Table Of Content**VolumeTitle**
ASPConferenceSeries,Vol.**VolumeNumber**
**Author**
(cid:13)c**CopyrightYear**AstronomicalSocietyofthePacific
Combining and comparing astrometricdata from different epochs:
A casestudy withHipparcos and Nano-JASMINE
2
1
0 DanielMichalik1,LennartLindegren1,DavidHobbs1,UweLammers2,and
2 Yoshiyuki Yamada3
n
1LundObservatory, LundUniversity, Box43,SE-22100 Lund,Sweden
a
J e-mail: daniel.michalik, lennart, [email protected]
3 2EuropeanSpaceAgency(ESA/ESAC),P.O.Box78,ES-28691Villanuevadela
1
Can˜ada,Madrid, Spain,e-mail: [email protected]
]
M 3DepartmentofPhysics,KyotoUniversity, Oiwake-choKita-Shirakaw
Sakyo-ku, Kyoto,606-8502 Japan,e-mail: [email protected]
I
.
h
p Abstract. TheHipparcosmission(1989-1993)resultedinthefirstspace-basedstel-
- lar catalogue including measurementsof positions, parallaxesand annualpropermo-
o
tionsaccuratetoaboutonemilli-arcsecond. Morespaceastrometrymissionswillfol-
r
t lowinthenearfuture.Theultra-smallJapanesemissionNano-JASMINE(launchinlate
s 2013)will determinepositionsand annualpropermotionswith some milli-arcsecond
a
accuracy.Inmid2013thenext-generationESAmissionGaiawilldeliversometensof
[
micro-arcsecondaccurateastrometricparameters.UntilthefinalGaiacatalogueispub-
1 lishedinearly2020thebestwayofimprovingpropermotionvaluesisthecombination
v ofpositionsfromdifferentmissionsseparatedbylongtimeintervals.Ratherthancom-
9
paringpositionsfromseparatelyreducedcatalogues,weproposeanoptimalmethodto
4
combinetheinformationfromthedifferentdatasetsbymakingajointastrometricso-
8
lution. Thisallowstoobtaingoodresultsevenwheneachdatasetaloneisinsufficient
2
for an accurate reduction. We demonstrate our method by combiningHipparcos and
.
1 simulatedNano-JASMINEdatainajointsolution.Weshowasignificantimprovement
0 overtheconventionalcataloguecombination.
2
1
:
v
i
X
1. Introduction
r
a
Stellar proper motions have traditionally been computed by comparing positional cat-
alogues based onobservations madeatdifferentepochs (typically separated byseveral
decades). Parallaxes were either ignored in this process, or determined by quite dif-
ferent instruments and methods. With the advent of Hipparcos and space astrometry,
it has become necessary to treat the determination of positions, proper motions, and
parallaxesinaunifiedmanner,i.e.,inasingleleast-squaressolution. Thiswastheprin-
ciple used for the construction of the Hipparcos and Tycho catalogues (ESA1997), as
wellasforthenewreduction oftheHipparcos data(vanLeeuwen2007),anditwillbe
usedfortheGaiamissionandotherplannedspaceastrometryprojects. Inthefuturewe
will therefore have access to several independent astrometric catalogues, one for each
spaceproject. Improvedpropermotionscanagainbecomputedbycomparingtheposi-
tionsincataloguesatdifferentepochs. Howeverratherthencombiningtheresultsofthe
1
2 D.Michalik,L.Lindegren,D.Hobbs,U.Lammers,andY.Yamada
catalogues intheconventional manner, wepropose tomakeajointsolution ofthedata
fromthemissions. Inthispaperweexplorepossibleadvantages ofthisapproach,com-
biningtheHipparcosCatalogueandsimulateddatafromtheJapaneseNano-JASMINE
missionasastudycase.
Nano-JASMINE (launch planned for late 2013) is an ultra-small Japanese satel-
lite to measure the astrometric parameters of about one million celestial sources to
12th magnitude. Theexpected accuracy ofthe positions, parallaxes and annual proper
motionsofmagnitude7.5objectsisabout3mas. Nano-JASMINEisaverysmalltech-
nological demonstrator for bigger follow-up missions. Its data will be reduced using
theAstrometricGlobalIterativeSolution(AGIS)beingdevelopedfortheGaiamission
atESA/ESACandLundObservatory (Lindegrenetal.2011).
2. Theory
The estimation of stellar astrometric parameters from observational data can be cast
as a linear least-squares problem. The optimum estimate of the unknowns x is ob-
tained by solving the normal equations Nx = b where the covariance of x is given
by N−1 and where b are the residuals. In the conventional approach of combining
twoastrometriccatalogues theleast-squares solution ofbothdatasetsisdoneindepen-
dently and the combination is done a posteriori. If σ and σ are the accuracies in
1 2
the two catalogues, positions and parallaxes improve as σ−2 = σ−2 +σ−2 by comput-
1 2
ing weighted means of the values of the two catalogues. Proper motions improve as
σ = (σ2 + σ2 )1/2/∆T by taking the position difference divided by the epoch
pm pos1 pos2
difference ∆T. Instead we propose to combine the normal equations of both missions
beforesolving
(N + N )x = b + b → Nx= b, (1)
1 2 1 2
allowing us to retrieve directly xˆ of the combined catalogues. This can also be
joint
understoodintermsofBayesianestimation(assumingmultivariateGaussianparameter
errors), with N ,b presenting thepriorinformation, N ,b thenewdata,and N,bthe
1 1 2 2
posteriorinformation. Evenifeachgroupisnotsolvableonitsown,thecombineddata
maybesolvable.
3. Simulations
SimulationsarecarriedoutusingAGISLab,asoftwarepackageaidingthedevelopment
ofalgorithmsforthedatareductionofGaia,developedatLundObservatory. Forsimu-
lations the Hipparcos catalogue isused to generate observations. Three catalogues are
created(seeFigure1):
TheHipparcoscatalogue contains thesourceparametersreadfromtheHipparcos
catalogue file including their covariance matrix. The Hipparcos data are available for
the reference epoch J1991.25. Tocombine them with simulated Nano-JASMINE data
thesource parameters andcovariance matrixhavetobepropagated tothemidmission
reference epoch(whichforNano-JASMINEisexpected tobeJ2015).
The noisy catalogue contains the starting values for the data processing. It is
createdbyperturbing theoriginal sourceswithrandom noiseofagivenamplitude.
The simulated “true” catalogue defines the sources used for creating the simu-
lated true observations (which may be perturbed subsequently). For the real mission
Combiningastrometriccatalogues 3
Hipparcoscatalogue
Astrometricparametersandcovari-
ancematricesreadfromHipparcos,
propagatedtonewmissionepoch
Puerntcuerrbteaidntbiyes sysRteamndatoimcnaonisde
Comparisonfor
Simulated”true”catalogue plottingoferrors Noisycatalogue
Usedtocreate Startingvaluesforthe
Nano-JASMINEobservations least-squaresolution
Figure1. Relationshipsbetweencatalogues
the true catalogue is not known. TheHipparcos sources give the observed astrometric
parameters and their uncertainties. The true parameters of the sources are expected to
be at an unknown position within the error space of the original sources. The five as-
trometricparametersofthesimulatedtruecataloguerequirefiveindependent Gaussian
variates scaled by the square root of the covariance matrix C. Tofind a unique square
root we use the lower triangular matrix L resulting from the Cholesky decomposition
C = LLT. Then e = Lg, where g is avector of fiveindependent unit Gaussian values
andeistheresulting vectoroferrorstobeappliedtotheastrometric parameters. Since
E(ggT) = I,where E(...)denotes theexpectation value and I istheidentity matrix, it
isreadily verifiedthatehasthedesiredcovariance E(eeT)= C.
InordertomakeajointsolutiontheHipparcosnormalequationsarereconstructed
from the covariance inverse, N = C−1 . The right-hand side of the Hipparcos
HIP HIP
normal equations needs to be chosen such that solving for the Hipparcos information
only,theupdatewouldbringthecurrentparametervaluesbacktotheoriginalHipparcos
values. Thiscanbedonebycalculatingavector ddefinedasthedifferencebetweenthe
original Hipparcos source parameters (subscript o)andthecurrent values (subscript c)
andchoosing theright-hand side b as
HIP
(α −α )cosδ
o c o
bHIP = NHIPd = C−H1IP µαµπ⋆δooo−−−−δπµµccα⋆c . (2)
δo δc
Solving N x = b gives x = d and the application of this update to the current
HIP HIP
parameters obviously recoverstheHipparcos parameters.
4. Results
Table 1 shows results of simulation runs. As expected the combination of Hipparcos
and Nano-JASMINE gives a great improvement in proper motions. Additionally we
showthatourproposedjointsolutionperformssignificantlybetterthantheconventional
catalogue combination method. This can be understood as follows. The astrometric
parameters in the Hipparcos (or Nano-JASMINE) catalogue are correlated. The large
improvement of the proper motions therefore brings some improvement also to the
4 D.Michalik,L.Lindegren,D.Hobbs,U.Lammers,andY.Yamada
otherparameters,providedthatthecorrelationsareproperlytakenintoaccount. Thisis
thecaseforthejointsolution, butnotfortheconventional combination.
Position@J2015 Parallax Propermotions
[mas] [mas] [mas/year]
mag< 7.5,∼ 15000stars α δ π µ µ
α⋆ δ
Hipparcosonly(Hip) 18.19 14.84 0.80 0.77 0.63
Nano-JASMINEonly(NJ) 2.56 2.54 3.05 4.65 4.50
Conventional combination Hip+NJ 2.54 2.51 0.77 0.111 0.110
JointsolutionHip+NJ 2.41 2.40 0.75 0.108 0.105
Improvementofjointsolution 5.2% 4.4% 3.5% 3.2% 4.4%
mag< 11.5,∼ 117000stars
Hipparcosonly(Hip) 27.06 22.35 1.18 1.14 0.94
Nano-JASMINEonly(NJ) 4.57 4.53 5.43 8.38 8.02
Conventional combination Hip+NJ 4.51 4.44 1.15 0.197 0.194
JointsolutionHip+NJ 4.43 4.26 1.11 0.188 0.185
Improvementofjointsolution 1.8% 3.9% 4.0% 4.5% 4.5%
Table1. Conventionalcataloguecombinationvs. ajointastrometricsolution,for
asubsetofbrightstarsandforallHipparcosstars. SimulationsofNano-JASMINE
arebasedonaconservativeobservationperformancemodelandanoptimalscanning
law. The positions from Hipparcos have been propagated to the Nano-JASMINE
mid-missionepochJ2015.
4.1. Futurework
Weare planning to extend our studies of catalogue combination by simulating the im-
provements that can be gained by applying our method to catalogues from simulated
GaiaandNano-JASMINEdatatogetherwiththeHipparcosandtheTycho-2catalogues.
Furthermore, the goodness-of-fit of the combined solution is sensitive to small
deviations ofthestellar motions from the assumed (rectilinear) model. Weareinvesti-
gatinghowthiscanbeusedtoidentifybinarycandidateswithorbitalperiodsofdecades
to centuries. This will contribute to the census of the binary population within a few
hundred parsecs from the sun by filling a difficult-to-observe gap between the shorter
periodspectroscopic binariesandthelongerperiodvisually resolvedsystems.
Acknowledgments. Our research is kindly supported by the Swedish National
SpaceBoardandtheEuropeanSpaceAgency.
References
ESA1997,TheHipparcosandTychoCatalogues.ESASP-1200
Lindegren, L., Lammers, U., Hobbs, D., O’Mullane, W., Bastian, U., & Hernandez, J. 2011,
Astronomy&Astrophysics.Submitted
vanLeeuwen,F.2007,Hipparcos,theNewReductionoftheRawData(Springer),Astrophysics
andSpaceScienceLibrary.Vol.350ed.
