Notes on the disentangling of spectra I. Enhancement in precision

0.06 MB·English

by  Petr Hadrava

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:[e-mail]
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 resam-
pling, 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 disen-
tangling, 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 harmon-
ics 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.ThecommentsbytherefereeSasˇaIlijic´arehighlyappre-
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