Table Of ContentJournal of the Korean Astronomical Society http://dx.doi.org/10.5303/JKAS.2014.00.0.1
00: 1 ∼ 99, 2014 March pISSN: 1225-4614 / eISSN: 2288-890X
(cid:13)c2014. The Korean Astronomical Society. All rights reserved. http://jkas.kas.org
AUGMENTING WFIRST MICROLENSING WITH A GROUND-BASED
TELESCOPE NETWORK
Wei Zhu1,2 & Andrew Gould3,1,2
1Department of Astronomy Ohio StateUniversity,140 W.18th Ave., Columbus, OH 43210, USA;
[email protected]
2Max-Planck-Institutefor Astronomy,K¨onigstuhl 17, 69117 Heidelberg, Germany
3Korea Astronomy and SpaceScience Institute,Daejon 305-348, Republicof Korea; [email protected]
6
Received —;accepted —
1
0 Abstract: Augmenting the WFIRSTmicrolensingcampaignswith intensiveobservationsfroma ground-
2
basednetworkofwide-fieldsurveytelescopeswouldhaveseveralmajoradvantages. First,itwouldenable
b full two-dimensional(2-D) vector microlens parallax measurements for a substantial fraction of low-mass
e lensesaswellasplanetaryandbinaryeventsthatshowcausticcrossingfeatures. Forasignificantfraction
F
ofthefree-floatingplanet(FFP)eventsandallcaustic-crossingplanetary/binaryevents,these2-Dparallax
8 measurements directly lead to complete solutions (mass, distance, transverse velocity) of the lens object
(or lens system). For even more events, the complementary ground-based observations will yield 1-D
] parallaxmeasurements. Togetherwiththe1-DparallaxesfromWFIRST alone,theycanprobetheentire
P
massrangeM &M . Forluminouslenses,such1-Dparallaxmeasurementscanbepromotedtocomplete
E
solutions (mass, dis⊕tance, transverse velocity) by high-resolution imaging. This would provide crucial
.
h information not only about the hosts of planets and other lenses, but also enable a much more precise
p Galactic model. Other benefits of such a survey include improvedunderstanding of binaries (particularly
- with low mass primaries), and sensitivity to distant ice-giant and gas-giant companions of WFIRST
o
lenses that cannot be detected by WFIRST itself due to its restricted observing windows. Existing
r
t ground-basedmicrolensingsurveyscanbe employedifWFIRST ispointedatlower-extinctionfields than
s
a iscurrentlyenvisaged. Thiswouldcomeatsomecosttotheeventrate. Thereforethebenefitsofimproved
[ characterizationof lenses must be weighed against these costs.
2 Keywords: astrometry – gravitationalmicrolensing – planets – stars: fundamental parameters (mass)
v
3
4 1. INTRODUCTION FFPs. Yee (2013) focused on the specific challenges of
0 conducting such parallax observations using WFIRST.
The proposed WFIRST mission (Spergel et al., 2015)
3
contains a significant microlensing component, which
0
. will plausibly consist of six roughly 72-day continu- Inthiswork,weprovideaquantitativeanalysisofhow
1
ous campaigns with 15 minute cadence covering about thesuccessofWFIRST microlensingexperimentcanbe
0
2.8deg2 ofthe GalacticBulge. The observationswillbe enhanced by complementary ground-based survey ob-
6
1 made fromL2orbitandeachcampaignwillbe centered servations. The value of the so-called one-dimensional
: onquadrature,i.e.,roughlyMarch21andSeptember21. (1-D) parallax measurements is discussed. These will
v TheobservationswillbecarriedoutinabroadH-band, be substantially more plentiful than the 2-D paral-
i
X which is substantially less affected by dust than optical laxes and, more importantly,measurable atallEinstein
bands. In principle, this permits observations closer to timescales. We also point out two additional benefits
r
a the Galactic plane where the microlensing event rate is thatsuchground-basedsurveyobservationscanprovide
almostcertainlyhigherthaninthelowest-latitudefields to WFIRST.
accessibletoground-basedI-bandmicrolensingsurveys.
The full power of a space-based survey at L2 can
only be realizedby complementingitwith adeep, high- Inourcalculations,weassumetheseground-basedob-
cadence, near continuous survey, as previous studies servations are taken by a survey similar to the Korean
have suggested. Gould et al. (2003), Han et al. (2004), Microlensing Telescope Network (KMTNet, Kim et al.,
andYee(2013)havepreviouslydiscussedthepossibility 2016;Henderson et al.,2014). Werecognizethatadopt-
of microlens parallax measurements by combining ob- ing such an optical survey comes with a loss in event
servations from Earth and L2 orbit. Han et al. (2004) rate, as it requires that WFIRST be pointed at lower-
demonstratedthatcombininglargeformatsurveysfrom extinction fields than is currently envisaged. Therefore,
thesetwolocationswouldyieldmicrolensparallaxesand the benefits of adopting such a survey must be weighed
somassmeasurementsforlow-massobjects,particularly against such a cost. Our methodology can be easily
adapted for any specified microlensing surveys, either
Corresponding author: WeiZhu ongoing or planned.
1
2 Wei Zhu & Andrew Gould
2. MICROLENSING PARALLAXES (Bulge lens + Bulge source) (Gould et al., 1994). That
is, its (inverse) amplitude is typically
The standard microlensing light curve normally only
yields one single observable that is of physical interest, 280km s 1 , Disk events
v˜= − (7)
the event timescale tE, 1000km s−1 , Bulge events
(cid:26)
θ Here we have adopted π = 0.12 mas and µ =
t E . (1) rel rel
E ≡ µrel 7 masyr−1 as typical values for disk events, and πrel =
0.02 mas and µ =4 masyr 1 for Bulge events.
rel −
Here µ = µ is the relative proper motion between In the satellite parallax method, Λ is more directly
rel | rel|
the lens and the source, and θ is the angular Einstein measured than π (Equation (4))
E E
radius
1
θ κM π , (2) Λ= (∆t ,t ∆u ) , (8)
E ≡ L rel D 0 E 0
sat,
twhheerleenκs-s≡ou4rcGe/r(ecl2aAtiUv)e appnadralπlarexl.=HAerUe(DDL−1an−dDDS−1)aries because ∆t0 is usually b⊥etter measured than tE. More-
L S over, as we will show below, for the special case of
the distances to the lens and the source, respectively.
L2 space parallaxes (as well as terrestrial parallaxes,
Under certain circumstances, the microlens parallax
Gould & Yee 2013) t ∆u ∆t where t u t ,
effect can be measured, which yields the microlens par- E 0 → eff eff ≡ 0 E
and t can be much better measured than t . This is
allax parameter, eff E
especially true for short timescale events, as t and u
E 0
µ π can be severely degenerate with each other as well as
πE ≡πEµrel; πE ≡ θrel . (3) other parameters such as the source flux Fs.
rel E
Thiscanbedonebyusingasingleacceleratingobserva- 3. WFIRST+GROUND PARALLAXES
tory (Gould, 1992), or by taking observations simulta- Whenever there are microlens parallax measurements
neously fromtwo separatedsites (Refsdal, 1966;Gould, from comparing the light curves of two observatories,
1994b; Holz & Wald, 1996; Gould, 1997). In the latter it is also possible to obtain complementary parallax in-
case, the microlens parallax vector is given by formation from the accelerated motion of one or both
observatories separately. In the present case, WFIRST
AU ∆t
π = 0,∆u , (4) orbitalparallaxwillplayamajorcomplementaryroleto
E 0
Dsat,⊥ (cid:18) tE (cid:19) the two-observatory parallaxes that are made possible
by a ground-based observatory (or network of ground-
where Dsat, is the projected separation between the based observatories). However, for clarity, we begin by
two observa⊥tories evaluated at the peak of the event, analyzing the parallax information that can be derived
and ∆t0 =t0,sat t0, and ∆u0 =u0,sat u0, are the by comparing the two light curves.
differences in the−pea⊕k times and impact−param⊕eters as WFIRST-Earth microlensing has some features that
seenfromthetwoobservatories(wehaveassumedEarth differ substantially from those two-observatory exper-
and one satellite), respectively. iments that have been carried out previously or that
In many cases, however, only the component of the are being carried out. First, since WFIRST is a ded-
vector πE parallel to the opposite direction of accelera- icated space-based photometry experiment, it will al-
tion (i.e., away from the Sun for Earth and WFIRST) most always have essentially perfect measurements rel-
projectedontotheskycanbemeasuredwithreasonable ative to the ground. Therefore, the errors in the paral-
precision. This component is denoted as πE, lax measurements are very well approximated as those
k
due to the ground observations. Second, for similar
πE, πE nˆ πEcosφπ , (5) reasons, WFIRST-selected events will be quite faint as
k ≡ · ≡
seen from Earth, and therefore the Earth-based pho-
where nˆ is the opposite direction of acceleration pro-
tometry errors can be treated as “below sky”, i.e., in-
jected on the sky, and φ is the angle between π and
π E dependent of flux. Third, since WFIRST will be at
nˆ.
L2, its projected motion relative to Earth will be ex-
The parametersπE, andπE areoftencalled1-Dand tremelyslow,substantiallylessthan1km s 1. Thiscan
2-D parallaxes,respecktively. −
be compared with typical lens-source projected veloci-
Tofacilitatelaterdiscussions,weintroducethevector ties v˜ (100km s 1). This means that the Einstein
microlensing parameter Λ (Dong et al., 2007) ∼ O −
timescales t are essentially identical as seen from the
E
twolocations. Inparticular,itimpliesthatthequantity
t π π /µ µ
Λ E E = rel rel rel, (6) entering Λ =tE∆u0/Dsat, can be simplified by
≡ AU AU µ ⊥ ⊥
rel
t ∆u =∆(u t ) u ∆t ∆t (9)
E 0 0 E 0 E eff
− →
whose amplitude Λ = 1/v˜ is the reciprocal of the pro-
jected transverse velocity. Because Λ is a purely kine- where teff ≡u0tE. That is, Equation (8) becomes
matic quantity, it can be used to distinguish between 1
Λ (∆t ,∆t ) , (10)
disk events (disk lens + Bulgesource)andBulge events → D 0 eff
sat,
⊥
WFIRST Plus Ground-based Microlensing Telescope Network 3
as we anticipated above. In addition, in most cases, That is, in this limiting regime, the parameters
t will in fact be measured from WFIRST even if it (u ,t ,F ) are degenerate. Hence, the only way to dis-
E 0 E s
cannot be measured from Earth, so that we can then tinguish them is fromthe wings ofthe lightcurve. This
convert π =(AU/t )Λ. From our standpoint, we will can be a serious problem for ground-basedobservations
E E
thereforeregardmeasurementofΛasthegoal,withthe of WFIRST targets, since they may be extremely faint
understanding that this itself will very often yield π . and noisy near baseline.
E
Andevenwhenitcannot,Λisthecrucialparameterfor However, as stated above we are not actually inter-
distinguishing populations in any case because it is a ested in directly measuring t from the ground. We
E
purely kinematic variable. Finally, since WFIRST will therefore rewrite Equations (11) and (12) in the high-
be at L2, WFIRST-Earth parallaxes are exceptionally magnification limit, which has only three parameters
sensitivetoshortevents,whichistraditionallythemost (Gould, 1996) a =(t ,t ,F )
i 0 eff peak
difficult regime, i.e., the regime of events generated by
very low-mass lenses. That is, such events do not last (t t0)2 −1/2
longenoughtomakeorbitalparallaxmeasurements,and F(t)=FpeakQ(t) ; Q(t)= −t2 +1 .
theirEinsteinradiiaretoosmalltopermitsimultaneous (cid:18) eff (cid:19)
(17)
observation from observatories separated by AU, like
Then
∼
Spitzer and Kepler. F Q3τ /t
∂F peak eff eff
We analyze the WFIRST-ground parallaxes using F Q3τ2 /t , (18)
Fisher matrices. The full point-lens equation is de- ∂ai → peak Qeff eff!
scribedby four parametersthatareofphysicalinterest,
where τ (t t )/t . We then evaluate the inverse
ai =(t0,u0,tE,Fs), and one nuisance parameter Fbase covarianecffe≡matr−ix,0 eff
where F(t)=Fs(A[u(t)]−1)+Fbase , (11) bij = π8ΓteffσF2p2eak t−e0ff2 3t0−eff2 4t−eff10Fp−e1ak ,
0 0 4t−eff1Fp−e1ak 8Fp−e2ak
A(u)= u2+2 ; u2 =τ2+u2; τ t−t0 . (12) (19)
u√u2+4 0 ≡ tE and thus the covariance matrix c=b−1
ThenuisanceparameterFbase isessentiallyuncorrelated 8 σ2 t2eff 0 0
wanitahlyositsh.eUrnpdaerarmtheetearsss,usmopwtieonigonforueniiftorinmtohbeseforlvlaotwioinngs cij = πFp2eak0Γteff 00 −tefftF2epffeak/2 −(t3e/ff8F)Fpep2aeka/k2 .
at a cadence Γ, the Fisher matrix (i.e., inverse of the (20)
covariance matrix) is then given by Thus the uncertainties on t and t are
0 eff
bij = σΓ02 Z−+∞∞dt∂∂Fa(it)∂∂Fa(jt) , (13) σ(t0)=rπ8 tΓEσF0su30/2g(u0) , (21)
where wehaveassumedthatthe observationsarebelow σ (t )= 8 tEσ0u3/2h (u ) . (22)
sky so that the flux error σ is independent of magnifi- i eff π Γ F 0 i 0
0 r s
cation. Here
The correction factors g(u ) and h (u ) (i=1,2) allow
0 i 0
us to extendthese formulaeto the generalcase,andare
F Aτ/(ut )
s ′ E
− derived in Appendix A. We provide two different forms
∂F F Au /u
∂ai = −FsAs ′τ′2/0(utE) , (14) obfasσe(dteiffn)f:ortmheatfiiornst, wishdileeritvheed sbeycounsdinigspduerreivlyedgrboyunads--
A 1
− suming perfect knowledge of t from WFIRST. These
E
threefunctions,g(u ),h (u ),andh (u )areillustrated
and 0 1 0 2 0
dA 8 in Figure 1 for 0 u0 1.
A′ = − . (15) In principle, t≤ ca≤nnot be known perfectly from
≡ du u2(u2+4)3/2 E
WFIRST for two reasons. First, WFIRST observa-
Although the Fisher matrix cannot be expressed in tions are not perfect, so the associated t measurement
E
closed form for the general case, it can be in the high has a statistical uncertainty σ (t ). Second, even if t
W E E
magnification regime, where A(u) = 1/u and A(u) = from WFIRST can be constrained extremely precisely,
′
1/u2. Below we derive these closed-form expressions t of the same event as seen from Earth is still uncer-
E
−
in this regime, and provide the analysis of the general tain to a limited level ∆t , due to the relative velocity
E
case in Appendix A. between WFIRST and Earth. However, as we show in
In the high-magnification limit, it can be seen that Appendix A, tE for ground-based observations can be
treated as “perfectly” knownas long as the uncertainty
∂F ∂F ∂F in t inferred from WFIRST is smaller, by a certain
E
u t +F +1 =0 . (16)
0∂u − E∂t s ∂F specified factor (Equation (57)), than the uncertainty
0 E (cid:18) s (cid:19)
4 Wei Zhu & Andrew Gould
We assume Γ = 240day−1, i.e., one observation per
102 2 minutes, for⊕four hours per night (which is the time
the Bulge is visible at the midpoint of the WFIRST
f(u )
0 campaigns)ateachofthree observatories,and33%bad
g(u0) weather. Wenormalizetheerrorsto0.05magnitudesat
1
h (u ) I =18. We then find
1 0
s
e h (u )
n valu101 2 0 (cid:20) σσ((ππEE,,⊥k)) (cid:21)= c0o.s52ψ (cid:18)dtaEy(cid:19)−1/210Is2−.518u30/2(cid:20) hg2((uu00)) (cid:21) .
o (23)
ti where ψ is the phase of the WFIRST orbit relative to
c
n quadrature at the peak of the event. We note that
u
F because the observations are centered at quadrature,
0.82 cosψ 1. At first sight this pre-factor does
≤ ≤
notlookespeciallypromising,particularlygiventhefact
that typical WFIRST microlensing sources will be sub-
100 stantiallyfainterthanIs =18. However,therearethree
0.0 0.2 0.4 0.6 0.8 1.0 points to keep in mind. First, we expect t 1day
u0 events to correspond to M M lenses, wEho∼se par-
Jup
∼
Figure 1. Functional forms of f(u0) (defined as Equa- allaxes would be πE 4 if they lay in the disk and
∼
tion (33)),g(u0)(definedasEquation(50)),h1(u0)(defined πE 1.5 in the bulge. Second, 10% of a “fair sample”
as Equation (52)), and u2(u0) (definedas Equation (54)) in of e∼vents will have u0 <0.1 and so errors that are &30
the range 0<u0 <1. times smaller. Third, the sample of events will not be
“fair”, but rather heavily biased toward fainter sources
in t from ground-based observations. We have further at high-magnification.
E
provedin Appendices A andB that this conditionis al- Regarding the first point, in order to make clear the
most always satisfied for both the WFIRST statistical measurability of parallax, it is better to express Equa-
uncertaintyσ (t )andtheWFIRST-Earthsystematic tion (23) in terms of Λ, since this is a purely kinematic
W E
offset ∆t . variable that does not vary with the lens mass
E
Therefore, the assumption that t is perfectly known
E
mgfrrooomsutnlyWd-wbFahIsRaetSdwToebascleamrnvoagstteitoanflwos.raythsehuonldcse,rtsaointthyatinσt2e(fftefffr)omis (cid:20) σσ((ΛΛ⊥k)) (cid:21)= 1000.300ksmecsψ−1 (cid:18)dtaEy(cid:19)1/210Is2−.518u30/2(cid:20) hg2((uu00)) (cid:21) .
(24)
Equations (21) and (22) have a number of important This shows that WFIRST-Earth parallaxes become
implications. First, the two terms entering Λ have ex- more sensitive at shorter timescales (at fixed Λ or pro-
actly the same errors in the high magnification limit, jected velocity v˜).
namely 8teff/πΓσ0/Fpeak. Second, the errors scale We illustrate this sensitivity in Figure 2 by show-
tsitornonfgalcytpowr.ithTmhiasginmifipclaietsioan,st∝roun30g/2mtaigmneifisctahteioncobrrieacs-, (inogr tπhEe/σn(uπmEb,er) o>f e5v)enftosrthtyapticsaatlisBfyulπgEe/lσe(nπsEes,k)(v˜>=5
sothatthemuchmorenumerousfaintpotentialsources AUµrel/πrel =⊥ 1000km s−1) and typical disk lenses
canrelativelyeasilyenterthe sampleathighmagnifica- (v˜=280km s−1). Here we have assumed AI =1.5 and
tion. The magnification bias is stronger for σ(t ) than then integrated over the Holtzman et al. (1998) lumi-
eff
for σ(t ), suggesting that π is always better deter- nosityfunction. TheFigureisnormalizedtothenumber
0 E,
mined than π . However, ckomparison of g(u ) and of events with WFIRST baseline photometric precision
E, 0
h (u ) shows th⊥at this superiority is relatively modest. better than 1%. This number can in turn be estimated
2 0
for any specific WFIRST strategy that is either con-
Third,t isnotcorrelatedwithotherparameters,and
0
sidered or adopted, and of course can be empirically
inparticularitisnotcorrelatedwitht . Thisissimply
eff
determined from the experiment itself.
duetothefactthat∂F/∂t isanoddfunctionoft,while
0
There are several important points regarding Fig-
the other derivativesareevenint. This is true for both
ure 2. First, by incorporating the t information from
Equations (14) and (18). E
WFIRST observations,weareabletoincreasethenum-
Fourth, t is correlated with other parameters. As
eff berof2-Dparallaxdetectionsbyafactor 1.2. Second,
thefirstindicationofwhythisisimportant,wenotethat ∼
the full parallax(π ) curves with and without incorpo-
even in the high-magnification limit, t remains sig- E
eff rating t information from WFIRST lie only about a
nificantly correlated with F (correlation coefficient E
peak factorof1.2and1.5belowthe π curves,respectively,
− 2/3). Hence,forexample,iftherewereindependent despitethemoreseriousdeteriorEat,kionsofhi(u0)relative
informationaboutthesourceflux,theerrorint could
eff tog(u )showninFigure1. Thisisbecausefirstthemea-
bepreduced by a factor up to √3. 0
surements are dominated by events with relatively low
Tomakeaquantitativeestimateofthemicrolenspar-
allax errors, we adopt parameters typical of KMTNet. 1TheVegamagnitudesystemisusedinthepresentwork.
WFIRST Plus Ground-based Microlensing Telescope Network 5
u .0.2forwhichtheaverageratio h (u )/g(u ) <2,
0 1 0 0
h i
andsecond,foragivensourcestar,thiscanbe compen-
sated by going lower in u by a factor 22/3 =1.6.
0
Figure 2 also shows that the 2-D parallax measure-
ments will be available for a substantial fraction of
Jupiter-mass FFPs (t . 1day) in the disk and for
E
Earth-mass FFPs (t .0.05day)in the Bulge. Shorter
E
eventsareingeneralpreferred,butthe WFIRST-Earth
parallaxmethodimplicitlysetsalowerlimitontheevent
timescale t that it can probe. This is because the
E
same eventmust be observablefromboth WFIRSTand
Earth, so that AU/π &D , or
E sat,
100 ⊥
1
v˜ −
t &0.02 day cosψ . (25)
E 103 km s 1
(cid:18) − (cid:19)
Therefore, Equations (23) and (24) are only valid for
)
E
(t disk events with tE & 0.06 days and Bulge events with
1
0.0 tE &0.02 days.
<
W
σphot,base, 10-1 We now4f.ocOuNsEa-tDteIMntEioNnSIoOnNA1L-DPApRaAraLlLlaAxXeEs.S As just
N
/ mentioned, these can be measured about 1.2 to 1.5
t)E Disk Events timesmorefrequentlythan2-Dparallaxesbycomparing
(
E,∥ Bulge Events WFIRST and ground-based lightcurves. However, our
π
N 1-D π primaryreasonforthisfocusisthatWFIRST can,byit-
E
2-D π self measure 1-D parallaxes for sufficiently long events.
E
2-D π (w/o t constraint) That is, WFIRST-Earth and WFIRST-only measure-
E E
Mass Measurements (2-D π + θ ) ments are complementary, being respectively most sen-
E E
10-2 sitive in the short t and long t regimes.
10-1 100 101 E E
Of course, the main disadvantage of 1-D parallaxes
t (days)
E is that they appear to be of little practical value. We
will show, however, that this assessment is far too pes-
Figure2.Thenormalized numbersof eventswith 1-D paral- simistic.
lax (πE,k) and 2-D parallax (πE) measured better than 5-σ
fortwosetsoftypicalevents,respectively. For“Diskevents”, 4.1. WFIRST-only1-DParallaxes
weveeandtso”p,twπereald=op0t.1π2rmela=sa0n.0d2µm=as7amnadsµyr=−14,amnadsfyorr−“1B.uFlgoer WebeginbymakinganestimateoftheWFIRST-only1-
D parallax errors via Fisher matrix. Because WFIRST
each set of typical events, we show two curves for the 2-D
parallaxmeasurements,onewithperfecttEinformationfrom is observing near quadrature, it is accelerating trans-
WFIRST (solidlines)andtheotherwithoutanyexternaltE verse to the line of sight at
information(dash-dottedlines). Theformerismorerealistic
a AUΩ2 cosψ; ψ Ω (t t ). (26)
(seeSection3). Wealsoshowthenumberofeventsforwhich quad
therearebothπEmeasurementsandθEmeasurements(from Here Ω⊥ ≃ 2π/y⊕r and t is≡the⊕epo−ch when the field
fimnailtiezastoiounrceisetffheectnsu),mabserdiosfcuesvseendtsinwSitehctWionFI5R.3S.TTbhaesenlionre- isatqu⊕ad≡rature. Intheqaupapdroximationthattheacceler-
ation is constant, this induces a quadratic deviation in
photometricprecision betterthan1%,whichcorrespondsto
a baseline magnitude of H = 20.2 (Gould, 2014b). We as- the lightcurve, which to lowest order implies a normal-
sumetheWFIRST microlensingsourcesfollowtheluminos- ized lens-source separation u(t),
ityfunctionofBulgestarsderivedbyHoltzman et al.(1998).
t t 1 2
[u(t)]2 = − 0 + π (Ω (t t ))2cosψ +u2 .
(cid:20) tE 2 E,k ⊕ − 0 (cid:21) 0
(27)
This leads to an asymmetric distortion in the magnifi-
cation (Gould et al., 1994). With this as well as Equa-
tion (11), one finds that
∂F F Ω2t2 cosψAτ3
∂πE, = s ⊕2E u′ (πE,k ≪1) . (28)
k
Because Equation(28) is odd in t, the only other mi-
crolensing parameter that it couples to is t . Thus, the
0
6 Wei Zhu & Andrew Gould
Fisher matrix is two-dimensional. To evaluate this, we 4.2. CombinedOrbitalandTwo-Observatory1-D
firstspecifythatWFIRST observationswillgenerallybe Parallaxes
above sky, so that the flux errors scale σ = σ A1/2,
0,W
Comparison of Equations (24) and (35) shows that the
whereσ istheerroratbaseline. AsinEquation(13),
0,W
two approaches to obtaining 1-D parallaxes are com-
we approximate the observations as being at a uniform
rate Γ , and find plementary, with precisions σ(Λ ) t1/2 for WFIRST-
W k ∝ E
bij = ΓσW2tE ∂∂aF ∂∂aF dAτ pEaarratlhlapxaersa.llaTxhees annodrmσa(lΛizka)ti∝ontsE−(3I/S2 f=or W18FiInRSthTe-ofinrlsyt
0,W Z i j and H =20 in the second) may appear deceptive, par-
= 4ΓσW02,WFst2E (cid:18) −42GηG01 −η22ηGG21 (cid:19) , (29) 0ti.c8u+laErsly(Ib−eHca)us∼e1t.y8paicsaslumsoiunrgcEes(Iw−illHh)av∼e1(.IH−owHe)ve∼r,
one must bear in mind that that the two-observatory
where η Ω2t3 cosψ, and
≡ ⊕ E formula has a very strong dependence on u0, whereas
G A2A 1u 2τ2dτ the WFIRST-only formula is, by comparison, almost
0 ′ − −
G ≡ A2A 1u 2τ4dτ . (30) flat in u0.
G1 ≡ R A′2A−1u−2τ6dτ To investigate this further, we consider the combined
2 ≡ R ′ − − impact of both measurements, assuming A = 0.5 and
H
Then the covariance maRtrix is (asbefore)aHoltzman et al.(1998)luminosityfunction
cij =b−ij1 = ΓσW02,FWs2tηE2G0G21−G21 (cid:18) 2ηη2GG21 24ηGG01 (cid:19) . abninWdinAegIsth=hoews1e.i5nt.wFWoigefoudrrmeenu3oltateheetbhnyeuσpmfrubelcle(irπssiEoo,nkf)de.vereinvtesdwbiythco1m-D-
(31)
parallaxmeasurementsbetterthanaspecified(absolute
The uncertainty in πE, is given by σ(πE, ) = c122/2. It and relative) precisionfor WFIRST-only andWFIRST
can be expressed analkytically only in thke high mag-
plus ground observations, respectively. The normaliza-
nification limit. Therefore, similarly to the case of tions are again to the number of events with WFIRST
WFIRST+ground 2-D parallaxes, we introduce a cor-
baseline precisions of 1%. Figure 3 demonstrates the
rection factor f(u ) that approaches unity as u ap-
0 0 importance of ground-based observations in measuring
proaches zero, and rewrite σ(πE, ) as 1-D parallaxes: without these observations, WFIRST
k
will not be able to make any meaningful 1-D parallax
σ secψ
0,W
σ(πE,k)=1.77 Fs Ω2t5/2Γ1/2f(u0) , (32) measurements for short timescale events.
E W
⊕
where 1.77=[14/3 321/2ln(1+√2)] 1/2 is the result 4.3. XallarapandLensOrbitalMotion
−
−
of an analytic calculation, and
Measurementsofπ madefromasingleobservatorycan
E
be corrupted by xallarap(motion of the source about a
G
f(u ) 1.13 0 . (33) companion) and lens orbitalmotion (motion of the lens
0 ≡ sG0G2−G21 about a companion), whereas those derived from com-
parisonofcontemporaneousmeasurementsfromtwoob-
ThisfunctionisalsoillustratedinFigure1. Thedeterio-
servatories cannot. This is because the basis of single-
rationtowardhigheru isprimarilyduetothefactthat
0 observatoryπ measurementsistheacceleratedmotion
thehigh-magpeakcontributesthemajorityoftheinfor- E
oftheobserver,whichcaninprinciplebeperfectlymim-
mationaboutπ . Secondarily,t becomesincreasingly
correlated withEπ,k at higher u .0 icked by accelerated motion of the source (or the lens).
AdoptingacadEe,nkceofΓW =1000day−1 andassuming B(i.eel.o,wxawllearoanplyeffcoecnts)i,debrutthoeuarccmeeletrhaotdioonlogoyf tahpeplsioeusrtcoe
0.01 mag errorsat H =20.2 (Gould, 2014b), this yields
the other case as well.
5/2 In the case of complete 2-D π measurements,
0.017 Hs−20 tE − E
σ(πE,k)= cosψ10 5 10day f(u0) . (34) xallarap-dominated acceleration effects can in principle
(cid:18) (cid:19) be distinguished fromthe parallaxeffects fromtheir or-
For an intuitive understanding of the relevance of this bital period and the direction of their implied angular
error bar to the parallax measurement, it is best to ex- momentum vector. Thatis, ifthe effects ofparallaxare
press it in terms of Λ mistakenly attributed to xallarap,then the xallarapso-
k lutionwillleadtoacompanionwitha1-yearperiodand
3/2
0.10 secψ Hs−20 tE − orbitalaxisthatisexactlyalignedtothatoftheEarth’s
σ(Λk)= 1000kms−110 5 (cid:18)10day(cid:19) f(u0) . (projected on the plane of the sky) (Poindexter et al.,
(35) 2005).
From Equations (34) and (35), it is clear that Thispurelyinternaltestfailscompletely,however,for
WFIRST will make very good π measurements for 1-D parallaxes, unless the period is so short that a 2-
E,
long events but will do much worske for short events. D xallarap solution can be reliably extracted from the
For example, for t = 2 days, the pre-factor in Equa- data. The contamination of 1-D parallaxes due to xal-
E
tion (35), goes from 0.10 to 1.1. laraphasneverpreviouslybeenestimated,probablydue
WFIRST Plus Ground-based Microlensing Telescope Network 7
101
WFIRST alone WFIRST alone
WFIRST+Ground,linear WFIRST+Ground,linear
WFIRST+Ground,full
WFIRST+Ground,full
) )
E E
(t (t
<0.01 100 <0.01
W W 100
Nσphot,base, σσ((ππEE,,∥∥))<<00..1003 Nσphot,base,
)/ σ(πE,∥)<0.01 )/
(tEπE,∥10-1 (tπEE,∥
N N
Disk Events
Bulge Events
10-2 10-1
10-1 100 101 10-1 100 101
t (days) t (days)
E E
Figure3.Thenormalizednumbersofeventswith1-Dparallax(π )measurementsbetterthangivenprecisionsforWFIRST-
E,k
onlyobservations(dashedlines),WFIRST plusground-basedsurveyobservations(dash-dottedlinesforlinearapproximation
andsolidlinesforfullderivation). Theleftpanelshowscurveswithabsolutemeasurementuncertainties(σ(π )). Theright
E,k
panel shows curves with 5-σ detections (πE,k/σ(πE,k) = 5) for typical disk events (πrel = 0.12 mas, µ = 7 masyr−1) and
Bulgeevents(πrel =0.02mas,µ=4masyr−1). ThenormalizationisthesameasthatusedinFigure2. Thatis,thenumber
of eventswith WFIRST baseline photometric precision seen by WFIRST better than 1%. The flattening for tE >10 days
is caused by our restriction that u0 ≤ 1, but it is very likely real for two reasons. First, πE,k measurements will be quite
difficult for u0 >1, as the correlation between t0 and πE,k becomes significant (correlation coefficient r(t0,πE,k)≥0.65 for
u0 ≥1); and second, many eventswith tE >10 dayswill not befully covered byWFIRST observations.
to the limited previous interest in 1-D parallaxes them- strict attention to the 53 companions with semi-major
selves. axes 0.2 < a/AU < 30 on the grounds that the hand-
The contamination due to xallarap can be rigorously ful of closer companions would be recognized as such
calculated under the assumption that the multiplicity from oscillations in the lightcurve, while those farther
properties(massesandsemi-majoraxes)ofcompanions away would induce asymmetries that are too small to
to the microlensed sources are similar to those of solar- measure. We allow φ to vary randomly over a circle
π
type starsin the solarneighborhood. We firstnote that and φ to vary randomly over a sphere. Figure 4 shows
ξ
theamplitudeofEarth’saccelerationrelativetothepro- thatxallarapcontaminationisveryserious(>100%)for
jected Einstein radius r˜ AU/π is (at quadrature) about 2% of Bulge lenses and is significant (>10%) for
E E
≡
A˜= (GM /AU2)/r˜ . The component of this accelera- about4.5%. Inaddition,wenotethatthereisacompa-
E
tion enteri⊙ng π is A˜cosφ . rable contamination from the acceleration of the lenses
E, π
Bythesametokken,theaccelerationduetoacompan- due to their companions(both withandwithoutbinary
lightcurve signatures).
ionofmassmandsemi-majoraxisarelativetotheEin-
steinradiusprojectedonthesourceplanerˆ D θ ,is Therefore,withWFIRST aloneupto9%ofallevents
E S E
Aˆ=(Gm/a2)/rˆ . Andthe componentthatc≡ontributes will show false parallax detections. Such contamination
E
to the asymmetry of the event is Aˆcosφ , where φ is would be removed by a complementary ground-based
ξ ξ survey for relatively short events (t . 3 days). For
the angle between the lens-source relative motion and E
longer events, this contamination cannot be removed
the instantaneous acceleration of the source about its
completely due to the limited power of WFIRST-Earth
companion. Hence, the ratio of the xallarap-to-parallax
1-Dparallax,butthestudyofthiscontaminationinthe
signals contributing to this asymmetry is
short event sample allows to better interpret the 1-D
|ξk| = Aˆ|cosφξ| = m a −2 DL |cosφξ|, parallax measurements for these longer events.
|πE,k| A˜|cosφπ| M⊙ (cid:18)AU(cid:19) DLS|cosφπ|(36) 4.4. ValueofπE,kMeasurements
where D D D is the distance between the lens A large sample of 1-D parallaxes with well-understood
LS S L
≡ −
and the source. selection has two major uses. First, it can be directly
To evaluate the distribution of this ratio for Bulge analyzedtosimultaneouslyderiveaGalacticmodeland
lenses, we first adopt D /D = 8. We then con- the lens mass function. Second, it allows one to de-
L LS
sider the ensemble of binary companions in Figure 11 rive the complete solution (namely individual masses,
of Raghavan et al. (2010), which is a complete sam- distances andtransversevelocities)once combinedwith
ple of companions for 454 G-dwarf primaries. We re- the measurement of the lens-source relative proper mo-
8 Wei Zhu & Andrew Gould
Han & Gould(1995)arguedthatindividuallensmass
0.12
anddistancewouldbemuchmoretightlyconstrainedby
measuring the full parallax vector π in addition to t ,
E E
0.10 and Calchi Novati et al. (2015) showed that this was in
factthecasefortheirsampleofmicrolenseswithSpitzer
parallaxes. Their work still incorporatedGalactic mod-
0.08
els,butthemass/distanceconstraintsweredramatically
improved compared the case with timescales alone. See
F
D0.06 also Gould (2000).
C
The 0-D parallaxesof Sumi et al. (2011) and the 2-D
parallaxesofCalchi Novati et al.(2015)havehaddiffer-
0.04
entapplications. Specifically,the greatvalueofthe first
is that it could make statistical statements about low-
0.02 mass objects, while that of the second was the greatly
improved precision of individual lenses. In particular,
the2-Dstudycouldnotmakeanystatementaboutlow-
0.00
103 102 101 100 10-1 10-2 10-3 10-4 mass objects because the Spitzer sample was strongly
ξ /π biased against short events.
| E,∥| | E,∥|
For WFIRST 1-D parallaxes, the situation is clearly
Figure4. The fraction of Bulge events (DL/DLS = 8) with intermediate between the 0-D parallaxes of Sumi et al.
parallax signalaffectedbythexallarap effecttoagivenpre- (2011) and the 2-D parallaxes of Calchi Novati et al.
cision. (2015). The greatpotential value of WFIRST 1-D par-
allaxes (i.e., much stronger statistical statement about
tion µ , which can be obtained for some events with BDsandFFPs)wouldbealmostcompletelylostifthese
rel
WFIRST alone and even more events with follow-up substellarobjectsweresystematicallyexcludedfromthe
imaging. These two applications are discussed in turn. 1-D parallax sample. Given the σ(Λ ) t−3/2 behav-
In both cases, we show that the addition of short tE ior of Equation (35) (also see Figurek3),∝thiEs is exactly
events from combining ground and space observations what would happen in the absence of a ground-based
will play a critical role in understanding the low-mass complementary survey. The addition of such a survey
endofthe massspectrum,i.e.,browndwarfs(BDs)and would enable 1-Dparallaxmeasurements acrossthe en-
FFPs. tire range of timescale t &2hrs for disk lenses.
E
4.4.2. CompleteSolutionsFromπ Plusµ
4.4.1. StatisticalStudiesUsing1-DParallaxSamples E,k rel
A complete solutionofthe lens (orthe lens system)can
At present, large microlensing samples with well-
be derived by combining the π measurement from
understood selection are characterized by a single mi- E,
the light curve and the µ meaksurement from high-
crolensing variable, the Einstein timescale t , i.e., no rel
E resolution imaging (Ghosh et al., 2004; Gould, 2014a),
parallax information. Although not usually thought of
basedonthedefinitionoft (Equation(1))andthefact
this way, we dub this case as a “0-D parallax measure- that π and µ are alongEthe same direction. 2
ment”inordertocontrastitto1-Dand2-Dparallaxes. E rel
The measurement of µ can be done by WFIRST
That is, in the three cases, the available information rel
alone, using its 40,000 high-resolution images,
conTshisetsstoafti(sttEic)a,l(itnEte,rπpEr,ekt)a,t(itoEn,oπfEa,k0,π-DE,⊥pa).rallaxsample for relatively lumin∼ous (& early M-dwarfs) lenses
(Bennett et al.,2007). Late-timehigh-resolutionfollow-
requires a Galactic model to constrain six of the input
upimagingcanextendthe domainofcoverageto alllu-
variables (lens and source distances and transverse ve-
minouslenses(i.e.,M >0.08M )andcanalsoimprove
locities) to obtain information about the seventh (lens
the precision, as well as resolv⊙ing certain ambiguities
mass function). This situation is not as bad as it may
thatwediscussinSection4.3. Suchimaginghasalready
first seem. The three source properties are well under-
achieved important results using HST (Alcock et al.,
stoodstatisticallyfromdirectobservations,evenifthese
2001; Bennett et al., 2015) and Keck (Batista et al.,
source properties are not measured for each individual
2015),butwillpotentiallybemuchmorepowerfulusing
event, and Galactic models are constructed based on a
next generation telescopes (Gould, 2014a; Henderson,
wide variety of very good data. Sumi et al. (2011) used
2015)
this technique to infer the existence of a population of
Before proceeding, one may ask why one would need
FFPs. Nevertheless, uncertainties in Galactic models
microlensing mass and distance measurements for a
remain considerable, and hence it would be valuable if
(subsequently) resolved lens, since photometric esti-
microlensingstudiescouldfurtherconstrainthemrather
mateswouldthenbeavailable. AspointedoutbyGould
than propagating them. Moreover, even if the Galactic
model were known perfectly, the statistical precision of
microlensing studies is greatly reduced by the require- 2Because πE,k is measured in the geocentric frame but µrel is
measured in the heliocentric frame, one must be careful when
ment of deconvolving three parameters to learn about
combiningthesetwomeasurements. SeeGould(2014a)formore
the one of greatest interest. details.
WFIRST Plus Ground-based Microlensing Telescope Network 9
(2014a), the answer is two-fold. First, one would al- 5. EINSTEIN RADIUS MEASUREMENTS: θE
ways prefer measurements to estimates. Second, high-
The particular interest of the θ measurement is that
resolution imaging alone may not identify the correct E
if there is also a measurement of π , then it yields a
microlenses. Because roughly 2/3 of all stars are in E
complete solution (i.e., mass, distance and transverse
binaries, and since the lensing cross section scales as
velocity).
θ M1/2, of order 1/3 of the events due to binaries
E
∝
will in fact be generated by the lower-mass (and gener- 5.1. WFIRST-onlyEinstein-RadiusMeasurements
ally fainter) companion. In the majority of such cases,
In the case of single-lens events, if the lens transits the
thestarthatactuallygeneratedtheeventwillnotbevis-
source, then the light curve is distorted by the finite
ible, because it is either unresolved or dark. Hence, for
source effect, which yields ρ θ /θ , the ratio of the
tens of percent of cases, the lens would be misidentified E
source radius to the angular≡Eins∗tein radius. Because
bysuchsimpleimaging. Gould(2014a)showshowthese
θ can be determined from the dereddened color and
casescanbeidentifiedandresolvedbyacombinationof
m∗agnitude of the source (Yoo et al., 2004), this yields
1-D parallaxes and imaging.
a measurement of θ and so also of the proper motion
E
AlargehomogeneoussampleofeventswithbothπE, µrel = θE/tE (Gould, 1994a). Such transits occur with
and µ measurements has several benefits. First, ikt probability ρ, 3 whichistypicallyoforder 3 10 3
rel −
would permit one to derive a Galactic model (rather for main-seq∼uencesources andand 3 10 ∼2 fo×r giant
−
than assuming one). Second, it would permit one to sources. ∼ ×
measurethelensmassfunctionovertherangeofmasses When the observational bias is taken into account,
that are probed. Third, for all the planetary events in the number of single-lens events with finite-source ef-
thissample,onewouldgainaprecisemassmeasurement fects should be more than what one would naively es-
of the host and thus (in almost all cases) of the planet. timate based purely on the above transit probability.
Single-lenseventswithfinite-sourceeffectscanreachex-
Even without a complementary ground-basedsurvey,
tremely high magnifications, so they can be detected
all of these benefits could be derived at least partially
even though the source stars at baseline are extremely
fromWFIRST observationsthemselves. However,these
faint. For example, an M6 dwarf in the Bulge has
observationsalonewouldprobeonlytheupperhalforso
H = 26 (assuming an extinction of A = 0.5) and
of the mass function, partly because WFIRST-only 1- H
an angular radius θ = 0.06µas, and can be magni-
D parallaxmeasurementsrequire long-timescaleevents, ⋆
fied in brightness by A = 2/ρ when a lens tran-
and partly because WFIRST proper motion measure- max
sits exactly through its center. For a Neptune-mass
ments require luminous lenses. Similarly, the masses of
Bulge lens, this event will reach H = 21 at its peak.
planets orbiting low-mass hosts would remain undeter-
WFIRST can therefore obtain on average one observa-
mined.
tion during the entire transit with a photometric preci-
Hence, complementary ground-based survey observa- sion of 1.4%, which would yield a precise measurement
tions are crucial to probe the low-mass stellar and sub- of ρ. 4 Hence if the lens is more massive than Nep-
stellar lenses. tune, then essentially all luminous stars in the Bulge
can serve as source stars for events with measurable
finite-source effects. For Earth-mass Bulge lenses, 2%
precision can be reached for individual measurements
4.5. Valueofπ Measurements
E of M4 (M 0.25M , R 0.25R ) sources, yield-
ing similar q∼uality θ ⊙deter∼mination⊙s (since there are
E
As discussed following Equation (24), the 2-D paral- on average 2 measurements per transit). Therefore, al-
lax measurements are biased toward substellar objects. thoughsmallstarsaredisfavoredbytheir smallangular
These are exactly the objects that are inaccessible to size, they contribute more significantly to the number
the conversion of 1-D parallaxes to complete solutions ofsingle-lenseventswithfinite-sourceeffects,thanthey
that was discussed in Section 4.4.2. These 2-D parallax do to the number of all detectable single-lens events.
measurements enable us to directly derive the complete
solutions purely from the microlensing light curves for 5.2. WFIRST+GroundEinstein-RadiusMeasurements
the roughly 50% of all planetary and binary events in
In this subsection, we wish to understand how the ad-
which θ can be measured (Zhu et al., 2014). See Sec-
E ditionofaground-basednetworkcontributesto the fre-
tion6fordetails. Whilemostofthese2-Dparallaxmea-
quency of such measurements. The answer is: quite
surements in single-lens events will not yield complete
solutions(butseeSection5),Calchi Novati et al.(2015) 3For ground-based observations, u0 < ρ is required in order to
haveshownthat2-Dparallaxmeasurements,whencom- detect the finite-source effect, but for WFIRST, because of its
much better photometry and the unblended source (almost al-
bined with a Galactic model, give tight constraints on
ways), the maximum allowed u0 can be somewhat bigger than
mass and distance, particularly for disk lenses. This is ρ.
important not only to further constrain the substellar 4We use Spitzer event OGLE-2015-BLG-0763 as an example to
mass function relative to the analysis that is possible prove this. There was only one Spitzer observation with pho-
tometric precision of 1.2% when the lens transited the source
based on 1-D parallaxes (Section 4.4.1), but also for a
of OGLE-2015-BLG-0763, but the uncertainty on ρ is already
more detailed understanding of individual objects. limitedto2.3%level(Zhuetal.,2015b).
10 Wei Zhu & Andrew Gould
modestly. There are two issues. First, an aggressive parameter as seen from WFIRST, u , is similar
0,W,max
ground-based network will only observe the WFIRST to u . Then the conditional probability can be
0, ,max
fieldabouthalfthetime. Sincethesourcecrossingstyp- estima⊕ted as
ically lastonly about 1–2hours,it is this instantaneous
ρ θ
coverage that matters rather than daily coverage. Sec- P = = ⋆ . (41)
u µ t u
ond, if the source size, projected on the observer plane 0,W,max rel E 0,W,max
(ρAU/π ) is larger than the Earth-satellite projected
E Thestellarangularradiusθ isalsorelatedtothestellar
⋆
separation Dsat, , then the probability that the source flux F : θ Fβ, and we take β 0.3 as a reasonable
will transit at le⊥ast one of the two observersis not sub- s ⋆ ∝ s ≈
value for I band. By only keeping F and t , we find
s E
stantially increased. That is, if the ground observatory
irsoutgahkliyngdooubbselervtahtiisonpsroabtabthileitytipmreovoidfetdhethpaetak, it can P ∝Fsβ−1/α t1E/(2α)−1 ≈Fs−0.3 tE−0.7 . (42)
Hence, this conditional probability is only weakly de-
ρAU θ pendent on source flux and has a substantially stronger
=100 ∗ secψ <1. (37)
π D π dependence on Einstein timescale.
E sat, rel
⊥ We adopt θ = 0.15µas100.12(23.5 I0) and use
−
That is, 3/2∗
σ (t ) u h (u ) to evaluate P numerically and
2 eff ∝ 0 2 0
show the results in Figure 2. In doing so, we do not
1
θ − count any events for which P > 1, since this implies
πrel >60µas ∗ secψ , (38)
0.6µas that the source is too big to permit high-enough mag-
(cid:18) (cid:19)
nification for t to be measured. Figure 2 shows that
eff
wherewehavenormalizedtotheangularradiusofatyp-
complete mass measurements peak at roughly 0.2 days
ical solar-typesource. Thus, for main sequence sources,
for Bulge lenses and 0.1 days for Disk lenses, where in
the simultaneous ground-based observations contribute
both cases they constitute roughly half of the 2-D par-
θ measurements for disk lenses but not Bulge lenses.
E allax measurements.
The situation is even less favorable for sub-giant and
giant sources, which are a small minority of all sources 6. COMPLETE SOLUTIONS FOR BINARYAND
but a larger fraction of all finite-source events due to PLANETARY EVENTS
their larger size (Zhu et al., 2015b). In brief, we expect
Finally,althoughthecompletesolutions(π plusθ )for
that the ground-based survey will add only 10–20% to E E
isolated stellar-mass lenses will be rare, such measure-
the rate of θ measurements. This is small enough to
E
ments aremuchmorecommonforplanetaryandbinary
ignore for present purposes.
events, because roughly half of the recognizable such
5.3. Valueofθ Measurements events will show steep features due to caustic crossings
E
and cusp crossings. For these, measurement of θ by
The main value of θ measurements, which come pri- E
E WFIRST will be virtually automatic. Moreover, these
marily from WFIRST observations alone, is derived
eventswillalsohavegreatlyenhanced2-Dparallaxmea-
fromcombiningthemwiththevectorπ measurements,
E surements by three different channels.
which come primarily from combining WFIRST and
First,An & Gould(2001)arguedthatcausticcrossing
ground-based survey data. We therefore evaluate the
events yield full 2-D parallaxes much more easily than
conditionalprobabilitythatθ ismeasurablegiventhat
E events without sharp features (either single-lens events
π is measured.
E or non-caustic-crossingmultiple lenses). The sharp fea-
The impactparameteras seenfromEarth, u , that
0, tures break the continuous degeneracies among the pa-
would allow for a π measurement is limited by⊕
E rameters that are even in t. They also break the sym-
π ΛD metry of the lightcurve,whichis the fundamentalcause
E sat,
(S/N)th ≤ σ(πE,⊥) = σ2(teff⊥) , (39) Sofmtihthe efotuarlt.h(-2o0rd03e)r tfoimresindgeplee-nlednesnceeveonftsπ.E,W⊥hfioluentdhebrye
has been no firm proof that this is the case, there is
where (S/N) is the threshold for claiming a reliable
th
substantial circumstantial evidence from the high frac-
detection. Byapproximatingu3/2h (u ) uα(α 1.7)
0 2 0 ∝ 0 ≈ tion of planetary lenses with well-determined paral-
in Equation (22), we can derive the maximum allowed
laxes relativeto single lens events of similar (u ,t ,f ).
0 E s
impact parameter u
0, ,max Hence, it is likely that WFIRST will by itself mea-
⊕
suremany2-Dparallaxesofcaustic-crossingbinaryand
1/α 1/α 1/(2α)
ΛDsat, Fs Γ planetary events, particularly for those with timescales
u0,⊕,max ∝(cid:18)(S/N)t⊥h(cid:19) (cid:18)σ0(cid:19) (cid:18)tE⊕(cid:19) . tE & 7days, for which the corresponding single-lens
(40) events show good 1-D parallax measurements. See Fig-
The impact parameter as seen from WFIRST is given ure 3.
by u =u +∆u , where ∆u is relatedto π by Second, events with two caustic crossings observed
0,W 0, 0 0 E,
Equation(4).⊕For 0.1 days t 3 days and below⊥sky from both WFIRST and the ground can also yield
E
≤ ≤
sources as seen from Earth, one can easily prove that 2-D parallax determinations based on two ∆t mea-
cc
∆u . u , so that the maximum allowed impact surements, one at each crossing. Here ∆t is the
0 0, ,max cc
⊕
