GianLuca Israel - - PowerPoint PPT Presentation

• GianLuca Israel gianluca@mporzio.astro.it

(INAF– OARoma)

withthankstoZ.Arzoumanian,C.Markwardt,T. Strohmayer (lecturesfrompreviousXray schooleditions)

• WhyshouldIbeinterested?
• Whatarethemethodsandtools?
• WhatshouldIdo?

timeseriesanalysisisaverybroadtopic,anddifficultto coverinonelecture.

presentthemostimportanttopics(partially)notdiscussedin thepreviousschooleditions.

• Timinganalysismayseema“magicbox”,sinceitcanrevealfeatures

thatarenotapparenttotheeyeintherawdata

• Timing“ analysis” isaroundsincealongtime:thinkabout

day/night,seasons,years,moonphases,etc.

• Therelevanceoftiminganalysis
• Basiclightcurveanalysis(r.m.s.)
• Fourierpowerspectralanalysis
• Powernormalizationsandsignalsearches
• Signaldetection,signalULandAsens
• Searchoptimization
• Aworkingsessionexample
• CrossCorrelation

• !" #
• Binaryorbits

– orbitalperiod – sizesofemissionregions andoccultingobjects – orbitalevolution

Accretionphenomena

broadbandvariability “quasiperiodic” oscillations(QPOs) bursts&“superbursts” Energydependentdelays(phaselags)

• Timing=>characteristictimescales=PHYSICS
• Timingmeasurementscanbeextremelyprecise!!

Spinaxis Magneticaxis Xrays

• Rotationofstellarbodies

– pulsationperiods – stabilityofrotation – torquesactingonsystem

$%&"

Isolatedpulsars(ms–10s) Xraybinarysystems Accretingpulsars(ms–10000s) Eclipses (10smin–days) Accretiondisks(~ms–years) Transientsorbitalperiods(daysmonths) Flaringstars&Xraybursters CataclysmicVariables(sdays) Magnetars (µss) Pulsating(nonradial)WDs (mindays)

• Therecouldbe

variableserendipitous sourcesinthefield, especiallyinChandra andXMMobservations Inshort,compactobjects(&supermassiveblackholes?)are, ingeneral,intrinsicallyvariable.

%$'&"

• Therootmeansquarevariability(thesameasstandarddeviation):
• Also,itiscommontoquotethe

fractionalr.m.s.,r.m.s./<RATE> …….moreover

• WemustrememberthatthelightcurvehasPoissoncounting

noise(i.e.Somerandomness),soweEXPECTsomevariation evenifthesourcehasaconstantintrinsicintensity.

• theabovedef.isbinsizedependent(i.e.Youmissany

variationssmallerthanyourtimebinsize)

(%)

– Hypothesis:thesourceisintrinsicallyconstant

• CanIrejectthishypothesis?
• Chisquarestatistic
• Ifmeasurementsaregaussian(!),thestatisticshouldhavea

chisquaredistributionwith(N1)degreesoffreedom.

• Wecancalculatethestatistic,comparetotabulatedvalues,and

computeconfidenceinourhypothesis. – AnalternativetestforvariabilityistheKStest Limits: Limits:

• Sofar,ouranalysishasfocusedonthetotalvariabilityinalightcurve.
• Thismethodcannotisolateparticulartimescalesofinterest.
• Ifweareinterestedinfastertimescales(higherfrequencies),wemust

makealightcurvewithsmallertimebins

• Theassumptionofgaussianstatisticseventuallyfails,whenthenumber
• fcountsperbinislessthan~10,andthismethodis

nolongeruseful.

Notethatalllightcurveshave50% fractionalr.m.s.variability

%*++

Light Curves

• TOTALvariability(r.m.s.)

doesnotcapturethefullinformation.Its time(scaleor(frequencyscale)is importantaswell.

,+

• Thisimportanttechniquecomesfromthetheoremthatanysignal can

bewrittenasasumofcomplexexponentials:

• Theak termsareknownasFouriercoefficients(oramplitudes),and

canbefoundbyusingtheFouriertransform(usuallyaFFT).Theyare complexvalued,containinganamplitudeandphase.

• OnceweknowtheFouriercoefficients,wehavedividedthetime

seriesintoitsdifferentfrequencycomponents,andhaveentered the frequency“domain.”

• Parsevalprovedthat:Var[fj]=

thelefthandsideisthetotal(r.m.s.)2 variance,summedintime;the righthandsideisthesametotalvariance,summedoverfrequencies. ThevaluesareknownasFourierpowers,andthesetofallFourier powersisaPOWERSPECTRUM(PSD).

,+-

Light Curves

Instrumentalnoisenotincluded!Whendealingwithnoiseonealsoneed astatisticaltooltohandleit.

($%$

• Powerspectraarecommonlynormalizedintwodifferentways.
• The“Leahy” normalizationisusefulforcomputingsignificances

(DETECTION).Inthefollowingwewillrefertoitasthedefault

• The“density” normalizationisusefulforcomputingfractionalr.m.s.

Variabilities(PHYSICS)

• Nph isthetotalnumberofphotons
• Withthisnormalization,thePoissonnoiselevelisdistributed

likeaχ2 withν =2NPSD degreesoffreedom(inunitsofcounts; NPSDisthenumberofaveragedPSD) – E[χ2|ν]=ν  2forNPSD=1 σ[χ2|ν]=sqrt(2ν) 2for NPSD=1 noisy

XTEJ1751305:accretingmspulsar.

• Perioddrifttestifiesofanorbit

withperiodof~40min

./$0+" '.

~2

./$01 2%

• Frequenciessaturate

toamaximum.Thisis likelythesignatureof the``innermoststablecircularorbit''around aneutronstar,aradiuspredictedby generalrelativityinsideofwhichmatter mustinexorablyspiraldowntosmallerradii

ScoX1

Submsoscillations seenfrom>20NS binaries

• SGR180620GiantFlare(Dic 2004)
• AsequenceofQPOfrequencieswas

detected:18,29,92and150,625, and1840Hz!

• Amplitudesinthe5– 11%range.

Asequenceof toroidalmodes

./$0' "1

Theshortestperiodin(Xray) astronomy ~2

%$$

IfyourlightcurvehasNbins,withbinsize∆tandtotaldurationT, (NOTeffectiveexposuretime)then:

• Thesmallestfrequencyyoucansampleisν =1/T :thisisalsothe

frequencyseparationbetweenpowersorfrequencyresolution)

• Thelargestfrequencyyoucansampleisν =1/(2∆t) (thisisthe

“Nyquist” limitingfrequency)

• ν andν canbechangedarbitrarlyinordertostudythe

continuumandnarrow(QPOs/coherentsignal)componentsofthePSD

• t=72s

ν=1/(2 t)~7e(3 Hz T=74000s ν=1/T~1.4e(5Hz T/ t~1024bins

$$

Theprocessofdetectingsomethinginapowerspectrumagainstthe backgroundofnoisehasseveralsteps:

• knowledgeoftheprobabilitydistributionofthenoisepowers
• Thedetectionlevel:Numberoftrials(frequenciesand/orsample)
• knowledgeoftheinteractionbetweenthenoiseandthesignalpowers

(determinationofthesignalupperlimit)

• Specificissuesrelatedtotheintrinsicsourcevariability(non

Poissonian noise)

• Specificissuesrelatedtoagiveninstrument/satellite(spurious

signals– spacecraftorbit,wobblemotion,largedatagaps,etc.)

2$"" "

Forawiderangeoftypeofnoise(includingthatofcountingphoton detectorsusedinXrayastronomy),thenoisepowersP follow aχ distributionwithν=2NPSD degreesoffreedom. However,forν=2itreducesto Correspondingly,thesignaldetection processresultsindefiningaP,such thattheprobabilityofhavingP > P issmallenough(accordingtothe

χ2

2 2 2 probabilitydistribution)

Rawdatafroma10 FF

• f1PSD

Theoreticalχ dist.

∫

+∞ − −

            Γ ≡

P P

dP e P

2 1 2 1 2 / 2

2 2 ) | (

ν ν

ν ν χ

• (

)

2 2 2

|

P

e

−

= χ

• 2

,

) ( Prob

thres

P thres noise j

e P P

−

= >

• apowerof44(inawhite

noisePSD)hasaprobabilityof e=3x10 ofbeingnoise.

2

Wedefineapriori aconfidencelevel(1ε)ofthesearch(typically 3.5σ),correspondingtoapowerP=P whichhasasmallprobability −ε tobeexceededbyanoisepower Acrucialconsideration,occasionallyoverlooked,isthenumber of trialpowersN overwhichthesearchhasbeencarriedout

• N =tothepowersinthePSDifalltheFourierfrequencyare

considered;

• N <thanthepowersinthePSDifasmallerrangeoffrequencies

hasbeenconsidered;

• N moltiplied bythenumberofPSDconsideredintheproject

2

) 2 | (

thres

P thres PSD trial

e P N N

−

= = ε

• thepreviousprobabilityof

3x10hastobemultipliedby 1.048.000trialfrequenciesand 1PSD Prob*N=3x10*1.048.000 =3x10 Stillsignificant!!

3 4.

IfnoP >P,itisusefultodetermineanupperlimittoanysignal powerbasedonthe properties.Thisisgivenby: !" ,whereP isthelargestactuallyobservedpowerinthe PSDandP" isapowerlevelwhichhasalargeprobabilitytobe exceededbyanyP# Itissometimesusefultopredictthecapabilitiesofaplanned experimentintermsofsensitivitytosignalpower.Thisiscalculated basedonthe$%& probability distributionofthenoise. !" Notethat P isinasensetheupper limittoP#'

=P

• P hastobeused

reportedinproposal.P isusedwhen reportinganondetectioninrawdata.

.+ $

YouneedtheIntensity(cts/s)ofthetargetandtheT(s)ofobs. correspondingtonetcountsN.Then,aconfidencelevelhastobeset (nσ) defines P BasedonknowPSDpropertiesonehas: relationshipbetweenA andtheN

( ) ( )( )

2 / 1 2 / 1 2

773 . 6 . 2 / sin / 6 . 2         ≈           − <

thres ph exceed thres ph sens

P N P P N j N N j A π π

• forasourceofIntensityof5ct/s,an

exposureofT=100ks =5e+5cts and~40for3.5σ c.l.(256000) A=[2.6*40/(0.773*5e+5)]^0.5=1.6%

• signaldetection

isnotpossibleforlessthan~200 photons!

!

ManydifferentclassesofX—raysourcesshowaperiodic variability whichtranslatesintononPoissonian noises(rednoise,bluenoise,low frequencynoise,shotnoise,etc.).

• powersarenotdistributedanymorelikeaχ withn

d.o.f. nostatisticaltoolstoassessthesignificanceofpowerpeaks. nostatisticaltoolstoassessthesignificanceofpowerpeaks.

! -

Threedifferentbutsimilarapproaches:(1) Rebin oftheoriginalPSD, (2) AverageofmorePSDbydividingthelightcurveintointervals, (3) EvaluationofthePSDcontinuumthroughsmoothing.Thecommonidea istousetheinformationofasufficientlyhighnumberofpowers suchthatitispossibletorelyuponaknowndistributionofthenew powersand/orcontinuumlevel(χ orGaussianorcombination). Notethatthethe processesabovemodifythePSDFourierresolution (1/T),butleaveunchangedthemaximumsampledν (1/2∆t)

IfMspectraareconsideredand/orWcontiguousfrequenciesare averaged,thenewvariable(incases1and2)willbedistributedlikea rescaledχ/MWwith2MWd.o.f.Inpractice,everythingisrescaled inordertohaveE[χ|2MW]=2MW/MW=2. Thereforeσ[χ|2MW]=sqrt(2MW)/MW lessnoisy!! NotethatforMW>30÷40theχ' Gaussian

• thenoisescatterislargelyreducedandfaintand

“extended” signalsmaybenowdetected.

! 5

$6

Thepresenceofthex/sinxtermintheamplituderelationshipimplies astrongcorrelationbetweensignalpoweranditslocation(intermsof Fourierν)withrespecttoν()* .Thepowersignalresponsefunction Decreasesof60%(from1to0.405)fromthe1st andlastfreq.

• Whensearchingforcoherentoquasi(coherentsignals

Itisimportanttousetheoriginal(ifbinnedtimeseries)orminimum(if arrivaltimeseries)timeresolution ν =const.

( ) ( )

2 / 1 2 2

/ sin / 773 . 4 1 2                 − > < = N j N j N MW P A

ph j

π π

$6-

( ) ( )

2 / 1 2 2

/ sin / 773 . 4 1 2                 − > < = N j N j N MW P A

ph j

π π Inthegreatestpartofthecasesthesignalfreq.ν+ isnotequalto theFourierfreq.ν.Thesignalpowerresponseasafunctionofthe differencebetweenν+ andtheclosestν ,isagainax/sinxterm whichvariesbetween1and0.5:foracoherentperiodicity1meansthat allthesignalpowerisrecoveredbythePSD,0.5meansthatthesignal powerisequallydistributedbetweentwoadjacentFourierfrequencies ν.

• Whensearchingforstrictlycoherentsignalsitisimportant

torelyupontheoriginal/maximumFourierresolution(1/T) donot dividethe observationintimesub(intervals.

$6$

Similarreasoningshowsthatthesignalpowerforafeaturewith finite width∆ν dropsproportionallyto1/MW whendegradingtheFourier resolution.However,aslongasfeaturewidthexceedsthefrequency resolution,∆ν > MW/T,thesignalpowerineachFourierfrequency withinthefeatureremainsapprox.constant. When∆ν < MW/Tthesignalpowerbeginstodrop.

• ThesearchforQPOs isathreestepinteractiveprocess.

Firstly,estimate(roughly)thefeaturewidth. Secondly,runagainaPSDbysetting theoptimalvalueofMWequalto theoptimalvalueofMWequalto ~T ~T∆ν ∆ν. .Twoorthreeiterationsare likelyneeded. Finally,useχ χ

hypothesistestingto

derivesignificanceofthefeature, itscentroidandr.m.s.

∆ν νο

ο ο ο

',# ',# Barycenterthedata:correctstoarrival timesatsolarsystem’scenterofmass(tools: fxbary/axbarydependingonthegivenmission). Correctforbinaryorbitalparam.(ifany) '# '# Createlightcurveswithlcurve for eachsourceinyourfieldofviewinspectfor features,e.g.,eclipses,dips,flares,large longperiodmodulations. lcstats givestatisticalinfoonthelightcurveproperties(includingr.m.s) '-# '-# Powerspectrum.Runpowspec or equivalentandsearchforpeaks. Ifnosignal calculateA (or A) Ifapeakisdetected inferν+ Onepeak likelysinusoidalpulseprofile Morepeaks complicatedprofile

• ν =0.18Hz =5.54s

T~48ks

'.# '.# Useefsearch (Pvsχ)torefinethe period.', ', ifyoualreadyknowtheperiod. Notethatefsearch usestheFourierperiod resolution(FPR),P/2T,asinputdefault. ItdependsfromP!!! ToinferthebestperiodtheFPRhastobe

• verestimatedbyafactorofseveral(ex.20).

FittheresultingpeakwithaGaussianandsave thecentralvalueandite uncertainty. OK OK forperiod,notgood notgood foritsuncertainty (whichistheFPR)

• forasignalat5.54sandT=48ks FPR=3.2e(4s

FPRinput=3.2e(4/20=1.6e(5s GC=((1.5±0.1)x10s(1σ c.l.) P= P=5.540368(0.000015=5.540353s 5.540353s FortheuncertaintyisoftenusedtheGCerrorx20(theoverestimation factorusedininput).∆ ∆P= P= 0.1x10 x20s=2x10 2x10

• s

s FinalBestPeriod: FinalBestPeriod:5.5404(2)s 5.5404(2)s (1 (1σ σ c.l c.l.) .)

5

'/# '/# Useefoldtoseethemodulation.Fitit withone ormoresinusoids.Inferthepulsed fraction(severaldefinitions)and/orthe r.m.s.RemovetheBG(itworkslikeunpulsed flux). '.0# '.0# Applyaphasefittingtechniquetoyour data(ifenoughphotons).Useefold andsave thesinusoidphaseofpulseprofilesobtainedin 4ormoretimeintervals.PlotandfitTimevs Phasewithalinearandquadraticcomponent Ifthelinearisconsistentwith0theinputP isOK IfalinearcomponentispresenttheinputPis wrong.Correctandapplyagainthetechnique.

min max min max

I I I I PF + − =

• 22

. 78 . 22 . 1 78 . 22 . 1 = + − = PF

• BestPeriod:

BestPeriod:5.54036(1)s 5.54036(1)s 1 1σ σ c.l c.l. . Afactorof~20moreaccuratethan efsearch

'.$

Itprovidesaphasecoherenttimingsolutionwhich canbeextendedinthefutureandinthepast withoutloosingtheinformationonthephase, therefore,providingatooltostudysmallchanges

• fsignalsonlongtimescales.
• Anegativequadratic

Anegativequadratic terminthephaseresiduals impliestheperiodisdecreasing

• Apositiveterm

Apositiveterm correspondstoanincreasing period Thismethodisoftenusedinradiopulsar astronomy.

• (1)ashrinkingbinary– orbitalperiod

decreasingatarateofdP/dt=1ms/yr≈(3x10s/s (2)Anisolatedneutronstarspinningdownata rateofdP/dt≈1.4x10s/s (1) (2)

(

Thecrosscorrelationmeasureshowcloselytwodifferentobservables arerelatedeachotheratthesameordifferingtimes.Italsogives informationonpossibledelaysoradvancesofonevariableswith respect totheother(inpracticalcasesonedealswithtimesorphases).

• CCFobtainedwithcrosscor.Twosimultaneouslightcurves
• fabinarysystemintwodifferentenergyintervals(softandhard).

TheCCFpeaksatpositivexandy:thetwo variablesarecorrelatedandthehard variabilityfollowsthesoftone.∆t=13±2s (1σ c.l.). ItisoftenusefultocrosschecktheCCF resultswiththespectralinformationor anyotherusefultimingresult.

(-

• Thefoldedlightcurvesinthe

soft(black)andhard(gray)bandsconfirm thepresenceofapossibledelay Thestudyoftheenergyspectrumclearly revealsthepresenceoftwodistinct components(BB+PL)inthesoft(S)and hard(H)energybandsconsideredforin theCCFanalysis. TheCCFresultisreliable/plausible!

S H

(5

! ! CCFmaybealso appliedtodatatakeninratherdifferentbands (i.e.opticalandX(ray)foragivensource.

• Samesourceasbefore,CCFobtained

fortheopticalandX(rayfoldedlightcurves (obtainedwithefold)overa4(yearsbaseline. Pseudo(simultaneousdata:samephase coherenttimesolutionused. TheCCFpeaksatpositivexandnegativey:the

• pticalandX(raydataareanti(correlatedwith

theopticaloneproceedingtheX—raysby0.16 inphase.

$

1'2'13' 1'2'13' aremostsensitivewhenno rebinning isdone(ie.,youwantthemaximumfrequencyresolution), andwhentheoriginalsamplingtimeisused(i.e.optimizingthe signal powerresponse).Alwayssearchinallserendipitoussources(N>300) 4' 4' needtobedonewithmultiplerebinning scales.In general,youaremostsensitivetoasignalwhenyourfrequency resolutionmatches(approximately)thefrequencywidthofthe signal. %%5 %%5 itisworthusingittostudytherelationamongdifferentenergies Crosscheckwithspectralinformation 6'7'+877'"1"'0) 6'7'+877'"1"'0) instrument,e.g.,CCDreadtime Pileup(washoutthesignal) (check/addkeywordTIMEDEL) Orbitalbinarymotion(“ ) Deadtime Theuseofuncorrect GTIs Orbitofspacecraft (forsingleandmergedsimult. Telescopemotion(wobble,etc.) lightcurves) Datagaps

$-

RightGTItable Wrong/noGTI table RightTIMEDEL keyword Wrong/noTIMEDEL keyword

%

vanderKlis,M.1989,“FourierTechinquesinXrayTiming”,inTiming NeutronStars,NATOASI282,eds.Ögelman&vandenHeuvel, Kluwer Superboverviewofspectraltechniques! Pressetal.,“NumericalRecipes” Clear,briefdiscussionsofmany numericaltopics Leahyetal.1983,ApJ,266,p.160 FFT&PSDStatistics Leahyetal.1983,ApJ,272,p.256 EpochFolding Davies1990,MNRAS,244,p.93 EpochFoldingStatistics Vaughanetal.1994,ApJ,435,p.362 NoiseStatistics Israel&Stella1996,ApJ,468,369– Signaldetectionin“noisy” PSD Nowaketal.1999,ApJ,510,874 Timingtutorial,coherence techniques Formorequestions:gianluca@mporzio.astro.it