A systematic fitting scheme for caustic-crossing microlensing events

Abstract

We outline a method for fitting binary-lens caustic-crossing microlensing events based on the alternative model parameterisation proposed and detailed in Cassan (2008). As an illustration of our methodology, we present an analysis of OGLE-2007-BLG-472, a double-peaked Galactic microlensing event with a source crossing the whole caustic structure in less than three days. In order to identify all possible models we conduct an extensive search of the parameter space, followed by a refinement of the parameters with a Markov Chain-Monte Carlo algorithm. We find a number of low-chi2 regions in the parameter space, which lead to several distinct competitive best models. We examine the parameters for each of them, and estimate their physical properties. We find that our fitting strategy locates several minima that are difficult to find with other modelling strategies and is therefore a more appropriate method to fit this type of events. Comment: 12 pages, 11 figures

Full text

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arXiv:0901.1285v1 [astro-ph.GA] 9 Jan 2009
Mon. Not. R. Astron. Soc. 000, 1–12 (2008)Printed January 23, 2009 (MN LATEX style file v2.2)
A systematic fitting scheme for caustic-crossing
microlensing events
N.Kains1,2,⋆, A. Cassan1,3, K.Horne1,2, M.D. Albrow1,4, S. Dieters1,5,
P. Fouqu´ e1,6, J. Greenhill1,7, A. Udalski8,9, M. Zub1,3†, D.P. Bennett1,10,
M. Dominik2‡, J. Donatowicz1,11, D. Kubas1,12, Y. Tsapras1,13, T. Anguita3,
V. Batista1,5, J.-P. Beaulieu1,5, S. Brillant1,12, M. Bode1,13, D.M. Bramich1,14,
M. Burgdorf1,13, J.A.R. Caldwell1,15, K.H. Cook1,16, Ch. Coutures1,17,
D. Dominis Prester1,18, U.G. Jørgensen1,19, S. Kane1,20, J.B. Marquette1,5,
R. Martin1,21, J. Menzies1,22, K.R. Pollard1,4, N. Rattenbury1,23, K.C. Sahu1,24,
C. Snodgrass1,12, I. Steele1,11, C. Vinter1,19, J. Wambsganss1,3, A. Williams1,21,
M. Kubiak8,9, G. Pietrzy´ nski8,9,25, I. Soszy´ nski8,9, O. Szewczyk8,9,25,
M.K. Szyma´ nski8,9, K. Ulaczyk8,9? L.Wyrzykowski13,26
1PLANET/RoboNet collaborations
2SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St Andrews, KY16 9SS, United Kingdom
3Astronomisches Rechen-Institut (ARI), Zentrum f¨ ur Astronomie (ZAH), Heidelberg University,
M¨ onchhofstraße 12-14, 69120 Heidelberg, Germany
4University of Canterbury, Department of Physics and Astronomy, Private Bag 4800, Christchurch, New Zealand
5Institut d’Astrophysique de Paris, UMR7095 CNRS, Universit´ e Pierre & Marie Curie, 98bis Boulevard Arago, 75014 Paris, France
6LATT, Universit´ e de Toulouse, CNRS, 14 avenue Edouard Belin, F-31400 Toulouse, France
7School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia
Accepted ... Received ... ; in original form ...
ABSTRACT
We outline a method for fitting binary-lens caustic-crossing microlensing events based
on the alternative model parameterisation proposed and detailed in Cassan (2008). As
an illustration of our methodology, we present an analysis of OGLE-2007-BLG-472,
a double-peaked Galactic microlensing event with a source crossing the whole caustic
structure in less than three days. In order to identify all possible models we conduct an
extensive search of the parameter space, followed by a refinement of the parameters
with a Markov Chain-Monte Carlo algorithm. We find a number of low-χ2regions
in the parameter space, which lead to several distinct competitive best models. We
examine the parameters for each of them, and estimate their physical properties. We
find that our fitting strategy locates several minima that are difficult to find with
other modelling strategies and is therefore a more appropriate method to fit this type
of events.
Key words: gravitational microlensing - data modelling - extrasolar planets - binary
stars - robotic telescopes
⋆email:nk87@st-andrews.ac.uk
† Member of International Max Planck Research School for As-
tronomy and Cosmic Physics at the University of Heidelberg
‡ Royal Society University Research Fellow

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2N. Kains et al.
1 INTRODUCTION
Gravitational microlensing (Paczy´ nski 1986) occurs when
the light from a source star is deflected by a massive com-
pact object between the source and the observer, leading
to an apparent brightening of the source, typically last-
ing a few days to a few weeks. When the deflecting body
has multiple components, such as a planet orbiting its
host star, there can be perturbations to the brightening
pattern of observed sources. These perturbations can be
large even when caused by low-mass objects, making them
detectable using small ground-based telescopes. Modelling
these lightcurve anomalies can lead to the detection of sub-
tle effects, allowing for measurements of properties such as
the source star limb-darkening coefficients (e.g. Cassan et al.
2004), the mass of stars with no visible companions (e.g.
Ghosh et al. 2004), and the detection of extrasolar planets,
as suggested by Mao & Paczy´ nski (1991) and first achieved
in 2003 (Bond et al. 2004).
Nevertheless, anomalous microlensing events usually re-
quire very detailed analysis for a full characterisation of their
nature to be possible, making them challenging. This applies
in particular to a class of microlensing events which display
caustic crossing features in their lightcurves. These events
are of primary interest, because they account for around
ten percent of the overall number of detected microlenses,
and they represent an important source of information on
physical properties of binary stars (Jaroszynski et al. 2006).
However there exist several degeneracies that affect the mod-
elling of this type of events: without a robust modelling
scheme and a full exploration of the parameter space, it
is impossible to pin down the true nature of a given event.
In addition to this, calculations of anomalous microlensing
models for extended sources are very demanding computa-
tionally.
Given these issues, brute force is not an option when
modelling caustic-crossing events, and one has to devise
ways of speeding up calculations, for example by exclud-
ing regions of parameter space which cannot reproduce fea-
tures that appear in data sets. A way to achieve this is to
use a non-standard parameterisation of the binary-lens mod-
els that ties them directly to data features, as proposed by
Cassan (2008), which we recall below.
In this paper, we present our method for exploring the
parameter space, and describe our approach to find all pos-
sible models for a given event (Sec. 2). We then use OGLE-
2007-BLG-472, a microlensing event observed in 2007 by
the OGLE and PLANET collaborations, as an illustration
of our methodology applied on a binary lens event which
intrinsically harbors many ambiguities (Sec. 3). We finally
discuss the implications of the individual competitive mod-
els that we find in order to discriminate between realistic
microlensing scenarii.
2BINARY-LENS EVENTS FITTING SCHEME
2.1Parameterisation of binary lens lightcurves
A static binary lens is usually described by the mass ratio
q < 1 of the two lens components and by their separation d,
expressed in units of the angular Einstein radius (Einstein
1936),
θE =
s
4GM
c2
„DS− DL
DSDL
«
,(1)
where M is the mass of the lens, and DL and DS are the
distances to the lens and the source respectively. Such a lens
produces caustics where the magnification of the source di-
verges to infinity for a perfect point source. The positions,
sizes and shapes of the caustics depend on d and q. For the
binary lens case, caustics can exist in three different topolo-
gies, usually referred as close, intermediate and wide; bifur-
cation values between these topologies are analytical expres-
sions relating d with q (Erdl & Schneider 1993). In the close
regime, there are three caustics: a central caustic near the
primary lens component, and two secondary caustics which
lie off the axis passing through both lens components. In the
intermediate case, there is only one large caustic on the axis.
In the wide case, there is a central as well as a secondary
caustic, both on the axis. The limits between these config-
urations are indicated as the dashed lines in e.g Fig. 2 (see
also Fig. 1 of Cassan (2008)).
The description of the lightcurve itself requires four
more geometrical parameters in addition to d and q. In the
current standard parameterisation of binary lens lightcurves,
these are the source trajectory’s angle α with the axis of
symmetry of the lens, the time of closest source-lens ap-
proach to the binary lens centre-of-mass t0, the Einstein
radius crossing time tE and the source-lens separation at
closest approach u0 (in units of θE). Finally for a uniformly
bright finite size source star, we add a further parameter,
the source size ρ∗ in units of θE. However, and as discussed
in Cassan (2008), this parameterisation is not well adapted
to conducting a full search of the parameter space, because
the value of the parameters cannot be directly related to
features present in the lightcurve, namely caustic crossings
for the type of events we are discussing in this paper. Conse-
quently, most of the probed models in a given fitting process
do not exhibit the most obvious features in the lightcurve,
leading to very inefficient modelling.
To avoid this drawback, Cassan (2008) introduced a
new parameterisation in place of α,t0,u0 and tE which is
closely related to the appearance of caustic crossing fea-
tures in the lightcurve. The caustic entry is then defined
by a date tentry when the source center crosses the caustic1
and its corresponding (two-dimensional) coordinate ζentryon
the source plane. However, since by definition this point is
located on a caustic line, Cassan (2008) introduced a (one-
dimensional) curvilinear abscissa s which locates the cross-
ing point directly on the caustic, so that ζentry ≡ ζ(sentry). A
given caustic structure is fully parameterised by 0 ≤ s ≤ 2.
The caustic entry is then characterised by a pair of param-
eters (tentry,sentry), and in the same way the caustic exit by
(texit,sexit). These four parameters (in addition to d,q and
ρ∗) which describe the caustic crossings therefore also define
an alternative parameterisation of the binary lens, far better
fitted to describing the problem at hand.
1Alternatively, any other point at a fixed position from the
source center can be defined as a reference.

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A systematic fitting scheme for caustic-crossing microlensing events3
2.2 Exploration of the parameter space
We start by exploring a wide region of the parameter space
with a (d,q) grid regularly sampled in logarithmic scale.
This choice comes from the fact that the size of the caustic
structures behave like power-laws of the lens separation and
mass ratio, and so do the corresponding lightcurve anoma-
lies. We fit for the remaining model parameters tentry, texit,
sentry, sexitand ρ∗, (d,q) being held fixed. From this, we then
build a χ2(d,q) map that we use to locate the best-fit (d,q)-
regions. As mentioned previously, there exist binary lens
configurations which involve central and secondary caustics.
In these cases (i.e. in the wide and close binary cases) and
following Cassan (2008), we study separately models where
the source crosses the central or the secondary caustic by
building two χ2(d,q) maps, corresponding to each configu-
ration.
In order to sample efficiently and extensively sentry and
sexit (which determine the source trajectory), we use a ge-
netic algorithm (e.g. Charbonneau 1995) that always keeps
the best model from one generation to the next one (elitism).
In fact, since we consider only models displaying caustics at
the right positions, there are a couple of local minima asso-
ciated with different (sentry,sexit) pairs which would usually
be missed by other minimisation methods, while a genetic
algorithm naturally solves this problem in an efficient way.
However, since such an algorithm never converges exactly
to the best model, we finally refine the model by perform-
ing a Markov-Chain Monte-Carlo (hereafter MCMC) fit: we
start several chains and use the criterion by Geweke (1992)
to assess convergence to a stationary posterior distribution
of the parameter probability densities.
From the obtained χ2maps, we then identify all the
local minima regions and use the corresponding best models
found on the (d,q) grid as starting points to refine the pa-
rameters, including (d,q) that we now allow to vary. Since
the fit is performed within a minimum χ2region, the fitting
process is very stable and fast.
3APPLICATION TO OGLE-2007-BLG-472
3.1Alert and photometric follow-up
On 19 August 2007, the OGLE Early Warning System
(Udalski 2003) flagged microlensing candidate event OGLE-
2007-BLG-472 at right ascension α2000.0 = 17:57:04.34, and
declination δ2000.0 = -28:22:02.1 or l = 1.77◦, b = −1.87◦.
The OGLE lightcurve has an instrumental baseline
magnitude I=16.00, which may differ from the calibrated
magnitude by as much as 0.5 magnitudes. Lensing by the
star in the point source-point lens (hereafter PSPL) approx-
imation accounts for a broad rise and fall in the lightcurve,
peaking around MHJD2=4334.0 with an apparent half-
width at half-peak of about 10 days (Fig. 1). Although the
observed OGLE flux rises only by 0.06 mag in the non-
anomalous part of the lightcurve, the shape of the curve
hints that blending is important for this target, with only ∼
12% of the baseline flux due to the un-magnified source.
On 19 August (MHJD=4331.5) an OGLE data point
2MHJD=HJD-2450000
Figure 1. OGLE, UTas and Danish data set for OGLE-2007-
BLG-472 data sets. Data points are plotted with 1-σ error bars.
. The x−axis is time in HJD-2450000.
TelescopeDataError bar rescaling factor
UTas 1.0m
Danish 1.54m
OGLE
34
84
857
1.79
1.55
1.21
Table 1. Datasets and error bar rescaling factors.
showed sudden brightening of the source, with subsequent
PLANET (UTas Mt. Canopus 1.0m telescope in Tasmania
and Danish 1.54m telescope at La Silla, Chile) and OGLE
data indicating what appears to be a fold caustic crossing by
the source, ending with a PLANET UTas data point on Au-
gust 21 (MHJD=4334.1). The caustic entry is observed by
a single OGLE point, while the caustic exit is well covered
by our UTas data set (Fig. 1). Treating the lightcurve as the
addition of an anomaly to a PSPL lightcurve, the underly-
ing PSPL curve then apparently reaches peak magnification
on August 22 (MHJD=4335.45). Particularly crucial in our
data set is the UTas observation taken within a few hours
of the caustic exit, which puts constraints on the position
of the caustic exit on the lightcurve, and on the size of the
source. Although some V-band observations were taken, the
V lightcurve of this event is flat and does not allow us to
place constraints on the V flux parameters.
3.2Data reduction
We reduced the PLANET data for this event using the data
reduction pipeline pysis3.0 (Albrow 2008). This pipeline uses
a kernel as a discrete pixel array, as proposed by Bramich
(2008), rather than a linear combination of basis func-
tions.This has the advantage that it removes the need for
the user to select basis functions manually, which can lead to
problems if inappropriate functions are chosen. In addition
to this, the pixel array kernel copes better with images that
are not optimally aligned. The result of using this pipeline is
a better reduction than was obtained with other methods.
We kept all points with seeing <3.5 arcseconds. Although
some dubious points remain with this simple cut, the size
of their associated error bars reflects their lack of certainty

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4N. Kains et al.
and ensures their weight in modelling procedures is appro-
priately reduced. Our final data set consists of 34 UTas data
points, 84 points from the Danish 1.54m telescope, and 857
points from OGLE (Table 1).
3.3Modelling OGLE-2007-BLG-472
After a first exploration of the parameter space, we find a
best model (close to model Cc, see below) which we use
as a basis to rescale our error bars. In fact, these can vary
rather widely from one telescope to another and are often
underestimated by photometry software. Ignoring this effect
would misrepresent the relative importance of the data sets.
From this step, we choose the rescaling factors shown in
Table 1, obtained by setting χ2/d.o.f. ≃ 1 for each data set.
We then use the rescaled data to perform a new parameter
space exploration.
We then apply the fitting scheme detailed in section 2
to our data sets. In particular, we choose a spacing between
the (d,q) grid points of 0.070 in logd and 0.275 in logq.
For the genetic algorithm fit, we use a model population
of 200 individuals evolving during 40 generations, which has
proven to be enough to safely locate the regions of minimum
χ2. Finite source effects are computed using the adaptive
contouring method of Dominik (2007).
The final χ2(d,q) maps that we obtain are plotted in
Fig. 2 for the intermediate and central caustic configura-
tions, and Fig. 3 for the intermediate and secondary caustic.
The red crosses show the underlying (d,q) grid, and the blue
shaded contours indicate values of ∆χ2= 5, 20, 50, 100, 250,
where the reference model is Cs, the global best-fitted model
(as obtained in Section 3.5).
3.4 Excluding minima
Fig. 4 shows a zoom of this region (secondary caustic and
close configuration d < 1), with an overplot of tEisocontours
(orange lines) roughly equally spaced on a logarithmic scale.
With this fitting approach, we put no initial constraints on
the Einstein time tE, though it will always remain physical
(tE > 0). Since we are not using any Bayesian prior for
this parameter, we find that very good fits to the data are
obtained with values of tE > 300 days, which correspond to
the minimum region in the left lower part of Fig. 3. Such long
Einstein times are not likely to happen commonly, and it
may happen that some of the values found for t0 correspond
to a lightcurve that reaches its peak well in the future; these
are very unlikely to be acceptable solutions. Hence, instead
of using a prior for tE in the fitting process, we adopt the
posterior distribution of Dominik (2006), from which we see
that tE > 400 days, well in the tail of the distribution, can be
used as a cut for a model to be physically plausible. Thus in
the following, we will not consider solutions with values of tE
greater than 400 days. This means that we will not include
the low-q (q ∼ 0.001) minima in the following discussion.
Although a very well-covered lightcurve generally en-
ables a good characterisation of the deviation caused by
the caustic approach or crossing, degeneracies make find-
ing a unique best-fitting model difficult. In particular,
Griest & Safizadeh (1998) and Dominik (1999) identified a
two-fold degeneracy in the projected lens components sep-
aration parameter d, under the change d ↔ 1/d, when
q ≪ 1. Moreover, Kubas et al. (2005) showed that very simi-
lar lightcurves could arise for a source crossing the secondary
caustic of a wide binary system and for the central caustic of
a close binary system. These degeneracies cause widely sep-
arated χ2minima in the parameter space, which must then
be located by exploring the parameter space thoroughly. In
addition to these degeneracies, imperfect sampling can in-
crease the number of local χ2minima; short event in partic-
ular are prone to under-sampling, leading to difficulties in
modelling. OGLE-2007-BLG-472 is no exception, as shown
in the next section.
3.5 Refining local minima
We see from Fig. 2 (intermediate and central caustic) that
there are three broad local minima in the region around the
white filled circles marked as Cc, I and Wc (“I”, “C” and
“W” for intermediate, close and wide models respectively,
and subscript “c” for central caustic). In Fig. 3 (intermediate
and secondary caustic), a best-fit region can easily be located
around the region marked Cs (subscript “s” for secondary
caustic), besides region I.
Now allowing for the parameters d and q to vary as well,
we use our MCMC algorithm to find the best solutions in
each of these local minimum regions. These are identified
with white filled circles on Fig. 2 and 3 and correspond to
the models listed in Table 2, and shown in Fig. 5, 6, 7 and
8. The best model lightcurve is dominated by strong caus-
tics, which all viable models must reproduce, with the low-
magnification base PSPL curve barely noticeable. All mod-
els have the first anomalous OGLE points on the descending
side of the caustic entry except for the worst model, model
Wc, which has this OGLE point on the ascending part of the
caustic entry. Statistically, the former case is more likely to
be observed since the ascending part of the caustic entry
happens much more rapidly than the descending side.
Our best model, Cs, has χ2= 949 for 975 data points,
with the other competitive models at ∆χ2= 13.2 (model
Cc), ∆χ2= 23.5 (model I) and ∆χ2= 39.6 (model Wc).
3.6Discussion
Fig. 4 shows that the models with a source crossing a sec-
ondary caustic have increasingly large values of tE as they
go towards lower values of the mass ratio. This is expected
since the time ∆t between tentryand texitis fixed by the data.
As the size of caustics scales with q1/2, and tE ∼ ∆t/q1/2,
the source must therefore cross the Einstein Ring over a
longer timescale in order to conserve the right timing for
sentry and sexit. In addition to this, blending decreases for
decreasing values of q , and therefore decreases with increas-
ing tE, contrary to what might be expected. Indeed, one
would expect the blending factor g = FB/FS (where FB and
FS are the blend and source flux respectively) to increase
with increasing tE in order to mask long timescales and re-
produce the observed timescale. However in this region of
parameter space, the caustics are weak, which means that
too much blending would not allow models to reproduce the
observed rise in the source magnitude at the caustic entry
and caustic exit. For a region of parameter space to contain
satisfactory models, there must therefore be a fine balance
between blending, timescale and mass ratio.

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A systematic fitting scheme for caustic-crossing microlensing events5
Figure 2. χ2(d,q) map for the intermediate and central caustic configurations. Contour lines and minima regions (in blue shades) are
plotted at ∆χ2= 5,20,50,100,250. The two dashed curves are the separation between the close, intermediate and wide regimes. The
models are labelled and marked with white filled circles.
Figure 3. Same as Fig. 2 for the the intermediate and secondary caustic configuration.
For models where the source crosses a central caus-
tic, the impact parameter u0 must decrease with decreasing
mass ratio, since the size of central caustic decreases with
decreasing mass ratio, and the range of allowed u0 decreases
if the source must cross the caustic. This means that for
smaller mass ratios, blending will have to increase in order
to mask the correspondingly higher PSPL magnification of
the source that results from the smaller impact parameter.

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6N. Kains et al.
Figure 4. Map of the value of tEin the (d, q) plane for converged models at each grid point, superimposed on the χ2map, zoomed in
on the close regime part of parameter space. Contours lines (orange) are labeled with their corresponding value of tEwhile χ2contour
lines are plotted at ∆χ2= 5, 20, 50, 100, 250 and filled with gradual shades of blue. The dashed curve is the separation between the
close and intermediate regimes. The models of Table 2 are labelled and marked with white filled circles.
3.7Physical properties of the models
3.7.1 Source characteristics
A colour-magnitude diagram of the field (Fig. 9) was pro-
duced extracting 1497 stars from I and V images at t =
4340.08 (I) and t = 4340.13 (V) taken at the Danish
1.54m telescope. The combination of the source and the
blend lies very slightly blueward of the red giant clump, at
(V − I)=2.43. All the models, however, are heavily blended
(Table 2). The actual source magnitude and blending mag-
nitude for each model can be found using the equations
Is = Ibase+ 2.5log(1 + g) and Ib= Is− 2.5log(g).
Using this equation, we find source magnitudes ranging
from 17.89 (model Cs) to 20.21 (model Cc) (see Table 2).
Our V-band data set does not allow us to determine the
source’s colour, but assuming that the source is a main se-
quence star we use the calculated I magnitude of the source
for each model to estimate a colour, using the results of
Holtzman et al. (1998). This then enables us to estimate the
source’s angular radius which we use in Section 3.7.2 to com-
pute probability densities of the lensing system’s properties.
We calibrate the baseline magnitude of our target
(source and blend combined) using the location of the red
clump as a reference. We find Ibase= 15.61 ± 0.10, which is
in agreement with the OGLE value of Ibase = 16.00 ± 0.50.
Comparing this to the location of the red clump, we can de-
rive an estimate for the reddening coefficient AI. From Hip-
parcos results, Stanek & Garnavich (1998) find an absolute
magnitude for the red clump at MI,RC = −0.23±0.03. Using
Figure 9. Colour - Magnitude diagram of the field. The target
OGLE-2007-BLG-472 is shown as a black triangle at (V −I,I) =
(2.43,15.61). The position of the deblended source for each model
is labeled and indicated by a dotted line and a coloured diamond,
with the blend for each model also plotted as a diamond in the
same colour.
a distance modulus to the galactic centre of µ = 14.41±0.09
(i.e. assuming DS = 7.6 kpc) (Eisenhauer et al. 2005),
this translates to a dereddened magnitude for this target
of Ibase = 14.18 ± 0.09. Hence using the relation AI =
Ibase− MI,RC − µ, we get a value for the I-band reddening
parameter of AI = 1.43±0.13. Alternatively, fitting 2MASS

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A systematic fitting scheme for caustic-crossing microlensing events7
Figure 5. Best-fitting binary lens model Cs with residuals and a zoom on the anomaly (left inset). Data points are plotted with 1-σ
error bars. The trajectory of the source in the lens plane with the caustics is plotted as an inset in the top right corner of the figure,
with the primary lens component located at the coordinate system’s origin.
Figure 6. Same as Fig. 5 for model I.

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8N. Kains et al.
Figure 7. Same as Fig. 5 for model Cc.
Figure 8. Same as Fig. 5 for model Wc.

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A systematic fitting scheme for caustic-crossing microlensing events9
Table 2. Best-fitting binary lens model parameters. The blending factor g(I) = FB(I)/FS(I) is given for the OGLE data (I-band).
The error bars were rescaled for each telescope by the factor given in Table 1, which lead to the rescaled χ2indicated here. Physical
parameters are also given for each model, for the case of a lens in the disk, and a lens in the bulge. These were calculated using the
procedure detailed in Sec. 3.7.2
ParameterModel Cs
Model Cc
Model IModel Wc
Units
χ2(rescaled σ)
∆χ2
χ2
UTas
χ2
Danish
χ2
OGLE
949.00
−
23.79
79.77
845.50
963.16
13.2
24.83
79.60
858.77
972.48
23.5
26.41
80.75
865.24
988.55
39.8
28.86
88.93
870.55
−
−
−
−
−
t0
tE
α
u0
4587.18 ± 0.80
213.82 ± 1.04
2.810 ± 0.006
−1.573 ± 0.013
0.34 ± 0.01
0.427 ± 0.002
0.078 ± 0.001
7.15 ± 0.013
17.89 ± 0.01
15.75 ± 0.01
1.80 ± 0.10
1.18 ± 0.24
4332.27 ± 0.29
52.00 ± 3.63
3.227 ± 0.030
0.091 ± 0.005
0.98 ± 0.19
0.673 ± 0.011
0.177 ± 0.017
68.11 ± 0.013
20.21 ± 0.01
15.63 ± 0.01
1.93 ± 0.11
0.46 ± 0.09
4332.10 ± 0.27
38.32 ± 2.60
3.305 ± 0.037
0.164 ± 0.019
1.55 ± 0.16
0.760 ± 0.015
0.236 ± 0.024
40.13 ± 0.09
19.65 ± 0.09
15.64 ± 0.09
1.91 ± 0.11
0.59 ± 0.12
4334.99 ± 0.28
53.46 ± 0.81
4.570 ± 0.018
0.277 ± 0.010
1.33 ± 0.05
2.158 ± 0.0169
0.288 ± 0.0096
56.98 ± 0.019
20.02 ± 0.01
15.63 ± 0.01
1.92 ± 0.12
0.50 ± 0.10
MHJD
days
rad
−
−
−
−
−
−
−
−
µas
ρ∗/10−3
d
q
g(I) = FB(I)/FS(I)
Is
Ib
(V − I)s
θ∗
Lens in the Disk
M1
1.50+1.85
−0.58
0.42+0.40
−0.22
0.34+0.36
−0.18
0.34+0.37
−0.18
M⊙
M2
0.12+0.14
−0.05
0.07+0.07
−0.04
0.08+0.08
−0.04
0.10+0.11
−0.05
M⊙
DL
1.00+0.95
−0.36
5.7+1.1
−1.5
6.1+1.1
−1.5
6.1+1.0
−1.5
kpc
v25+24
−9
80+15
−21
93+16
−22
67+11
−16
kms−1
Lens in the Bulge
M1
41+14
−14
1.25+1.47
−0.59
0.79+0.93
−0.35
0.79+0.94
−0.36
M⊙
M2
3.2+1.1
−1.1
0.22+0.26
−0.10
0.19+0.22
−0.08
0.23+0.27
−0.10
M⊙
DL
6.7+0.4
−0.7
7.3+0.6
−0.8
7.3+0.6
−0.8
7.3+0.6
−0.8
kpc
v167+10
−17
102+8
−12
111+10
−12
79+6
−8
kms−1
isochrones to our CMD, we obtain a value AI = 1.46 ± 0.08
and E(V − I) = 1.46 ± 0.11. We use these values of red-
dening to determine dereddened magnitudes and colours for
the source of each model. These, together with the surface
brightness relations from Kervella & Fouqu´ e (2008), allow
us to calculate the apparent angular radius of the source θ∗
for each of the models, given in Table 2.
3.7.2Lens characteristics
Although the characteristics of any microlensing event de-
pend on various properties of the lensing system, including
the mass of the lenses, the only measurable quantity that
can be directly related to physical properties of the lens is
the timescale of the event tE. While the physical properties
of the lensing system can be fully constrained when the pho-
tometry is affected by both finite source-size effects and par-
allax, when these are not measured, such as is the case with
our analysis OGLE-2007-BLG-472, we can still use Bayesian
inference to determine probability densities of physical prop-
erties of the lens, based on a chosen Galactic model. We have
chosen not to include parallax in our analysis because its ef-
fect would be very small for such a low-magnification event;
in addition to this, we are only seeking a first-order analy-
sis of binary-lens events with our current method, although
second-order effects such as parallax and lens rotation will
be taken into account in future work.
We use our fitted value of the source size parameter ρ∗
to place constraints on the mass of the lens, which can be
expressed as a function of fractional distance x = DL/DS
and the source size ρ∗ as (e.g. Dominik 1998)
M(x)
M⊙
=
c2
4GM⊙
DSθ2
ρ2
∗
∗
x
1 − x,
(2)

Page 10

10N. Kains et al.
Figure 10. Probability densities for the mass of the primary lens star and the fractional distance DL/DS, for a lens in the disk (left
side) and a lens in the bulge (right side). The values quoted in Tables 2 & 2 are the median value and the limits of the 68.3% confidence
interval. On each plot, the probability densities are plotted for model Cs (red), model Cc (green), model I (dark blue), and model Wc
(light blue).
where M is the mass of the lens, θ∗ is the angular radius
of the source, the value of which is given in Table 2, and
other quantities are defined as before. The mass-distance
curve showing constraints from this equation is plotted on
Fig. 11.
However, since we cannot measure parallax for this
event, we use a probabilistic approach following that of
Dominik (2006) to derive probability densities for physical
properties of lens components. The Galactic model used here
is a piecewise mass spectrum (e.g. Chabrier 2003), two dou-
ble exponentials for the disk mass density and a barred bulge
tilted at an angle of 20◦with the direction to the Galactic
centre (Dwek et al. 1995), and the distribution of effective
transverse velocities used in Dominik (2006).
Using these galactic models, we infer properties for the
lensing system, separating the cases where the lens is in
the Galactic disk and in the Galactic bulge. For a lens in
the disk, we find a primary mass 1.50+1.85
ondary mass of 0.12+0.14
with a lens velocity of 25+24
we find a primary mass 41+14
−0.58M⊙ and a sec-
−0.05M⊙, at a distance of 1.00+0.95
−9kms−1. For a lens in the bulge,
−14M⊙ and a secondary mass of
−0.36kpc
3.2+1.1
ity 167+10
the lowest-χ2model (model Cs). The values of these phys-
ical parameters for the other models are given in Table 2.
Probabilty densities of these properties for all models are
plotted on Fig. 10.
−1.1M⊙, at a distance of 6.7+0.4
−17kms−1. These are the physical lens properties for
−0.7kpc with a lens veloc-
3.7.3Discussion
For our lowest-χ2model, the parameters we find imply very
unusual properties of the lensing system. As discussed in
Sec. 3.4, the fact that we find these types of models is a con-
sequence of the fitting approach we are taking. Traditional
fitting methods would struggle to find these minima, since
most of them require providing a starting point in parameter
space. This is an issue when solely using an MCMC algo-
rithm to fit microlensing events: although an MCMC run
may be able to make its way through parameter space to
find minima reasonably far away from its starting point, it
is highly unlikely that a chain will be able to reach a mini-
mum that has parameters different from the starting point

Page 11

A systematic fitting scheme for caustic-crossing microlensing events 11
Figure 11. Mass-distance diagram showing the constraint on the
lens mass from the source size, given by Eq. (2), for each model.
The curves are labeled with the name of the model to which they
correspond.
by more than one order of magnitude. As we see from Fig. 4,
there exist minima in many parts of parameter space, with
values of tEthat are different by almost two orders of magni-
tude. These parameters are non-intuitive, since they cannot
be guessed only by looking at the lightcurve. As a result, it
is improbable that this kind of parameters will be used as
starting points for ”classic” fitting algorithms.
We solve this problem for the static binary-lens case
by resorting to the method described in Sec. 2.2. Using this
approach, we manage to systematically locate minima in
parameter space. However, one then has to be careful with
interpreting the significance of the obtained model param-
eters. The shape of probability densities shown in Fig. 10
for model Cs indicates that our value of tE push the lens
mass towards the end of the adopted mass spectrum in the
Galactic model we have adopted. This results in the abrupt
transients seen on Fig. 10. Similarly, the mass-distance curve
for model Cs on Fig. 11 shows that the mass of the lens in-
creases very rapidly for lenses above ∼ 1 kpc. These unusual
curves are caused by a value of tE ∼ 200 days. Models with
tE ∼ 3000 days (corresponding to the low-q minimum visible
on Fig. 3 & 4) are obviously not acceptable, but how can we
formally reject them? Finding these models from minima in
the χ2surface shows the limits of using χ2as a strong crite-
rion for favouring models. A solution to this would be to use
prior distributions on as many of the parameters as we can.
During the MCMC part of our fitting process, this would
mean that we obtain posterior distributions that are differ-
ent from the ones obtained without using prior distributions
on the parameters, or, equivalently, assuming uniform pri-
ors for all parameters. Such priors can be obtained in var-
ious ways, such as looking at the distribution of timescales
for past microlensing events or calculating these distribu-
tions from Galactic models (e.g. Dominik 2006), or by using
luminosity functions of the Galactic bulge to find a prior
for the blending factor g (e.g. Holtzman et al. 1998). Such
work requires careful consideration of which priors are most
appropriate to use, and is beyond the scope of this paper.
Using these priors in combination with our method to find
minima will lead to more robust determination of minima
by taking into account our knowledge of physical parameter
distributions.
4SUMMARY AND PROSPECTS
Our analysis of OGLE-2007-BLG-472 is a good illustration
of the importance and power of using parameters that are re-
lated to actual observed features. Indeed, despite incomplete
coverage of the caustic entry and high blending, a few cru-
cial data points and an appropriate choice of non-standard
parameters enable us to find several good binary-lens model
fits to our data for this event by exploring the parameter
space systematically. Some of the good fits that we identify
have unphysical parameters, and we must then reject them.
However using this parameterisation allows us to be certain
that the parameter space has been thoroughly explored. We
find four models with different parameters: two close binary
models, one intermediate configuration, and a wide binary
model. The lowest-χ2model corresponds to a G dwarf star
being lensed by a binary system with component masses
1.50+1.85
secondary), which are compatible with our blending values.
However it is obvious from physical parameter distributions
that using χ2as a sole criterion for determining the best
model is insufficient, because it does not take into account
our knowledge of the distributions of physical parameters.
Since the approach presented in this paper can form
the basis for a systematic, wide ranging exploration of the
parameter space to localise all possible models for a given
data set, it is particularly relevant to current efforts to au-
tomatise real-time fitting of binary-lens events. This could
prove useful to provide faster feedback on the events be-
ing observed and prioritise observing schedules, especially
on robotic telescopes. Expanding robotic telescope networks
controlled by automated intelligent algorithms are expected
to play an increasingly important role in microlensing sur-
veys in the coming years (e.g. Tsapras et al. 2008). Fitting
methods such as the one described in this paper are essen-
tial for making sure any anomalies are interpreted correctly,
and that minima are located in as large a part of parameter
space as possible.
−0.58M⊙ (for the primary) and 0.12+0.14
−0.05M⊙ (for the
ACKNOWLEDGEMENTS
NK acknowledges STFC studentship PA/S/S/2006/04497
and an STFC travel grant covering his observing run at La
Silla. We thank David Warren for financial support for the
Mt Canopus Observatory. NK thanks Pascal Fouqu´ e for or-
ganising a workshop in Toulouse in November 2007, and
Joachim Wambsganss and Arnaud Cassan for their invita-
tion to visit the Astronomisches Rechen-Institut in Heidel-
berg in April 2008. We would like to thank the anonymous
referee for his helpful comments on the manuscript. We also
thank the University of Tasmania for access to their TPAC
supercomputer on which part of the calculations were car-
ried out. PF expresses his gratitude to ESO for a two months
invitation at Santiago headquarters, Chile in October and
November 2008. The OGLE project is partially supported
by the Polish MNiSW grant N20303032/4275.

Page 12

12N. Kains et al.
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