Astrometric imaging of crowded stellar fields with only two SIM pointings

9 Pages·0.44 MB·English

by  Neal Dalal

DraftversionFebruary1,2008
PreprinttypesetusingLATEXstyleemulateapjv.14/09/00
ASTROMETRIC IMAGING OF CROWDED STELLAR FIELDS WITH ONLY TWO SIM
POINTINGS
Neal Dalal & Kim Griest
PhysicsDept.,UniversityofCalifornia,SanDiego,CA92093
[email protected], [email protected]
Draft version February 1, 2008
ABSTRACT
The Space Interferometry Mission (SIM) will observe sources in crowded fields. Recent work has
shown that source crowding can induce significantpositional errorsin SIM’s astrometricmeasurements,
1
0 evenfortargetsmanymagnitudesbrighterthanallothercrowdingsources. Hereweinvestigatewhether
0 the spectral decomposition of the fringe pattern may be used to disentangle the overlapping fringes
2 from multiple blended sources, effectively by performing synthesis imaging with two baselines. We
find that spectrally dispersed fringes enable SIM to identify and localize a limited number of field
n
sources quite robustly, thereby removing their effect from SIM astrometry and reducing astrometry
a
J errors to near photon noise levels. We simulate SIM measurements of the LMC, and show that (a) SIM
astrometrywillnotbe corruptedby blendingand(b)extremelypreciseimagingofmildly crowdedfields
4
1 may be performed using only two orthogonal baseline orientations, allowing microarcsecond positional
measurements. Welastlyillustratethemethod’spotentialwiththeexampleofastrometricmicrolensing,

1 showing that SIM’s mass and distance measurements of lenses will be untainted by crowding.
v
7
1. introduction tothoseexpectedfromphotonnoiseandsystematicerror,
1
and could pose a serious threat to astrometric measure-
2 The Space Interferometry Mission (SIM) will perform
1 astrometrywithunprecedentedprecision,measuringposi- ments in crowded fields if uncorrected. Brighter sources,
0 such as binary companions or highly luminous field stars
tions at the microarcsecond level. This will allow numer-
1 &5%inbrightness,couldcompletelydestroySIMastrom-
ous astrophysical experiments to be performed that pre-
0 etry. To correct these measurements, one would like to
viously were impossible. Even though its 10 meter base-
/ know the positions and brightnesses of all the important
h line allows resolution only of 10 mas, SIM achieves mi-
p croarcsecond astrometric prec∼ision by centroiding fringes sources in the field of view – that is, one would like an
- to better than 1 part in a thousand. If SIM can meet image. A possibility is to obtain, prior to SIM observa-
o tions, either space telescope or groundbasedadaptive op-
its technical specifications,the primarysource of errorfor
r tics(AO)snapshotsofthefield. Unfortunately,asweshow
t single star astrometry would be photon noise, which can
s below, the rough positions obtainable ( 40 mas) from
a be beat down as 1/√N until the systematic error floor is ∼
such images are of insufficient precision to rectify SIM
: reached. For multiple stars in the field of view, however,
v astrometry. Another alternative is to perform synthesis
i other errorsmay be incurred. For example, the extra flux imagingwithSIM.Inanalogytoradiointerferometry,one
X
ofphotonsfromfieldsourcesgivesadditionalphotonnoise,
might orient SIM along numerous baselines, using multi-
r which increases the amount of on-source integration time
a ple siderostat pairs to fill in the (u,v) plane (Allen 2000).
toachieveagivenastrometricprecision. Anothersourceof
Note,however,thatapossibleSIMdesigncurrentlyunder
error,whichcannotbereducedbyfurtherintegrationtime,
considerationinvolvesjust one 10 meter baseline. A third
is the distortion in the measured fringe pattern caused
alternative,which we discuss here, is to use SIM in its as-
by the superposition of fringes from multiple incoherent
trometry mode. If the sources all lie several fringes away
sources. Since SIM achieves its revolutionary precision
from each other then the (u,v) coverage obtained from
by fitting the observed fringe pattern in exquisite detail,
just two baselines, using multiple frequency channels, is
evensmallscaledeviationsfromthe single starfringepat-
sufficient to disentangle the sources from each other and
tern can lead to large astrometric errors. Several possible
measureall oftheir positions to few µas precision. This is
SIM experiments, such as microlensing, globular cluster
amarkedincreasein astrometricprecisionrelativeto that
dynamics,orstellarmotionsinexternalgalaxieslikeM31,
possible from SIM imaging, 250µas (Allen 2000), and
requireSIMastrometryincrowdedfields, andsoit isnec- ∼
thusallowssignificantlyimprovedpropermotionmeasure-
essary to determine whether or not crowding will corrupt
ments. We show that, for the currently expected charac-
SIM measurements irretrievably.
teristicsoftheSIMsatellite,andfortypicallevelsofcrowd-
OnemightimaginethatSIMastrometryoftargetsmany
ing in the LMC, Bulge, and the outer regions of globular
magnitudesbrighterthanothersourcesinthefieldofview
clusters, these two orthogonal baselines are adequate to
wouldbeunaffectedbythepresenceofthosesources. How-
achieve excellent astrometric or proper motion precision.
ever, Rajagopal et al. (2001) have recently shown that
Note that similar ideas have been used previously in sim-
source confusion in crowded fields can lead to significant
ulations of closure phase measurements (Schloerb 1990)
errors ( few µas) in individual position measurements,
∼ and in visibility measurements of the binary star Capella
even when the confusing field sources are just a few per-
(Koechlin et al. 1979).
centofthetarget’sbrightness. Sucherrorsarecomparable
1
2
To see why just two baselines are sufficient, consider a 2. fitting dispersed fringes
synthesis imaging measurement of several point sources.
This involves measuring the complex visibility at nu- The above procedure of Fourier transforming along the
merous points in the (u,v) plane, where u =V B /λ, u or v direction is equivalent to taking the spectrally
x
v = B /λ are spatial frequencies, and B and B are dispersed 1D fringe pattern for each orthogonal base-
y x y
the projections of the baseline separation along the x,y line orientation, and Fourier transforming with respect
axes respectively. The visibility (u,v) and brightness to frequency k. This is the same procedure that is of-
distribution F(x,y) are related bVy Fourier transforma- ten used for single source phase and group delay esti-
tion, e.g. (u,v)= dxdyF(x,y)exp( 2πi[ux+vy]), via mation (Lawson 2000, Chapter 8). Let F(k) equal the
the van CVittert-Zernike theorem (see fo−r example Lawson product of source spectrum, throughput, and detector re-
2000, Chapter 2). RFor an arbitrary brightness distribu- sponse. Note that since CCD’s count numbers of pho-
tion F(x,y) sufficiently many u,v points must be sam- tons, not power, F has units of numbers. The interfer-
pled to permit accurate Fourier inversion of the visibil- ometer response at frequency k to a source offset from
ity. A typical astrometry measurement relies upon just the phase center by an angle θ along the baseline B is
two baseline orientations, which would appear insufficient F(k)(1+coskd)= F(k)(1+coskBθ) where d is the phys-
2 2
to localize more than one source position (x and y co- ical delay. Fourier transforming with respect to k gives
ordinates). Note, however, that each frequency channel [2F˜(l)+F˜(l d)+F˜(l+d)]/4, with l the Fourier conju-
at a given baseline orientation constitutes an indepen- gatetok and−F˜(l)the FouriertransformofF(k). Inmost
dent u,v measurement, since u = kBx/2π, v = kBy/2π cases, F(k) is slowly varying, and so its Fourier trans-
(Schloerb 1990). The multifrequency u,v coverage from form has power mainly at low frequencies. For example,
two orthogonal SIM baselines is limited to points on the for F(k)=const, F˜(l) = δ(l) and the Fourier transform of
u,v axes (see Figure 1), and is clearly insufficient to re-
the fringe pattern becomes [2δ(l)+δ(l d)+δ(l+d)]/4.
construct an arbitrary brightness distribution. For point −
Likewise,forarectangularbandpass,wewouldhaveasinc
sources,ontheotherhand,thiscoveragesuffices. Consider
function. Forsimplicity,wewillfocusonrectangularF(k),
F(x,y) = F (k)δ(x x )δ(y y ) corresponding to a
i i − i − i although the generalizationis clear.
sumofpointsources,eachwithspectrumF (k). Takingfor
i As expected, each source in the field of view appears
now each FPindependent of k gives the measured visibil-
i as a narrowpeak in the Fourier transformspace (actually
ity as (u,v)= F exp( 2πi[x u+y v]). Now, instead
V i i − i i a pair since the brightness is a real quantity). Also note
of the usual 2D Fourier transform to recover the bright-
that half of the power lies at zero frequency, due to the
ness distributionP, consider a single 1D Fourier transform,
1/2 which offsets the average fringe intensity from zero.
say along the u direction. This gives f (v)δ(x x ),
i i − i Figure 2 illustrates the pattern for two point sources.
and similarly for the 1D transform along the v direction.
Clearly, the sources’x coordinates are mPeasurable just by
80
sampling the visibility at multiple u along a line of con-
stant v, and similarly for the y positions. By placing each
source on a color-magnitude diagram (CMD) using the
F (k),thexsourcescanbematchedupwiththeysources,
i
giving the unique 2D coordinates of each bright source in
the field (Dyck et al. 1995). If the pairing is not obvious 60
fromtheCMD,thenathirdbaselineorientationisrequired
to match up the x and y sources, but note that this need
be done only once, at any time. In theory, therefore, the
two orthogonal SIM baselines required to perform single
source astrometry are all that are necessary to perform 40
imaging of point sources. In practice, finite bandwidth,
Nyquistsampling andphotonnoisewill limitpointsource
resolution. We explore these effects in the next section.
v v v
20
u u u
0
0 20 40 60 80
Fig. 2.— Fourier transform of the channeled spectrum with two
equallybrightsources,oneatthephasecenter andtheother at250
mas. The abscissa is the Fourier conjugate to wavenumber, the or-
Fig. 1.— Schematic depictions of (u,v) coverage for (a) synthe- dinate is delay, and the numbers along the axes label pixels. Note
sis imaging with many baseline orientations, (b) astrometry with that sources appear as lines in the Fourier transform, which cross
twobaselineorientations,and(c)astrometrywithtwobaselinesand at the position (delay) of the respective source. Additional sources
channeledspectrum. Onlyhalfofthe(u,v)planeisillustratedsince shouldappearasadditional,distinctlines. Thisfigureneglectspho-
thebrightnessdistributionisreal;redundantpointsarenotplotted. tonnoise,whichcanhidethesignalfromfaintsources.
3
Now, assume that SIM uses n frequency channels. For expect centroiding errors σ = 1.87N−1/2 mas for scans
c θ
simplicity, we take each channel to have the same fre- covering 11 mas and the wavelength range 0.4 1µm.
−
quency width. This assumption allows the use of the Using the white-light fringe pattern (n =1) we expect
c
FFT, otherwise a more generaldiscrete Fourier transform σ =2.91N−1/2 mas. So for N =340,000 photons we ex-
θ
or Lomb-Scargle periodogram would be used. With this pect 3.2 µas and 5 µas errors for the spectrally dispersed
assumption, the Nyquist theorem tells us that each bin in and white-light fringe pattern, respectively. This analytic
Fourierspacehasasizeequalingthewhite-lightcoherence estimate compares well with a Monte Carlo calculation of
envelopesize λmaxλmin/[B(λmax λmin)]whichfora rect- the expected astrometric errorsusing the above fringe fit-
−
angular 0.4 1.0µm bandpass and 10 meter baseline is ting procedure. Using bins of size 1 mas (n 11) gives
roughly 13.7−5 mas. This makes sense, since SIM should errors of 3.8 µas and 5 µas, in agreement wdith≈the MVB
be able to resolve sources separated by more than one estimate for white-light fringes and somewhat larger than
coherence envelope size. Clearly, decreasing the effective the MVB estimate for the spectrally dispersed fringe pat-
bandwidth should diminish SIM’s ability to resolve point tern. Increasing the number of bins to 100 gives Monte
sources. Varying the number of channels should affect CarloestimatesinagreementwiththeM∼VBestimate. One
onlyaliasing. Withfewer,broaderchannels,asourcenear might conclude from this that it is always better to cen-
the edge of the field of view has reduced fringe contrast troidusingspectrally dispersedfringesratherthan broad-
and would be mistaken for the combination of a fainter band, white-light fringes since the errors are smaller for a
source near the middle of the FOV and uniform surface given number of photons. However, keep in mind that we
brightness. This could corrupt astrometric measurements have neglected additional errors such as read noise, and
of sources towards the edge of the field of view. there may be a reduction in throughput when the fringes
We have simulated SIM astrometric measurements of are spectrally dispersed.
crowded fields with the above considerations. We assume
thatSIMscansrepeatedlyoverasmallrangeindelay,cor- 3. fitting multiple sources
respondingtothecentralfringesizeatthemeanfrequency,
Having confirmed that our Monte Carlo errors agree
which for a 10 m baseline and 0.4 1.0µm bandpass is
− with analytic estimates for single sources,we now explore
11.8 mas. We divide this range into n bins 1 mas large,
d
errors for multiple sources. As mentioned earlier, addi-
andintegratethe fringepatternf(k,d)=F (1+coskd)/2
i
tional sources in the field of view can increase astrometry
over n n total bins, where n = 64 is the number of
d c c
× errorsby contributing extra photon noise, and by distort-
wavelength channels spanning the bandpass. We sum the
ing the fringe shape. Fitting for multiple sources cannot
contributions for all sources in the field of view, and ap-
remove photon noise, but it can mitigate the distortion
proximate photon noise by adding to each bin a random
arising from superimposed fringes. To ease the compari-
number of photons drawnfrom a Gaussianwith σ =√N.
son with unblended centroiding as described in the pre-
Weneglectallinstrumentalnoise,suchasreadnoise,dark
vious section, we set the on-source integration time to
current, thermal drift, etc., basically assuming that SIM
equal the time necessary to accumulate 340,000 photons
is a perfect astrometer. These additional errors may be-
fromthe science target. In terms ofa signal-to-noiseratio
come important for the faintest sources. To simulate SIM
(S/N), this holds fixed the signal while letting the noise
astrometryandphotometry,wefittheresultingfringepat-
vary. When gauging how well fitting can remove the ef-
tern to the form
fects of blending, we should compare to a case with the
1+coskB(θ θ )
N =N + N − i . sameratioofsignal(photons fromsciencetarget)to noise
model 0 i
2 (square root of total number of photons). For example,
i
X consider centroiding a single star in the presence of uni-
The number of sources, their positions and brightnesses
form surface brightness that contributes an equal number
are first estimated using the Fourier transform method
of photons as the star. Since the S/N is smaller by a fac-
described above. Starting with these initial values we fit
by minimizing the error function χ2 = (N torof√2relativetothe isolatedstarcase,thecentroiding
i,j ij,data − errors will be greater by a factor of √2.
N )2/N . The fitted parameters are each
ij,model ij,data P As our first example, we consider measurements of a
source’s brightness N and position θ , as well as uniform
i i brightmicrolensingeventinthe LMC. We place a V =19
surface brightness N . This assumes that the product of
0 target at the phase center, and randomly place field stars
thesourcespectrum,throughput,anddetectorresponseis
in the 800 mas FOV. The number of field stars, and their
constant with respect to wavelength,but it is not difficult
luminosity function (LF), are chosen to match the crowd-
to generalize to include simple spectral dependence.
ing and LF of the LMC derived by the MACHO project
Anestimateofthecentroidingprecisionpossiblebysuch
(Alcock et al. 2001). On average, MACHO finds 10-16
fringe fitting may be computed using the minimum vari-
fieldstarswithin1-2”ofeachobservedstar. Tobeconser-
ance bound, hereafter MVB (Gould 1995). When fitting
vative,wehaveused16fieldstarsinallofoursimulations.
data to a function f(x) with normalization and one ad-
TheaveragebrightnessofastardrawnfromthisLFisap-
ditional parameter x , the minimum variance expected in
0 proximately 1.4% the brightness of a V = 19 star, so by
the fitted x is
0 adding 16 of them, we add approximately 22% more pho-
1 dlnf 2 dlnf 2 −1 tons. We thus expect a roughly 10% increase in astrom-
var(x0)≡σx20 = N "*(cid:18) dx0 (cid:19) +−(cid:28) dx0 (cid:29) # ethtreyoenr-rsoorusrcfreoombspehrvoatotinonnotiimseeatloongei.veSuinsce3.8weµansorpmhoatliozne
where g ( fgdx)/( fdx). Using the MVB formula, noise errors from the V = 19 source alone, we expect 4.2
h i≡
andtakingN totalphotonsfromasinglepointsource,we µas errors from photon noise. Any additional errors will
R R
4
12000
4000
(a)
(b)
3500 10000
3000
8000
2500
6000
2000
1500
4000
1000
2000
500
0 0
−0.04 −0.02 0 0.02 0.04 −0.04 −0.02 0 0.02 0.04
position error [mas] position error [mas]
Fig. 3.— Histogram showing position residuals for multiple source fitting to V = 19 source and 16 stars drawn from LMC luminosity
function. The first panel shows the histogram for random placement of stars. The peaks at the edge of the plot are due to events falling
outsidethehistogramrange. Thesecondpanelshowsthehistogramforthesamecalculation,restrictingsourcestoliemorethan25masfrom
thetargetatthephasecenter. Seethetextformoredetails.
be due to source confusion (Rajagopal et al. 2001). binary companions), and their brightnesses vary during
We follow the same fitting procedure described above, thecourseofthelensingevent. Therefore,thecancellation
allowing for multiple sources at different positions and of errors that occurs for typical proper motion measure-
with different brightnesses. Note, however, that faint ments will not occur here, and we expect proper motion
sources may not be detectable above photon noise, so we errors of the same magnitude as the large position errors
will only be able to remove the brightest sources. Since foundabove. Fortunately,asweshowbelow,itisstillpos-
in this example we are only interested in the measure- sibletodisentangletheblendeffectusingthecompleteset
ment errors for our single target, we only reportmeasure- of position measurements. Thus, even in this most badly
ment errors for the V = 19 target. Panel (a) of Figure affectedcase,it is still possible to recoveraccuratemasses
3 shows the histogram of position residuals for this sim- and distances from astrometric microlensing events.
ulation. Overall, the standard deviation of this error dis- Generally, blend stars in SIM observations will be far
tribution is σ = 10.5µas, much larger than the 4.2 µas fainterthantheintendedsciencetargets,whicharechosen
θ
errors expected from photon noise. Note, however, that fortheir brightness. However,itisstillworthwhiletocon-
the histogram may not be well described by a single “av- sider what effects comparably bright sources would have.
erage” error value. We find that the error distribution is To quantify this, we simulate SIM measurements of two
characterized by a narrow core superimposed on a slowly equallybrightstarsinthe fieldofview. Againwe normal-
decayingtailofoutliers. Thishistogramisquitesimilarto ize the on-sourceintegrationtime to give 340,000photons
that found by Rajagopalet al.(2001), althoughfor differ- from the science target. Photon noise alone should thus
entreasons. Intheir simulationsofwhite-lightastrometry giveuserrorsof√2 3.8µas=5.4µas. Weplotthe results
·
in crowded fields in M31, the LMC and Galactic Bulge, of our simulations in Figure 4. We can distinguish three
the long tail was due to the occasional bright field star. kinds of blends from this figure. First, note that bright
Here, in contrast, bright field stars are easily identified blends falling outside the central fringe are resolved by
and removedusing dispersed fringes. The long tail here is SIM and can be removed. The offsets due to such sources
due to the occasional star landing nearby our target and areconsistentwith zero,andthe dispersions( 6µas)are
≈
becoming unresolved from it. To demonstrate this point, not much larger than what is expected from photon noise
panel (b) of Figure 3 shows the histogram for the same alone. Clearly,suchblends pose no threatto SIM astrom-
simulation,butherewerestrictfieldstarstoliemorethan etry using spectrally dispersed fringes. The averageoffset
25masfromourtarget. Notethatthelongtailshavevan- is not zero, however, for sources falling within the central
ished, and that the distribution’s standard deviation has fringe. Sources at distances 2 mas are completely unre-
≤
significantly diminished, to 5.3 µas, near to the value ex- solved by SIM. As expected, then, the measured centroid
pected from photon noise. Clearly, almost all of the 10.5 falls at the center of light, which is 0.5 mas for an equally
µas errorsin this casearise fromunresolvedor marginally bright blend at 1 mas, and 1 mas for an equally bright
unresolved stars falling close to the target. Rajagopal et blend at 2 mas. Note that although the offset is large,
al. (2001) have shown that these quite significant errors the standard deviation is small. This is just what we ex-
largelycancelforpropermotionmeasurements,andsofor pect; two equally bright sources at 0 and 1 mas should
many SIM applications they are not a worry. For mi- appear to SIM to be a source twice as bright located at
crolensing, however, confusion errors will be a problem in 0.5 mas, and so the errors should be smaller by a factor
crowded fields. This is because microlensed stars undergo of 2−1/2 compared to single source astrometry. The only
differentpropermotionsthanotherstarsinthefield(even blend positions giving significant errorsare those that are
5
marginallyunresolved,i.e. fallingbarelyinsidethecentral For the simulation plotted in panel (a), the astrometric
fringe, at separations of 3-4 mas. There would seem to be errors were σ = 6 µas. In panel (b), we plot the scaling
no way to handle such blends with a single baseline ori- of errors with brightness. Using the above arguments on
entation, however an additional pointing along a different S/N,weexpecttheerrorstoscaleinverselywitheachstar’s
baseline orientation could resolve the two sources. brightness, and this is verified in Figure 5(b). Clearly, far
It is apparent from Figure 4 that the vast majority of smaller errors are attainable using dispersed fringes than
bright field stars will pose no threat to SIM astrometry. the 250µas errors expected from synthesis imaging of
∼
The fact that the fit residuals are so small, comparable to many baseline orientations (although keep in mind that
photon noise, means that fitting can reliably remove the we have neglected errors such as thermal drift). Thus, it
fringes from blending sources in the field of view. Since ispossiblethathighlypreciseastrometricimagingmaybe
we are able to recover µas astrometry even in the pres- achieved using only two SIM pointings.
ence of blends, this must mean that we are able to fit the
blend positions to microarcseconds themselves. Since we 4. application: astrometric microlensing
aredoingthis forallimportantblends inthe fieldofview,
we are basically imaging the field of view. It is therefore One possible application of SIM astrometry in crowded
interesting to explore the imaging capabilities of SIM us- fields is microlensing. There are numerous astrophysical
ing just the spectrally dispersed fringes measured along applications of SIM observations of microlensing events
two orthogonal baselines. From the discussion above, we (Paczyn´ski1998; Boden et al. 1998;Safizadeh et al. 1999;
expect that we can get reliable positions as long as the Gould&Salim1999;Salim& Gould2000),mostofwhich
sourcesare separatedby more thanone white-lightcoher- entail crowded fields. For ordinary proper motion, Ra-
enceenvelope. IntermsoftheFouriertransformspace,we jagopal et al. (2001) have shown that astrometric errors
expect to measure reliable positions for sources separated arising from source confusion largely cancel. This is be-
by more thanone pixel. With n wavelengthchannels,we cause any offset caused by additional sources remains ba-
c
have n /2 distinct pixels in the Fourierconjugate variable sically constant as the target executes its proper motion,
c
since the brightness is real. We have assumed 64 chan- andcancelswhenthe relativemotionofthe targetismea-
nels, so we have 32 pixels. Assuming that sources should sured. However, this is not true for microlensing. The
be separated by 3 pixels for reliable positions means that reasonforthisisthatmicrolensingchangesthetarget’sap-
we expect that for up to 10 sources in the FOV, we can parent brightness while also causing the apparent proper
measurepositionsreliably. Totestthis,wesimulatedSIM motion. The astrometric offset caused by blending, there-
astrometric measurements of 10 stars in the field of view. fore, does not remain constant and will not cancel. Thus,
We firstsimulated10starsofequalbrightness;the results wemightexpectthatblendingcouldseriouslydamagemi-
are shown in panel (a) of Figure 5. We find that the as- crolensingmeasurements,renderingSIMobservationsuse-
trometric errors scale like t−1/2 where t is the on-source less in crowded fields. Fortunately, the results discussed
in previous sections indicate that blending by brightstars
integration time, so they are mainly due to photon noise.
should not corrupt SIM astrometry, except when those
sources fall within the white-light coherence envelope of
the target star. However, binary companions to lensed
starscouldeasilyfallinsidethecentralfringe. Sincefringe
fitting could never detect such stars, just as fringe fitting
cannot resolve the two microlensed images, our concerns
about crowding could still hold. Fortunately, as we de-
scribe below, evensourcesfalling within the centralfringe
may be detected and removed, not from individual SIM
measurements, but by using the entire set of observations
of the microlensing event.
As discussed in the previous section, field sources may
be detected and essentially removed as long as they lie
outside the central fringe of the target. In fact, even
marginally resolved sources falling barely within the cen-
tralfringemay be detected andremovedif brightenough,
by the distortionthey causein the fringe shape. The only
sources undetectable are those which do not distort the
fringe shape, but merely shift its centroid. These sources,
lying well inside the central fringe, shift the fringe cen-
troid to the center of light of the target and blend, and
add all of their light to the target. What SIM would de-
tect would be a star of brightness equaling the sum of the
target and blend’s brightness, located at their center of
light. For this reason, multiple blends falling inside the
Fig. 4.— Fit residuals for astrometric measurements of a single central fringe are equivalent to a single blend, and so we
star,withanequallybrightblendinthefieldofview,as afunction
consideronlythe singleblend case. Suchblends affectthe
of distance totheblend. Thethincurveisthe average offset ofthe
fitsourcepositionfromtheinputpositionandthethickcurveisthe microlensinglightcurveandastrometricmotioninasimple
standarddeviationofthoseoffsets. way (Han & Kim 1999), and we show that SIM observa-
6
80
(a)
70
60
50
40
30
20
10
0
−0.03 −0.02 −0.01 0 0.01 0.02 0.03
astrometric error [mas]
Fig. 5.— Simulated astrometric imaging errors for 10 sources. Panel (a) show the error histogram from a simulation of imaging of 10
equally bright sources in the 800 mas field of view. The standard deviation of this histogram is σ = 6µas. Panel (b) shows the scaling of
errorwithbrightnessforasimulationof10sourceswithvaryingbrightnesses. Plottedpointsareerrorsasafunctionofbrightnessrelativeto
thebrighteststarinthefield. Thedashedlineisnotafit,but showstheexpected scaling: errorinverselyproportionaltobrightness.
tions may be corrected without degeneracy to extract the of the photometric lightcurve into lensed and unlensed
desired measurables. (blend) light. That is, F , F , and A may be measured
s b
It is first necessary to review astrometric microlensing; from the photometry alone. Given the blend fraction, we
see Boden et al. (1998) for more details. Microlensing of now show that it is easy to fit the astrometric data to
a single source by a single lens gives two images of the recover the microlensing parameters θ , t ~u˙ −1, etc.,
E E
source. If the source is located at position ~xs =θE~u rela- which are then used to determine the lens m≡as|s|and dis-
tive to the lens, then the two images appear at positions tance.
with magnifications
~x =θ (u u2+4)/2
1,2 E
±
u2+p2 1
A = .
1,2
2u√u2+4 ± 2
Here,θ istheangularEinsteinradius,u= ~u isthelens-
E
| |
source separation in units of θ (not the u from the u,v
E
plane),andpositionsarerelativetothelensposition. SIM
cannot resolvethese images,so in the absence of blending
it detects a source with position ~x=~x +∆~x and magni-
s
fication A given by
~u u2+2
∆~x=θ , A= .
Eu2+2 u√u2+4
Note that ∆~x traces out an ellipse.
It is now easy to see how unresolved blends affect this.
If the lensed source has unlensed brightness F , and the
s
blendhas brightnessF andposition~x , thenSIMdetects
b b
a source at position ~x with brightness F given by
obs tot
AF (~x +∆~x)+F ~x
s s b b
F =AF +F , ~x =
tot s b obs
F
tot Fig. 6.— Microlensing source and motion of lens and image
Anexample ofthis is giveninFigure6. Fromthe illustra- center-of-light. The large dot shows the location of the blend star,
tion,itmayappearthatblendingwouldruinSIMastrom- the small dot the location of the target star. The straight arrow
shows the location and relative motion of the lens near the peak of
etry,sincethe observedmotionbearslittle resemblanceto
the event, with x’s placed every week. tE =20 days and θE =0.32
the unblended motion. However, our simulations indicate
mas. Theellipticalcurveasymptotingtothetargetsourceshowsthe
that it will be possible to recover the unblended motion. unblendedastrometricmotionoftheimagecenteroflight,whilethe
This is largelybecauseSIM’s exquisitephotometry (recall irregularcurveistheobservedmotionofthecenteroflightincluding
theblendstarlight. Arrowsshowthedirectionofmotion.
N & 340,000) permits the nondegenerate decomposition
7
0.2 The blended microlensing event is parametrized by 11
quantities: theabovementionedθ ,t ,~x ,~x ,F ,andF ,
E E s b s b
0
as well as the normalized impact parameter u , time of
min
closest approach t , and relative proper motion direction
−0.2 0
φ. For simplicity we assume that the same blend bright-
−0.4 ness Fb applies for both the x and y astrometric measure-
s] ments. This will not be true in general(a field star whose
y
da−0.6 projected position relative to the target happens to fall
∆t [E inside the central fringe under one orientation need not
−0.8 fall inside the central fringe along the orthogonal orienta-
tion). However,it is trivial to generalize our treatment to
−1
includetwodifferentblendsforthexandymeasurements.
We have also assumed that the blend’s proper motion is
−1.2
negligible for simplicity, but again it should not be diffi-
−1.4 cultto generalizetomovingblends. Now,the photometry
−0.06 −0.04 −0.02 ∆0θ [ma0s.0]2 0.04 0.06 0.08 alone gives us t0,tE,umin,Fs, and Fb. The remaining 6
E parameters are to be derived from the astrometry. First
0.4
note thatthe unblended astrometricmotionalongthe rel-
ative proper motion vector is antisymmetric in time with
0.2
respect to t , while the unblended motion perpendicular
0
to it is symmetric (see Figure 6). The blend position and
0 brightness are constant, and therefore symmetric. So if
wemultiply~x byF ,andantisymmetrizeintimewith
obs tot
ays]−0.2 respect to t0, then the blend parameters drop out, and
∆t [dE−0.4 tthioenrveesuctltoirn.gSmoowtieonimismaedloinatgeltyhegeutnφb.lenWdeedcapnroepsetrimmatoe-
θ from t , u and the velocity ~x˙ at peak. Again, note
E E min
−0.6 that the blend parameters make a constant contribution
to the productF ~x , soby taking the difference ofthis
tot obs
−0.8 quantity evaluated at two different times, the blend pa-
rameters drop out. We can thus estimate the unlensed
−1 source position ~x by using the estimates for the other
−0.06 −0.04 −0.02 0 0.02 0.04 s
∆θ [mas] parameters, and the change in F ~x from peak to the
E tot obs
end of the event. Now we have estimates for all parame-
3
ters except the blend position ~x , which we can guess by
b
comparing the measured F ~x with the contributions
2 tot obs
fromtheotherterms. Thus,wecansimplyandaccurately
guess values for all of the microlensing parameters, and
1
use these as initial estimates for fitting routines. This is
also true if there are different blend fractions for the two
ys] 0 baseline orientations.
a
∆t [dE−1 SIMTooqbusearnvtaiftyionthseofamboicvreolsetnastienmgeenvtesn,tws.eEhaacvheosfiomuurlasitmed-
ulated events has θ = 0.32 mas and t = 20 days. The
E E
−2 otherparameters,suchas u , φand~x werepickedran-
min s
domly. We measure 25 points during the event, evenly
−3 spaced in time, starting when the apparent magnification
exceeds a threshold A > A = 1.5, and lasting 5t from
th E
−4 the trigger. This is probably not an optimal observing
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15
∆θ [mas] strategy, but it is not our intention here to optimize the
E
strategy. Eachastrometricmeasurementismadefollowing
Fig. 7.—Fit residualsfor microlensingevent. Each point inthe
scatter plot represents a different simulated event; plotted are the the procedure described in previous sections – the fringe
residuals in fitted θE and tE. 10,000 simulations of unblended mi- pattern is simulated and fitted to recover the source po-
crolensing events are plotted in the first panel; the standard de- sition and brightness. We assume that the x and y mea-
viations are σθ = 0.009 mas and σt = 0.1 days, for input values surements occur at the same time, although this assump-
θE=0.32masandtE =20days. Inthesecondpanel,weplot1000
simulations of microlensing events with a V = 19 target near the tion has no effect. If SIM measures the orthogonal base-
phase center and 16 additional stars drawn from LMC luminosity linesatdifferenttimes,saytominimizeitsoverallslewing,
functionplacedrandomlyinFOV.Here,thestandarddeviationsare no information should be lost. Even if the baselines are
σθ = 0.008 mas and σt =0.14 days for the same input parameters not perfectly orthogonal, measurements of the microlens-
astheunblendedcase. Thelastpanelplotserrorsfor10,000simula-
tionsofmicrolensingevents with16fieldstars,butwithwhite-light ing parameters should not be seriously degraded.
fringefitting. Here,θEissystematicallyunderestimatedby0.03mas To measure how badly crowding affects microlensing
(roughly 10%), with a dispersion of 0.017 mas, while tE has a dis- measurements,wefirstseehowwelltheparametersmaybe
persionof0.44days.
8
recoveredfor a single, unblended source. To save on com- Weshouldbecareful,however. Mostfieldblendsdrawn
putation time, we used white-light fringe measurements, fromtheLMCluminosityfunctionarelessthanafewper-
which as noted above should not be much worse than cent of the brightness of a V = 19 star. A comparably
spectrally dispersed measurements for unblended sources. brightstar landing in the field ofview could corruptmea-
Now, to measure the lens mass and distance, the param- surementsbaseduponwhite-lightfringes(Rajagopaletal.
eters of interest are θ , t , and φ. In the first panel of 2001). Toillustratethis,wesimulatewhitelightSIMmea-
E E
Figure7weplotfitresidualsfortheaboveobservingstrat- surements of microlensing events with an equally bright
egy. Note that these fits utilize both SIM astrometricand source in the field of view. Now, it should be clear that
photometricmeasurementsofthemicrolensingevents. We sources within a few fringes of the target will hopelessly
find that θ can be measured to 2.84%, and t to 0.5%, corrupt our measurements. However, sources falling out-
E E
for input values of θ =0.32 mas and t =20 days. side the coherence envelope have significantly diminished
E E
Now, we add blend stars to see how badly they affect fringecontrast,andonemighthopethattheireffectwould
microlensing mass and distance measurements. Here, we be minimal. These hopes are unfounded, as figure 8 illus-
clearly need to use the full multifrequency fringe fit to re- trates. This figure plots the errors in θ as a function of
E
solve bright blend stars. From the individual fringe fits, source separation(both in mas). Even sources lying more
whichgaveerrors 10µas(twiceaslargeastheunblended than 100 mas away can induce 20% errors in the fit. A
∼
errors)wemightthink thattheerrorsherewouldbetwice caveatto this statement is that the assumption of rectan-
as large. We plot the fit residuals in the second panel of gularbandpassoverestimatesthe effectof distantsources.
Figure 7. We obtain errors σ = 0.008 mas, σ = 0.0235, ThediscontinuousjumpsinF(k)attheedgesoftheband-
θ φ
and σ = 0.142 days. Normalized to θ = 0.32 mas and passgivesidelobesinthewhite-lightfringeenvelope,which
t E
t = 20 days, these are fractional errors of 2.5%, 2.4%, would be suppressed with a more smooth decline. Never-
E
and 0.71% respectively, as good as or better than the un- theless,distantbrightsourcesareclearlycauseforconcern
blendedwhitelighterrors! Again,thereasonthattheseer- whencentroidingwhitelightfringes. Now,spacetelescope
rorsarenottwiceasbigastheunblendederrorsisthatthe or ground based AO snapshots could detect such sources.
relativelylargesingle measurementresiduals mainly arose However, the positions available from such snapshots are
from unresolved sources falling inside the target’s central rough by SIM standards, 20 40 mas, which is larger
∼ −
fringe, which as we discussed may easily be removed. than a fringe size and larger than the length scale of the
Another important question to ask is how badly we variations exhibited in Figure 8. Thus, such high reso-
woulddo withoutthe spectraldecompositionofthe fringe lution snapshots would not be helpful for fitting multiple
pattern. To address this question, we repeated the above sources.
simulations, but this time used only white-light fringes. Howmuchbetter dowedoifwehavedispersedfringes?
The resulting errors are plotted in Figure 7. We find that As previous sections showed, bright sources landing out-
the error distribution is bimodal, and neither blob is cen- side the central fringe are easily detected and removed.
tered on the correct answer! The bulk of the fits underes- Therefore,suchsourceswillnot corruptSIM microlensing
timate θ by roughly 10%, with a scatter of 5%, al- events beyond adding photon noise, which can be allevi-
E
∼ ±
thoughasubsetofeventsgivesomewhatoverestimatedθ , ated by integrating longer. We might worry about closer
E
about 5% too large,with a scatter of 3%. This “more sources, however. Figure 9 shows results from the same
∼±
correct”subset,however,overestimatest byroughly8%. simulation as above, but for blends within 50 mas, using
E
So if we were to fit a white light measurement of a mi- the channeled spectrum. The errors are all small ( 3%)
∼
crolensing event in the LMC, we could be reasonably cer- except for marginally resolvedblends at about 3 mas sep-
tainthatthederivedmassesanddistanceswouldbeincor- aration which have 10-20% errors. Bright sources at this
rect by about 20%. While 20% distances would certainly distancefallwithinthecentralfringeandcannotbeclearly
be good enough to resolve the question of the location of resolved,but are not well enough inside the central fringe
the LMClenses,itisclearlypreferabletoutilizethe chan- to be treated as completely unresolved. In contrast, faint
neled spectrum when possible. sources at this distance are effectively unresolved.
This shows the extent to which spectrally dispersed
ΣΘ Θ =0.32, blend fraction f=0.5
E fringes are useful. If the spectral decomposition of the
0.14 fringe pattern is unavailable or costly to transmit, and we
must rely upon white-light fringes, then high resolution
0.12
space telescope or AO images will be required before fol-
0.1 lowing up a microlensing event with SIM. Detection of a
bright field source in such snapshots would not help us
0.08
remove the blend’s effect, but instead would tell us that
0.06 such events are hopelessly corrupted and should not be
followedby SIM.Since brighteventsin the LMC arerare,
0.04
andsnapshotswouldtellusnothingaboutpossibleblends
0.02 . 40 mas from the target which would also corrupt SIM
measurements, we hope that spectrally dispersed fringes
Dx
100 150 200 250 will be available in the final SIM design. High resolution
Fig. 8.— Fit residuals for microlensing event with an equally snapshots may be useful with dispersed fringes as well,
bright blend in FOV, fitting to the white-light fringe pattern. We sincetheycouldbeusedtoavoidorientationsthatproject
onlyplotfordistances largecomparedtothe fringesize; closersep- bright field stars inside the target’s central fringe.
arationsgivemuchlargererrors. σθ isinmas.
9
mensional (delay vs. wavelength) fringe pattern, thereby
restoring SIM astrometry to the precision (few µas) ex-
pected for isolated sources. We have further shown that
in mildly crowded stellar fields, precise astrometric imag-
ing is possible using fringes in the channeled spectrum,
with positional precision comparable to that expected for
single star astrometry. Although the details will depend
uponSIM’sprecisecharacteristics,weexpectthatimaging
should be possible for up to 10 comparablybright sources
in the field of view.
We then illustrated our method with the specific ex-
ample of astrometric microlensing. We showed that, al-
thoughunresolvedsourcesfallingwithinthe centralfringe
could corruptindividual astrometric measurements,accu-
rate masses may still be measured by fitting the entire
microlensing event. In this manner, confusion or blending
errorsmay be brought below photon noise errors. For the
worst case, i.e. LMC events which are the most crowded
and have relatively faint sources, 5% mass measurements
are still easily attainable. We also consideredthe effect of
exceptionally brightfield stars or binary companions, and
showed that for all separations except about 3 mas, such
brightsourcesalsodonotadverselyaffectSIMastrometry.
Chanceseparationsof3mas,however,canleadto20%er-
Fig.9.—Fitresidualsformicrolensingeventwithequallybright
rorsinmassanddistancemeasurementsfrommicrolensing
blend star and 15 other blends drawn from LMC luminosity func-
tion. The15faintLMCblendsareplacedrandomlyinfieldofview, events.
and errors are plotted as a function of the bright blend position. In conclusion, we find that SIM astrometry will not be
ComparethiscurvewithFigure4. corrupted by blending, and extremely precise imaging of
mildly crowded fields may be performed using only two
orthogonal baseline orientations, allowing microarcsecond
5. summary
positional measurements.
We have shown that the spectral decomposition of the
fringe pattern,whichis oftenusedforgroupdelayestima- WethankAndyBoden,AndyGould,andAndreasQuir-
tion,alsoallowshighlypreciseastrometrytobeperformed renbachformanyhelpfuldiscussionsandsuggestions,and
incrowdedfields. Evenwhenwhite-lightfringesarehope- JayadevRajagopalforhelpfuldiscussions,suggestionsand
lessly corrupted, dispersed fringes allow microarcsecond for sending us an advance copy of his paper. This work
positions to be measured. We find that any significant was supported in part by the U.S. Dept. of Energy under
sourceoutsideofthecentralfringe(>6mas)ofthetarget grantDEFG0390ER40546. NDwasalsosupportedinpart
star may be detected and removed by fitting the two di- by the ARCS Foundation.
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