehemisphereatmaximumactiv-
– foraspotgroupatalatitudeof15deg,K ∝ f0.90,
RV s itylevel).InFig.7weshowthecalculatedRVvariationobtained
– foraspotgroupatalatitudeof22.5deg,K ∝ f0.94.
RV s fordifferentvaluesofthetotalspotnumber.Aquadraticfittothe
These results are consistent with the relation presented in datapointsindicatesa RV variationof51cms−1 for100spots,
Saar&Donahue (1997), where the authors have found that which is very close to the value cited in Meunieretal. (2010).
K ∝ f0.90. We will use this relation in our simulation for Thus, our simulation, starting from very simple characteristics
RV s
alllatitudes. Supposingthatbesidesthe amplitude,the RV sig- ofspotgroupssuchasthelifetimedistribution,thespotnumber
nal has the same shape for any spot group size, we calculate accordingto the activity level, and the Poisson process of spot
the proportionality coefficient between two signals, which is groupappearance,managesverywelltorepresenttheRVeffect
K /K = (f /f )0.9.MultiplyingtheRVeffectinducedby induced by spot groups. Note that the inhibition of convection
RV2 RV1 s2 s1
a spot group with f = 0.001 by this coefficient yields the RV in active regions is not yet included in our simulation, and ac-
s
effectinducedbyanyspotgroupsize. cordingto Meunieretal.(2010),thiseffectcouldbedominant.
Thesimulationdoesnotconsiderspotgroupswithalifetime Therefore,theRVeffectoftheactivityisnotyetfullysimulated
shorter than one day, because the maximumfilling factors will andcouldbeunderestimated.Theverygoodagreementcompar-
beverysmallanthereforetheRVcontributionnegligible. ing the spot flux effect between real Sun observations and the
WepresenthereasimulationfortheSun,whichisseenequa- simulationindicatesthatoursimplemodelofspotgroupsisre-
tor on, thereforethe inclinationangle betweenthe line of sight alistic.Includingtheinhibitionofconvectioninthesimulationis
andtherotationaxisofthestar,i,equals90◦.Forotherstars,this inprogressandwillbethetopicofaforthcomingpaper.Ourac-
assumptioncannotbemadebecauseforthemajorityofthemwe tivitysimulationpresentsanadvantagecomparedwiththework
do not have access to the i angle. Moreover,when i decreases, doneby Meunieretal. (2010) in the sense thatwe can adaptit
theRVwhichisequaltovsinidecreasesaswell.ThustheRV to other stars than the Sun, if basic characteristics of the spot
effect of spot groups crossing the visible stellar disc, which is groupsareknown.
proportionaltovsini, willbelower.Inconclusion,considering According to Quelozetal. (2009) and Boisseetal. (2010),
i = 90◦ inthe simulationgivesusthemaximumRV effectthat the RV signal caused by stellar spots is found at the rotational
spotgroupscaninduce. period of the star, P , and its two first harmonics.When con-
rot
sideringfewbigspotsatthesurfaceofthestarthatcanchangein
size but notdisappearrapidly,this assumptionis valid because
2.4.Comparisonwithotherworks
the activity signal stays in phase with rotation. In the two pa-
Anexampleofsimulationsforlog(R′ )=-4.9and-4.75canbe perscitedabove,theauthorsconsideractivestarsthatareknown
HK
seen in Fig. 5. The amplitudesfoundcan be comparedto a re- to be dominated by few big spots that can live for several ro-
cent result from Meunieretal. (2010), who calculated the RV tational periods. In our simulation, were we model non-active
inducedbythe real distributionofspot groupsduringthe solar stars, we consider a lot of spots that are allowed to disappear
sunspotcycle23.Formaximumactivity,theyfindaRVvariation rapidlyandreappearatotherplacesanytime.Thisrandombe-
6 Dumusqueetal.:Activitynoiseandplanetarydetection
days,theideaistoaverageouttheseeffectsbyanadequatesam-
plingofthevariationtimescale.
InPaper I, a goodobservationalstrategyto averageoutos-
cillationandgranulationnoisewastotakethree10-minutemea-
surementspernightwithtwohoursofspacing.Thisstrategywas
appliedtotheactualcalendarofHD69830,whichisoneofthe
most followed stars by HARPS. This calendar, besides obser-
vationgapsowingtobadweather,isverysimilarto10consecu-
tivenightsofobservationeverymonth.Thisidealisticstrategyof
threemeasurementspernightof10minuteseach,on10consec-
utive nights every month, will be referredto below as the 3N1
strategy. Because the typical time scales of activity noises are
longer than 10 days (see Fig. 5), the 3N1 strategy is not opti-
mizedtoaverageoutthiskindofperturbation.Inordertoaver-
ageoutallthreekindsofnoiseatthesametime,wehavetocover
Fig.7. Evolution of the RV variation as a function of the total
aperiodofamonthwithmeasurements,whilekeepingtheben-
numberofspots.Eachpointcorrespondsto100simulations.
efit of the higher frequencysampling. Taking into accountthis
last argument,we simulate the effect of two new observational
strategies.Keepingthethreemeasurementspernightof10min-
havior,ruledbyaPoissonlaw,killstheactivitysignalphaseeven utesandthe10nightsofobservationpermonth,wechangethe
ontimescalessimilartotherotationalperiodofthestar.TheRV separation between each observational night. The new consid-
signal however keeps a kind of periodicity on a rotational pe- ered observational strategy will measure the star every second
riodtimescale,buttheperiodcanbe differentform Prot andits night (hereafter 3N2 strategy) and every third night (hereafter
twofirstharmonics.InFig.6weplottheperiodogramsandthe 3N3strategy).Forthe 3N1,3N2,and3N3strategiesthe calen-
RVs for 31 day slices (1.2× Prot) of the simulation showed in darofHD69830wasnotsuitableanymoreandwehadtocreate
Fig. 5 with an activity level of log(R′ ) = −4.75. Looking at newcalendars,askingforthefollowingcriteria:
HK
the periodograms, we see that sometimes important peaks are
presentatP ,P /2andP /3(plotontheright),assuspected – Thetotalobservationaltimespanisfouryears.
rot rot rot
byQuelozetal.(2009)andBoisseetal.(2010).Atothertimes, – The star disappears from the sky during four months each
thepeakatP isclearbutnothingshowsupatP /2andP /3 year.
rot rot rot
(plotontheleft)orattherotationalperiodandthetwofirsthar- – Each month of observation, the beginning of the measure-
monics(plotonthecenter).LookingnowattheRVplots,wesee mentsisrandomlyset.
that the modelof three sine waves with period P , P /2 and
fit fit
P /3canfitthesimulatedRVsnicely.Thecomparisonbetween In addition, we randomly suppress 20% of nights to take into
fit
the rmsof the simulated RVs and the correctedone (simulated accountbadweatherortechnicalproblems.Thesecalendarsrep-
RVs minus fitted sine waves) shows that the three sine waves resentatotalof256nightsofmeasurementsoveratimespanof
modelreducestheactivitynoiseby70%.Thereforethismodel four years. In Fig. 8 (left panel) we can compare these differ-
can be used to correct short-term activity noise. However, if a entobservationalstrategies. The RV variation,inducedby spot
small mass planet orbits the stars with a period close to P , groups for a maximum level of activity (log(R′ ) = −4.75),
rot HK
the three sine waves model will kill this signal too. We have reaches a level of approximately 60cms−1. By using the 3N3
thereforetobeverycarefulwhenusingthesemodels,andonly strategy, we manage to reduce this level of RV variation to
thestudyofotherCCFparameterssuchasthefullwidthathalf 10cms−1whenaveragingtheeffectin10daybins.Forthesame
maximum(FWHM) or the bissector span (BIS) will be able to binning,the3N1strategyreducesthislevelto30cms−1.
give us clues on the nature of the signal: short-term activity or Wecannowcalculaterms asafunctionofthebinningin
RVb
planet. This work will be done once the inhibition of the con- days,includingalltypesofnoise:oscillation,granulation,activ-
vectiveblueshiftinactiveregionswillbeimplementedintoour ity, and instrumental(80cms−1). The result for αCenB(K1V)
model.Inaddition,theestimateoftherotationalperiodusingthe canbeseeninFig.8(rightpanel).
threesinewavesmodel,asshowninQuelozetal.(2009),isno Wenoticethatthe3N3strategyaveragesoutmuchmoreac-
longervalid.Indeed,weseethatthefittedperiod,Pfit,canvary tivitynoiseafterbinningthanthe3N1strategy.Withoutactivity
between22to34days,whereasthetruerotationalperiodofthe (log(R′ ) = −5)thetwostrategiesareidentical.TheRVvaria-
HK
starisfixedto26.4daysinthesimulation. tionsreachableusing10daysbins(suitableforlongperiodplan-
ets) with the best considered strategy are 22cms−1, 25cms−1,
and28cms−1foralog(R′ )equalto-5,-4.9and-4.75,respec-
3. Efficientobservationalstrategies HK
tively.Ifthereishighactivity,theimprovementbroughtbythis
TheRV signalinducedbyaspotgroupcanbedescribedasthe optimalstrategyreaches30%comparedtothe3N1strategy.
successionofnullcontributionwhenthespotgroupisinthehid- Inconclusion,samplingthecompleterotationalperiodofthe
den hemisphere, and as sinusoidalshapes when the spot group starwiththesamenumberofmeasurements,butwithmoresep-
movesacrossthevisiblehemisphere(Fig.3).ThisRVsignalis aration between them, averagesoutthe activity noise to a con-
regularintimeandiscomposedofsinusoidalsignalswithperi- siderable extent. Thus, the strategy needs to be adjusted to the
odsequaltotherotationalperiodofthestaranditsfirstharmon- stellar rotational period. Here for the Sun, which has a 26 day
ics(Boisseetal.2010).Ifthereareseveralspotgroups,theseRV rotational period, the 3N3 strategy is the optimal one with 10
signalswilladdorcancelthemselves,leadingtoaquasi-periodic measurementspermonth.Forastarwitha20dayrotationalpe-
signalwithunpredictablebutlimitedamplitude.Becausethepe- riod,the3N2strategywouldaverageoutthestellarnoisebetter.
riod of the RV signal varies between approximately 10 to 30 Inaddition,increasingthegapbetweenobservationalnightsby
Dumusqueetal.:Activitynoiseandplanetarydetection 7
Fig.6.Periodograms(top)andcorrespondingRVs(bottom)forthreedifferentslicesof31daysfortheactivitysimulationshownin
Fig.5(log(R′ ) = −4.75).Intheperiodogramsthehorizontallinecorrespondsto the1%false-alarmprobabilityandthevertical
HK
dashed lines to the rotational period of the star, P , as well as P /2 and P /3. On the RV plots, the dots correspond to the
rot rot rot
simulatedRVsandthetrianglestoafittedfunctionwiththreesinwavesofperiodP ,P /2andP /3(P isafreeparameter).
fit fit fit fit
P aswellasthermsofthesimulatedRVscanbefoundintheupperleftpartoftheplot.P aswellasthermsofthecorrected
rot fit
RVs(simulatedminusfittedsinewaves)canbefoundintheupperrightpartoftheplot.
Fig.8. Evolution of the binned RV variation, rms , owing to activity-related group spots (left panel) and to all kind of stellar
RVb
noises(rightpanel)asafunctionofthebinningindays.Thethreestrategiesshownfordifferentlog(R′ )(givenforeachlines)use
HK
threemeasurementspernightof10minutesand10nightsofobservationpermonth.Theonlydifferenceliesinthespacingbetween
eachnightofobservation.Thickslinescorrespondtothe3N1strategy,thinlinestothe3N2strategy,anddashedlinestothe3N3
strategy.
morethanthreedaysforslowrotatorsisnotrecommended,be- tal, and photon noises as we did in the previous sections. The
causeitwillinduceapoorsamplingofshortperiodplanets. calendars used depend on the selected strategy, but all have a
totalof256observationalnightsoveratimespanoffouryears
(seeSect.3).Thenwecarryout1000bootstraprandomizations
4. Detectionlimitsinthemass-perioddiagram
oftheseRVsandcalculatethecorrespondingperiodograms.For
Inordertoderivedetectionlimitsintermsofplanetmassandpe- each periodogram, we select the highest peak and construct a
riodaccessiblewiththedifferentstudiedstrategies,wecalculate distribution of these 1000 highest peaks. The 1% FAP corre-
the false-alarm probability (FAP) of detection using bootstrap sponds to the power, which is only reached 1% of the time.
randomization(Endletal. 2001; Efron&Tibshirani1998). We The second step consists in adding a sinusoidal signal with a
first simulate a synthetic RV set containing stellar, instrumen- givenperiodtothesyntheticRVs.Wecalculatetheperiodogram
8 Dumusqueetal.:Activitynoiseandplanetarydetection
Fig.9. Detection limits for αCenB(K1V)for differentactivity levels and strategies. The thick line at the top correspondsto the
presentHARPSobservationalstrategy(onemeasurementpernightof15minutes,10consecutivedayseachmonth)foranactivity
levellog(R′ )=−4.75.Thesecondthicklineattheverybottomcorrespondstothedetectionlimit,usingESPRESSO,forthe3N3
HK
strategyandanactivitylevellog(R′ )=−5.Besidesthesetwothicklines,fromtoptobottom,thedashedthinlinescorrespondto
HK
the3N1strategyusingHARPS,foranactivitylevellog(R′ )=−4.75,−4.9and−5,respectively.The3N3strategyusingHARPS
HK
correspondsto thecontinuousthinlinesonthesame orderofactivitylevel.Finally,the smalldotsrepresenttheplanetsexpected
fromBern’smodelwitharandomsiniforeachbody.Weseethatthecut-offinperiodaround12daysofthemodelintroducesan
anomalyintheplanetdistribution.ThehabitablezoneisderivedusingthemodelofSelsisetal.(2007)appliedtothetemperature
andtheluminosityofaK1Vdwarf,calculatedusingtheresultsofdeJager&Nieuwenhuijzen(1987).
ofthesenewRVs andcomparetheheightoftheobservedpeak level detectionlimits expectedfor the 3N1 and the 3N3 strate-
withthe1%FAP.Wethenadjustthesemi-amplitudeofthesig- giesusingHARPS.
nal until the power of the peak is equal to the 1% FAP. The The thick line in top part of Fig. 9 corresponds to the de-
obtainedsemi-amplitudecorrespondstothedetectionlimitofa tection limit for a maximum-activitylevel(log(R′ ) = −4.75)
HK
null eccentricity planet with a confidence level of 99%. To be using the present HARPS observational strategy (1N1 strategy
conservative,we test 10 different phases and select the highest = one measurementof 15 minutes on 10 consecutive days per
semi-amplitudevalue. month, see Paper I for details). Obviously, there is a great im-
In practice,we simulate 100RV sets for each activitylevel provementbroughtbythe3N1and3N3strategiescomparedto
(log(R′ )=-5, -4.9 and -4.75) and strategy, and to be conser- whatisdoneatpresentonHARPS.
HK
vative,onlythe 10 ”worst” cases per strategywere considered. We can also compare the level of the detection limits ob-
It is important to note that for each strategy we simulate the tained for the 3N1 and 3N3 strategies. For example, at a 100
RVsaccordingtotherespectivecalendar(10measurementsper day period, the improvement brought by the 3N3 strategy is
months on consecutive nights or every second or third night, 4%, 10% and 17%, for an activity level of log(R′ ) = −5,
HK
eightmonthsayearoverfouryears)andweremovedrandomly −4.9and−4.75,respectively.Asexpected,theimprovementin-
20 % of the nights to simulate bad weather or technical prob- creases with a rising activity level because the 3N3 strategy is
lems.Theobservationalcalendarsarethereforerealistic.Figure optimizedtoreduceactivitynoise.Forminimumactivity,which
9 presents for different levels of activity the 99% confidence correspondsto no spot groupsand consequentlyno noise from
Dumusqueetal.:Activitynoiseandplanetarydetection 9
activityinoursimulation,thedetectionlimitsobtainedwiththe 6. Detectionofasimulatedplanet
twostrategiesshouldbeidentical.The4%differencecanbeex-
To show that the calculated detection limits are realistic, we
plained by a more regular sampling (over all months with the
present in this section an example of planet detection. For
3N3 strategy and only 10 consecutive days per month for the
αCenB(K1V),wefirstcreatesomesyntheticRVsincludingall
3N1). In conclusion, the 3N3 strategy seems to be an efficient
types of noise, oscillations, granulation phenomena, and activ-
strategy to average out all kinds of noise. For example, for a
ity. The strategy used is the 3N3 and the activity levelis set to
period of 200 days, which corresponds to the habitable region
log(R′ ) = −4.9.Secondly,weaddtheRVsignalinducedbya
aroundaK1dwarfsuchasαCenB,thedetectionlimitsforthe HK
smallplanetinthehabitablezone(Msini = 2.5M ,a208day
3N3strategyrangesbetween2.5and3.5 M dependingonthe ⊕
⊕
period).Tocheckwhethertheplanetisdetectedornot,wecalcu-
activitylevel.Followingtheseresults,itseemsclearthatevenin-
latetheperiodogramandthe1%and0.1%FAP,usingbootstrap
cludingtheactivitynoiserelatedtospotgroups,planetssmaller
randomization(seeSect.4).
than 5 M in habitableregionscouldbe detectedwith HARPS
⊕
In Fig. 11, top panels, we show the raw RVs, stellar noise
usinganappropriateobservationalstrategy.
plus planet, for the 3N3 strategy as well as the added planet.
Future instruments, like ESPRESSO at the VLT, will reach
aprecisionlevelof10cms−1.Thethicklineattheverybottom Theperiodogramwiththe1%FAPforeachperiodisalsorepre-
sented.Thepeakat200daysisfarabovethe0.1%FAP,which
of Fig. 9 shows the improvement when we change the instru-
mental noise from 80 to 10cms−1. Evidently the improvement proves that the planet is easily detected. To compare this with
theactualmeasurementsmadeonHARPSforthehigh-precision
is significant, reducing approximately the mass detection limit
program, we apply the same process for the 1N1 strategy (1N
by1 M .Wenote,however,thatbecauseweusedHARPSdata
⊕
strategyinPaperI).Thisstrategyconsistsofmeasuringthestar
asabasisforoursimulations,itispossiblethatweoverestimate
15minutespernighton10consecutivedayspermonth.Thebot-
the noise of the ESPRESSO-like radial-velocities. These latter
tompanelsofFig.11showtheresults.Evidentlyheretheplanet
valuesmustthenbeseenasupperlimits.
isfarfrombeingdetectable.
Looking at Fig. 9, we see that the mass detection limit for
5. PlanetdetectionaccordingtoBern’smodel 208daysofperiodandforanactivitylevellog(R′ ) = −4.9is
HK
more than 3M for the 3N3 strategy. In this example, we see
The detection limits give us an idea of the smallest planet that ⊕
thatusingthesamestrategyandconfiguration,a2.5earthmass
could be found with an appropriatestrategy. Nevertheless, this
planetcouldeasilybedetected.Thusthedetectionlimitsderived
doesnotinformusontheproportionofsmallmassplanetsthat
inSec.4areveryconservative.Thisisbecauseweconsideronly
couldbedetected.Thisrequiresamodelofplanetaryformation
the10”worst”caseswhenderivingthedetectionlimits.
thatcanpredictthe”true”populationofsmallplanets.Weused
Bern’s model (e.g. Mordasinietal. 2009a,b), which is a good
proxyof planetpopulation.In their simulation they solve as in
7. Concludingremarks
classical core accretion models the internal structure equation
for the forming giant planet, but include at the same time disk InPaperIwehadproposedanefficientobservationalstrategyto
evolutionandtypeIandIIplanetarymigration. reducethestellarnoisegeneratedbyoscillationandgranulation
Using the detection limits derived in Sect. 4, we calculate asmuchaspossible.Thisstrategy,requiringthreemeasurements
thenumberofplanetspredictedbyBern’smodelthatareabove per night of 10 minutes over 10 consecutive days each month,
agivendetectablelevel.Theproportionofexpectedplanetsbe- improvesthe averagingout of stellar noise coming from oscil-
tween1and5M thatcouldbefoundwiththe3N3strategyon lationandgranulationby30%,withanobservationalcostonly
⊕
HARPSisrepresentedintheleftpanelofFig.10.Inthispanel, multipliedby a factor2.Inthe presentpaper,we addthenoise
we clearly see that the proportionof detection decreases when inducedbyactivity-relatedstellarspotgroups.
theactivitylevelincreases.Fortheperiodrangebetween100and To study the radial-velocity (RV) effect caused by stellar
200days,theproportionofplanetswith1M < M sini<5M spot groups, we consider three different activity levels. Based
⊕ ⊕ ⊕
that would be found with HARPS changes from 35% to 15% on Sun observations, we simulate the minimum solar activity
when the log(R′ ) varies from −5 to −4.75. Although the ac- level(log(R′ ) = −5),the maximumone(log(R′ ) = −4.75),
HK HK HK
tivitygreatlyreducesthenumberofdetectableplanets,itisstill andanintermediateone(log(R′ ) = −4.9).ComparingtheRV
HK
possible to find some in the habitable region of K dwarfs (200 variationatmaximumactivitygivenbyoursimulationwiththe
days)forthehighestcaseofactivity(log(R′ )=−4.75forlow- onecalculatedbyrealpositionandsizeofspotgroupsfromcy-
HK
activitystars,seeSect.2). cle 23 (Meunieretal. 2010), we find a very good agreement,
We notefora similaractivitylevelthatthe valueofthebin 51cms−1 and 48cms−1, respectively. We note that the RV ef-
for 300 to 400 days is lower than the one for 400 to 500 days. fect of activity is not fully simulated because the inhibition of
Because the mass detectionlimits increase with period,we ex- convectioninactiveregionsisnotyetimplemented.According
pecttheopposite.Thiseffectis owingto the observationalcal- toMeunieretal.(2010),thiseffectcouldbeimportantandcon-
endar used. Indeed, because stars in the sky can generally be sequently, the detection limits calculated here could be under-
followedduringeightmonthsperyearbeforetheydisappear(an estimated.ComparedtotheresultsofMeunieretal.(2010),our
effecttakenintoaccountinourobservationalcalendar),theone simulation is more generalin the sense that we can predictthe
year period is not well sampled, which complicates the detec- RV effectofactivityrelatedspotsgroupsforotherstars,know-
tion.ThisiswellillustratedinFig.9,wereapeaknearoneyear ingafewcharacteristics(meanspotnumber,distributionofspot
appearsforeachdetectionlimit. groupslifetime, presenceof activelongitudes,differentialrota-
A comparisonwith whatcouldbe foundusing ESPRESSO tion). This will be very important when a better knowledge of
isshownintherightpanelofFig.10.Iftheactivitylevelisset activity phenomenaof other stars than the Sun will be known.
tolog(R′ )=−5,ESPRESSOcouldfind80%oftheplanetsin TheKeplermissionshouldgiveussomecluesinthenearfuture.
HK
amassrangefrom1to5M andaperiodbetween100and200 Modeling the short-term activity with a three sine waves
⊕
days,whereasHARPScouldonlyfind35%ofthem. function with period P , P /2 and P /3 can greatly reduce
fit fit fit
10 Dumusqueetal.:Activitynoiseandplanetarydetection
Fig.10. Left panel: Proportion of expected planets with 1 ≤ Msini ≤ 5 M that could be found with HARPS around early-K
⊕
dwarfs,usingthe3N3strategy.Foreachbinfromtoptobottomwehavethedetectionproportionforlog(R′ )equalto−5,−4.9
HK
and−4.75.Rightpanel:Proportionofexpectedplanetswith1≤ Msini ≤ 5 M thatcouldbefoundwithHARPS(darkorblue)
⊕
andESPRESSO(grayorcyan)aroundearly-Kdwarfs,usingthe3N3strategy.Theactivitylevelissettoalog(R′ )equalto−5.
HK
Fig.11.Detectionsimulationfora 2.5M planetin thehabitablezone(P = 208days)ofa K1dwarfpresentingan activitylevel
⊕
log(R′ ) = −4.9.Top panels: Detection simulation for the 3N3 strategy.On the left, we show the raw RVs includingthe stellar
HK
noise(oscillation,granulationphenomena,andactivity)plustheinjectedplanet(redcurve).Inthemiddle,theperiodogramofthe
stellarnoiseonly(redfilledcurve)andthestellarnoiseplusplanet(blackline)isrepresented.Thehorizontallinescorrespondfrom
toptobottomtothe0,1%andthe1%FAP.FinallyweshowontherighttherawRVsbinnedon1monthandfoldedinphasewith
theperiodoftheplanet.Bottompanels:Thesamebutforthe1N1strategy(strategypresentlyusedforthehigh-precisionHARPS
program,onemeasurementpernightof15minuteson10consecutivedayspermonth).
the spot-induced noise by approximately 70%. However, we After simulating the effect of several observational strate-
have to be verycarefulwith this methodbecause P can vary gies, the most efficient one to average out all kinds of noise is
fit
in a non negligible way from the rotational period of the star. the3N3.Thisstrategyconsistsinmeasuringthestarthreetimes
Thus,signalofsmallmassplanetswithperiodsimilartothero- 10minutespernight,withaspacingoftwohours.Thenthemea-
tational period of the star will be killed by this type of model. surementsaretakeneverythirdnight,10dayseverymonth.With
OnlyastudyofotherCCFparameterssuchastheFWHMorthe this strategy it would be possible to find planets of 2.5 to 3.5
BIScangiveuscluesonthetruenatureofthesignal:short-term Earth mass with HARPS in the habitable region of K dwarfs
activityorplanet. (200 days). The first mass value correspondsto a case without