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Astronomy&Astrophysicsmanuscriptno.aa10810 (cid:13)c ESO2009 January27,2009 Notes on the disentangling of spectra I. Enhancement in precision P.Hadrava 9 0 AstronomicalInstitute,AcademyofSciences,Bocˇn´ıII1401,CZ-14131Praha4,CzechRepublic 0 e-mail:[email protected] 2 Received15August2008/Accepted16October2008 n a ABSTRACT J 7 Context.Thetechniqueofdisentanglinghasbeenappliedtonumeroushigh-precisionstudiesofspectroscopicbinariesandmultiple 2 stars.Although,itspossibilitieshavenotyetbeenfullyunderstoodandexploited. Aims.Theoretical background aspectsof themethod, itslatestimprovements andhintsforitsuseinpracticeareexplainedinthis ] seriesofpapers. M Methods.Inthisfirstpaperoftheseries,wediscussspectral-resolutionlimitationsduetoadiscreterepresentationoftheobserved spectraandintroduceanewmethodhowtoachieveaprecisionhigherthanthestepofinput-databinning. I . Results.Basedonthisprinciple,thelatestversionoftheKORELcodeforFourierdisentanglingachievesanincreaseinprecisionfor h anorderofmagnitude. p - Keywords.Line:profiles–Techniques:spectroscopic–Stars:binaries:spectroscopic o r t s a 1. Introduction whereλ0isanarbitrarilychosenreferencewavelength.Itshould [ benotedthatthesameassumptioniscommonlyimposedonthe Disentanglingspectraofbinaryandmultiplestarsenablesusto solution in the wavelength domain as well, as it has been de- 1 determineefficientlythe orbitalparametersand simultaneously scribedbySimonandSturm(1994)anditisalsoobviousfrom v to separate the spectra of the componentstars. This numerical theexplanationinFig.1ofHensbergeetal.(2008)ortheirex- 2 technique performed in either the wavelength domain (Simon ampleinAppendixofthesamepaper. 6 andSturm1994)orits Fourierimage(Hadrava1995)hasbeen 2 Theuniformsamplingof inputdata simplifies the solution, applied successfully in numerous studies of individual stellar 4 but this assumption may be avoided in both methods of disen- systems. However, some users failed in their attempts or were . tangling.IntheFourierview,itisobviousthattheFouriertrans- 1 unabletotakefulladvantageofthemethod,partlyduetoamis- forms I˜(y) of the observed spectra I(x) in chosen (equidistant) 0 understanding of its principles. A review of the Fourier disen- samplingfrequenciesy maybecalculateddirectlyaccordingto 9 k tanglinghasbeenprovidedbyHadrava(2004)togetherwiththe 0 thedefinition releasein2004oftheauthor’scodeKOREL,but,regardingnew : v improvementsofthemethod,thisreviewisalreadyoutofdate. I˜(y )= I(x)exp(iy x)dx (2) i The purpose of the present series of papers is to explain some k Z k X commonmistakes,providepracticalhintsforusingthemethod, r fromanyoriginal(evennon-equidistant)binning x ifthefunc- a andpresentitsnewdevelopments. l tion I(x) is suitably interpolated, e.g. by the simple linear for- In this paper, consequencesof the discretization of the ob- mula served spectra are discussed in Sect. 2. A new method for en- hancementofthespectralresolutionindisentanglingspectrais x −x x−x I(x) = I(x) l+1 +I(x ) l (3) introducedin Sect. 3. Results and their implicationsare briefly l x −x l+1 x −x l+1 l l+1 l summarizedinSect.4. for x∈(x,x ). l l+1 The common practice of interpolating I(x) first to the equidis- 2. Samplingoftheinputspectra tantgridpointsandthenusingtheFastFourierTransformsaves thecomputertime(atsomeexpenseofaccuracy),butisnotin- In their study of disentangling, Hensberge et al. (2008) specu- evitableinFourierdisentangling. lated about“expense”at which the computationalefficiencyof In the wavelength-domain solution, the single off-diagonal the spectral method dealing with the Fourier image surpasses matrices(N in notation of Simon and Sturm 1994)shifting the the method of singular-value decomposition in the wavelength spectraofcomponentstarstotheirappropriatepositionsinindi- domain.Theysuggestedthat,amongotherthings,itmaybethe vidualexposuresmaybereplacedbywiderbandmatricesifthe needofhavingtheinputobservedspectrasampledonacommon observedspectraarenotsampledinthesameequidistantsetof gridequidistantinthelogarithmicwavelengthscale the logarithmicwavelengths into which the componentspectra are to be separated (Simon and Sturm 1994, p. 287:“The sub- x=clnλ/λ , (1) matricesofM,N andN ,arerectangularbandmatriceswith 0 Ai Bi 2 P.Hadrava:Notesonthedisentanglingofspectra a bandwidth depending on the differences in dispersion of the wavelengthscalesofc andx”).A simplepossibilityisto usea 1.0 matrixN with two non-zeroelementsin each columngivenby Eq. (3). This is, however,again equivalentto a suitable resampling, and a subsequent solution in the convenient representa- tionoftheobservedspectra.Anadditionalsignificanceofusing bandwith matrices to refine the model is discussed in the next Section. Althoughanyresamplingimpliessomesmoothingofthein- putsignal and thusa loss of informationin the high-frequency modes,itisinevitableprovidedtheobservationsarenotdirectly obtainedintherequireddatabins.Thequestionisthereforenot about performing a resampling, but how it can be performed best. This problem, which is common to any method of disentangling, is related to the more general task of optimal data- processingofobservedspectra(cf.Hensberge2004)anditsas- pectsindisentanglingwillbestudiedindetailelsewhere. Aconsequenceofdiscretizingobservedspectraislimitation of the accuracy at which the radial velocities are determined. Until now, a common practice in both the wavelength-domain and Fourier disentangling was to round the expected Doppler x shifttoanintegermultipleoftheradial-velocitystep.Thislim- ited resolution of Doppler shifts in the individual spectra also limits the precision of the disentangled orbital elements and the sharpness of the separated spectra of component stars. A Fig.1. Discretization of a Lorentzian profile I(x) (the smooth straightforward means of improvement appears to be a choice thickline)centeredonthepixelpositionshouldyieldasymmet- of a smaller sampling step. However, the resolution is limited ricdistributionofcountsD[I](x)inneighbouringbins(thethick bythedetectorsin anycase. Ontheotherhand,itisintuitively stepfunction).AslightlyshiftedprofileI′(x)(for0.2pixel-width evident that if we have a set of spectra with mutually shifted inthisfigure–seethethinlines)resultsinanasymmetryofthe sampling, we can also reconstruct details on a sub-pixel scale. counts D[I′](x), which in turn enables us to determine the line Forinstance,ifaverynarrowlinewithasub-pixelwidthmoves positionataprecisionbelowthepixelwidth. fromagivenpixelinsomeexposurestotheneighbouringpixel inotherexposures,thenitspositionmaybefoundpreciselyfrom the time and its width from the durationof this transition. It is However, owing to the resolution in the digitalized values thusworthinvestigatingthelimitsofresolutionindetails. ofintensityreadfromindividualdetectorpixels,thepositionof spectrallineswiderthanthesamplingstepcanbededucedwith anaccuracyexceedingthestepwidth(cf.Fig1).Alternativelyto 3. Anincreaseofspectralresolution a convolutionwith the shifted δ-function,a shift of a spectrum Inmyspectralmethod,theshiftinthespectraI(x)ofeachcom- I(x)forvaluevcanbeexpressedasaTaylorexpansion ponent(whichwewishtoseparatefromtheobservedsuperposi- tions),inthelogarithmicwavelengthscale x definedbyEq.(1) ∞ 1 foravaluevoftheradialvelocity,isgivenbytheconvolutionof I(x−v)= I(j)(x)(−v)j , (6) thespectrawiththeshiftedDiracdelta-functionδ(x−v), Xj=0 j! I′(x)= I(x−v)= I(x)∗δ(x−v), (4) whichusuallyconvergesrapidlyforsmallvaluesofv. Ina dis- creteequidistantrepresentationx withthestep∆ ,thefirsttwo whichimplies,intheFouriertransform(x→y),amultiplication k x derivativescanbeapproximatedbyfinitedifferences byafunctionexp(iyv), I˜′(y)= I˜(y)exp(iyv) (5) I(1)(x ) ≃ 1 (I(x )−I(x )), (7) k 2∆ k+1 k−1 x (cf. I in Eqs. (1) and(2) of Hadrava1995).Thissimple expo- j 1 nentialfunctioncanbeevaluatedpreciselyateachfrequencyy. I(2)(xk) ≃ ∆2(I(xk+1)−2I(xk)+I(xk−1)). (8) However,dueto thelimitednumber N ofthe modestakeninto x account, its inverse Fourier transform will generally produce a Thereforeinthevicinityofthegridpointx asmallshiftofIcan widerpeakwithsomeghostsonitssidesresemblinginterference k beexpressedintermsofvaluesinthisandthetwoneighbouring fringes. Only in the special case of v being an integer multiple pointsas of the gridstep, the periodof functionexp(iyv) is in resonance withtheintervallengthiny-representationandasharpshiftedδ- 1 functioncoincidingwithagridpointofthex-representationcan I(x −v) ≃ I(x )−I(1)(x )v+ I(2)(x )v2+o(v3)≃ k k k k 2 be reproduced.For that reason the radial velocity was rounded v tothenearestgridpointintheFourierdisentanglingalsoandit ≃ I(x )− (I(x )−I(x ))+ (9) k 2∆ k+1 k−1 explains why the radial velocities or their residuals calculated x by the originalKOREL-codewere quantizeddependingon the v2 + (I(x )−2I(x )+I(x ))+o(v3). radial-velocitystep. 2∆2 k+1 k k−1 x P.Hadrava:Notesonthedisentanglingofspectra 3 Theabovedescribedprocedureofreconstructingcomponent spectrafromalargesetofobservationsshouldnotbeconfused withasimpleinterpolation(givene.g.byEq.(3))inasingleob- servationorbetweengridpointsofsometheoreticalmodels.For instance,inmodelatmospheres,thedependentvariable(e.g.the 1.0 specificintensity)iscalculatedusuallyforchosenexactvaluesof theindependentvariable(wavelength)fromwhichitcanbein- terpolatedtoothervaluesorintegratedoversomeregionsofthe independent variable. On the other hand, in true observations, the values read at individual detector pixels are the quantities integratedoversomeintervalofthewavelength,whichprovide someconstraintonlyontheinnerdistribution.Withoutanyaddi- tionalinformation,thesevaluesmaybeusedinasingleexposure asanestimateofthevariableforthemiddleoftheinterval,while y 0.0 closertoitsedgesthevalueinterpolatedbetweentheneighbour- ing bins is more appropriate. However, for a set of exposures, the informationcan be combined to reveal partly also the sub- pixelstructure,or,usingtheabovedescribedprocedure,tofind subpixelmutualshiftsbetweentheexposures. As an example,we show a resultof disentanglingof simu- lateddata.Twentyspectrauniformlycoveringoneperiod(which istakentobeaunitoftime)ofadouble-linebinaryoncircular orbitwithchosenradial-velocitysemi-amplitudesK = 50km/s 1 fortheprimary,and K = 100km/sforthesecondarywerecal- 2 Fig.2. FouriertransformofthediscretizedshiftedprofileD[I′] culated. For each component, one line with a Lorentzian pro- correspondstotheFouriertransformofthecenteredprofileD[I] file (withcentraldepths0.3 and0.2of the commoncontinuum multipliedbyacorrection,therealandimaginarypartsofwhich and half-widths equal to 30 and 40 km/s for the primary and are drawn by the upper and lower (distorted sinusoidal) thick secondary,respectively) was included. A pseudo-randomnoise lines, resp., while their approximations (11) are drawn by the scaled to amplitudes n = 0%, 0.5%, 1.%, or 2.% of the con- thinsinusoidallines. tinuumlevelwasadded.Thespectraweresampledbyintegrat- ing in bins of width corresponding to 10 km/s. The results of disentangling obtained using the KOREL code in its old ver- Thisimpliesthattheoperatorδ(x−v)oftheshiftisapproximated sion(KOREL04releasedbytheauthorin 2004)andin itsnew by version (KOREL08) of enhanced precision, are compared in v Table 1. In thisTable, S denotesthe integratedsquareof spec- δ(x−v) ≃ δ(x)− 2∆ (δ(x+∆x)−δ(x−∆x))+ (10) traresiduals,and∆T0 isthedifferenceinunitsoftheperiodbe- x tweenthesolvedepochofperiastron(definedbyfixedperiastron + v2 (δ(x+∆ )−2δ(x)+δ(x−∆ ))+o(v3) longitude)and its true value T0 = 0 chosen for the simulation. 2∆2 x x Similarly, ∆K are the differences between the calculated and x 1 true radial-velocitysemi-amplitudesof the primary and ∆q for anditsFouriertransform themassratio(q= M /M = K /K =0.5). 2 1 1 2 v ItcanbeseenfromtheresultsthatthesquaresS oftheresid- exp(iyv) ≃ 1− (exp(−iy∆ )−exp(iy∆ ))+ 2∆ x x uals consist of a part approximatelyproportionalto the square x n2 of the noise, as can be supposed, but also an other addi- v2 + (exp(−iy∆ )−2+exp(iy∆ ))+o(v3)= tive,almostconstantpart,whichiscomparabletothe1%noise 2∆2x x x in the solution with the classical KOREL04, but is suppressed iv v2 for at least two orders in the super-resolution KOREL08. This = 1+ ∆ sin(y∆x)+ ∆2(cos(y∆x)−1)+o(v3). (11) partisobviouslydue to thediscrepanciesbetweensamplingof x x the componentspectra in differentexposuresshifted by a non- ItcanbeseeninFig.2thataratioofthetwoFouriertrans- integermultipleofthesamplingstep.Thiscontributiondepends formsof mutuallyshifted profilesis approximatedwell bythis on the shape of the spectrum and its importance on the level simple sinusoid for small values of y, while for the higher- of the noise. This explains why in preliminary applications to frequency modes (drawn closer to the middle of the figure) realdatathenewmethodyieldedsignificantlysuperiorresultsin higher harmonics contribute significantly. This is an obvious somecases,butonlyanegligibleimprovementinothercases. consequence of the fact that the approximations(7) and (8) of Similarly, the errors in orbital parameters have a part that thederivativesaremoreaccurateforthelowermodes,whichdo increaseswiththenoiseanda noise-independentpart,whichis notchangesignificantlyon the scale of ∆ . The exactshape of significantlysmallerinthesolutionbasedonthenewKOREL08. x theshiftoperatordependsonthespectrumtobeshifted,unless visanintegermultipleof∆ .Itmeansthatthevalueofvcannot x 4. Conclusions bedisentangledwithunlimitedprecisionfromroughlysampled, unknown spectra. However, already the application of the cor- The correctionprovidedin Eq. (11) for the residualpart of the rection(11)improvestheprecisionofthedisentanglingsignifi- radial velocities over an integer multiple of the sampling step cantly,andtheaccuracycouldbeevenhigherfordisentangling improves the Fourier disentangling significantly. With this re- constrainedbyatemplatespectrum. sult andotherimprovementscompletedby the authorto recent 4 P.Hadrava:Notesonthedisentanglingofspectra Table1.ComparisonoftheoldandnewKORELsolutions. Parameter n[%] KOREL04 KOREL08 S 0 3.61 0.012 0.5 4.98 1.22 1. 8.76 4.85 2. 23.59 19.33 ∆T [Period] 0 4.9×10−4 4.×10−9 0 0.5 4.9×10−4 6.6×10−6 1. 4.9×10−4 2.3×10−4 2. 7.2×10−4 5.0×10−4 ∆K [km/s] 0 0.49 0.07 1 0.5 0.49 0.03 1. 0.49 0.06 2. 0.33 0.18 ∆q 0 0.010 0.0005 0.5 0.010 0.0002 1. 0.010 0.0014 2. 0.013 0.0040 versionsof the KOREL code, the version of 2004 is no longer supported, and we recommend using for true applications the versionof2008. DuetotheequivalenceofEqs.(4)and(5),thewavelength- domain solution could be improved similarly if the single off- diagonalmatrixNwouldbereplacedbyathree-(off-)diagonal matrix(10),orevenbya morecomplicatedmatrix,if itshould alsoincludeaninterpolationfromanon-uniformsamplingofthe inputdata.Apossibilityforusingband-matriceswasmentioned by Simon and Sturm (1994), and in more detail explained by Sturm(1994)(cf.alsoHensbergeetal.,2008). The method described here could be improved to achieve anevenhigherprecisionforknowncomponentspectra (i.e.for the constrained disentangling), for which the higher harmonics of the approximationgivenby Eq. (11) couldbe estimated. AnalogousnumericalrefinementeitherindirectorFourierspace could be useful also in other methods in spectroscopy (e.g. in methodsusing the broadeningfunction)as well as in data pro- cessinginotherfieldsofastrophysics. Acknowledgements. This work has been completed in the framework of the Center for Theoretical Astrophysics (ref. LC06014) with a support of grant GACˇR202/06/0041.ThecommentsbytherefereeSasˇaIlijicárehighlyappre- ciated. References Hadrava,P.1995,A&AS114,393 Hadrava,P.2004,Publ.Astron.Inst.ASCR92,15 Hensberge,H.2004,ASPConf.Ser.318,43 Hensberge,H.,Ilijic´,S.&Torres,K.B.V.2008,A&A482,1031 Simon,K.P.&Sturm,E.1994,A&A281,286 Sturm,E.1994,Ph.D.Thesis,Ludwig-Maximilians-Universita¨t,Mu¨nchen