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arXiv:0907.3471v3 [astro-ph.EP] 22 Jul 2009
Astronomy & Astrophysics manuscript no. OB050x
July 22, 2009
c ? ESO 2009
Mass measurement of a single unseen star
and planetary detection efficiency for
OGLE 2007-BLG-050
V. Batista1,51, Subo Dong2,52, A. Gould2,52, J.P. Beaulieu1,51, A. Cassan3,51, G.W. Christie25,52, C. Han30,52, A.
Udalski24,53
and
W. Allen28, D.L. DePoy2, A. Gal-Yam29, B.S. Gaudi2, B. Johnson32, S. Kaspi43, C.U. Lee35, D. Maoz43,
J. McCormick33, I. McGreer31, B. Monard28, T. Natusch50, E. Ofek34, B.-G. Park35, R.W. Pogge2, D. Polishook43,
A. Shporer43
(The µFUN Collaboration),
M.D. Albrow5, D. P. Bennett4,54, S. Brillant6, M. Bode7, D.M. Bramich8, M. Burgdorf49,50, J.A.R. Caldwell9, H.
Calitz10, A. Cole13, K. H. Cook11, Ch. Coutures12, S. Dieters1,13, M. Dominik14⋆, D. Dominis Prester15,
J. Donatowicz16, P. Fouqu´ e17, J. Greenhill13, M. Hoffman10, K. Horne14, U.G. Jørgensen18, N. Kains14,
S. Kane19,D. Kubas1,6, J.B. Marquette1, R. Martin20, P. Meintjes10, J. Menzies21, K.R. Pollard5, K.C. Sahu22,
C. Snodgrass6, I. Steele7, Y. Tsapras23, J. Wambsganss3, A. Williams20, M. Zub3
(The PLANET/RoboNet Collaboration),
Ł. Wyrzykowski24,27, M. Kubiak24, M.K. Szyma´ nski24, G. Pietrzy´ nski24,26, I. Soszy´ nski24, O. Szewczyk26,24, K.
Ulaczyk24
(The OGLE Collaboration),
F. Abe36, I.A. Bond37, A. Fukui36, K. Furusawa36, J.B. Hearnshaw5, S. Holderness38, Y. Itow36, K. Kamiya36, P.M.
Kilmartin40, A. Korpela40, W. Lin37,C.H. Ling37, K. Masuda36, Y. Matsubara36, N. Miyake36, Y. Muraki41, M.
Nagaya36, K. Ohnishi42, T. Okumura36, Y.C. Perrott46, N. Rattenbury46, To. Saito44, T. Sako36, L. Skuljan37, D.
Sullivan39, T. Sumi36, W.L. Sweatman37, P.J. Tristram40, P.C.M. Yock46
(The MOA Collaboration)
(Affiliations can be found after the references)
Submitted
ABSTRACT
Aims. We analyze OGLE-2007-BLG-050, a high magnification microlensing event (A ∼ 432) whose peak occurred on 2 May, 2007, with
pronounced finite-source and parallax effects. We compute planet detection efficiencies for this event in order to determine its sensitivity to the
presence of planets around the lens star.
Methods. Both finite-source and parallax effects permit a measurement of the angular Einstein radius θE= 0.48 ± 0.01 mas and the parallax
πE= 0.12 ± 0.03, leading to an estimate of the lens mass M = 0.50 ± 0.14 M⊙and its distance to the observer DL= 5.5 ± 0.4 kpc. This is only
the second determination of a reasonably precise (< 30%) mass estimate for an isolated unseen object, using any method. This allows us to
calculate the planetary detection efficiency in physical units (r⊥,mp), where r⊥is the projected planet-star separation and mpis the planet mass.
Results. When computing planet detection efficiency, we did not find any planetary signature and our detection efficiency results reveal signif-
icant sensitivity to Neptune-mass planets, and to a lesser extent Earth-mass planets in some configurations. Indeed, Jupiter and Neptune-mass
planets are excluded with a high confidence for a large projected separation range between the planet and the lens star, respectively [0.6 - 10]
and [1.4 - 4] AU, and Earth-mass planets are excluded with a 10% confidence in the lensing zone, i.e. [1.8 - 3.1] AU.
Key words. extrasolar planets - gravitational microlensing
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2 V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency
1. Introduction
Over the last decade, microlensing events have been intensively followed in order to detect extrasolar planets around lens stars
and to measure their abundance in our Galaxy. This is one of the few planet-detection techniques that is sensitive to very low
mass planets, and microlensing discoveries have twice the record for the lowest mass planet to orbit a star other than a stellar
remnant (Beaulieu et al. 2006; Bennett et al. 2008). During a microlensing event, i.e. when a background source passes close to
the line of sight to a foregroundlens star, the observedsource flux is magnified by the gravitational field of the lens. The presence
of a companion around the lens star introduces two kinds of caustics into the magnification pattern : one or two “planetary
caustics” associated with the planet and a “central caustic” close to the primary lens projected on the source plane. When the
source crosses or approaches one of these features, deviations appear from a single point-lens light curve (Mao & Paczy´ nski
1991; Gould & Loeb 1992).
1.1. Central caustic and detection efficiency
Significant effort has been expendedon the observationand modelingof highmagnificationeventsbecause they probethe central
caustic (Griest & Safizadeh 1998; Rhie et al. 2000; Rattenbury et al. 2002). Any planets in the system are highly likely to affect
the central caustic, resulting in potentially high sensitivity to the presence of even low-mass planets.
Indeed, a major advantage of the central caustic is that it is possible to predict in advance when the source passes close to
the line of sight of the lens and so when there is the greatest chance of detecting planets. Thus observations can be intensified,
further improving the sensitivity to planetary-inducedanomalies in the lightcurve.
In these specific cases, for which the impact parameter can be very small, finite-source effects might strongly affect and
diminish a possible planetary signal (e.g., Dong et al. 2009b, Bennett et Rhie 1996). In the absence of any deviation from a
finite-source single point-lens model, one can still compute the planet detection efficiency in order to derive upper limits on the
probability that the lens harbors a planet (Gaudi & Sackett 2000). It also allows to combine statistically the detection efficiencies
computed from observed events to estimate the frequency of planetary companions to the lens (Gaudi et al. 2002).
The extremely high magnification microlensing event OGLE-2007-BLG-050 was well followed and is a good candidate for
analyzing the sensitivity of such an event with pronounced finite-source effects to the presence of a planetary companion. In this
study, we compute the planetary detection efficiency for this event, following the Gaudi & Sackett (2000) method. To perform
the calculations of binary light curves, we use the binary-lens finite-source algorithm developed by Dong et al. (2006) and the
formalism of Yoo et al. (2004a) for the single-lens finite-source effects.
1.2. Mass and distance estimates of the lens star
OGLE-2007-BLE-050 is also one of the rare events that can potentially be completely solved by measuring both the microlens
Einstein angular radius θEand the microlens parallax πE. Indeed, after the first microlenses were detected (Alcock et al. 1993;
Udalski et al. 1993), several authors showed that the microlens Einstein angular radius θE,
θE=θ∗
ρ∗
could be measured from deviations relative to the standard point-lens (Paczy´ nski 1986) lightcurve, due to finite-source effects
(Gould et al. 1994; Nemiroff & Wickramasinghe1994; Witt & Mao 1994). The measuredparameterassociated with these effects
is ρ∗, corresponding to the angular size of the source θ∗in units of θE. The measurement of θEconstrains the physical properties
of the lens and so leads to the first part of a full solution for an event (Gould 2000),
θE=
?
κMπrel,κ ≡
4G
c2AU≈ 8mas M−1
⊙,
where M is the lens mass and πrelis the lens-source relative parallax. For most events, the only measured parameter that depends
on the mass M is the Einstein timescale, tE, which is a degenerate combination of the lens mass M, the lens-source relative
parallax πreland the proper motion µrel. It can be expressed as :
tE=
θE
µrel
Gould (1992) showed that if one measures both θEand the microlens parallax, πE, which is derived from the distortion of the
microlens light curve induced by the accelerated motion of the Earth, one can determine
πE=
?πrel
κM,
⋆Royal Society University Research Fellow
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V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency3
and so determine the lens mass and lens-source relative parallax as well,
M =
θE
κπE;
πrel= πEθE.
After thousands of single-lens microlensing events discovered to date, measurements of both θEand πEstill remain a chal-
lenge. The microlens parallax πEhas been measured for more than twenty single lenses (Alcock et al. 1995 [the first parallax
measurement], Poindexter et al. 2005 and references therein), while the angular Einstein radius θEhas been measured for only
few cases of single lenses (Alcock et al. 1997, 2001 ; Smith et al. 2003b ; Yoo et al. 2004a ; Jiang et al. 2004 ; Cassan et al. 2006
; Gould et al. 2009).
However, reliable mass estimates for isolated stars have been determined with microlensing only twice. Alcock et al. (2001)
andGould et al. (2009)eachmeasuredbothθEandπErespectivelyforMACHO LMC-5andOGLE2007-BLG-224.ForMACHO
LMC-5, good measurements of πreland µrelwere obtained with the original photometric data and additional high resolution
photometryof the lens (HST observations).Only for OGLE 2007-BLG-224has there been a reliable mass estimate derivedusing
only ground-based photometric data.
All other good microlens stellar mass measurements to date have been obtained for binary (or planetary) lens events : EROS
BLG-2000-5 (An et al. 2002), OGLE 2006-BLG-109 (Gaudi et al. 2008), OGLE 2007-BLG-071 (Dong et al. 2009a), OGLE
2003-BLG-267 (Jaroszynski et al. 2005), OGLE 2002-BLG-069 (Kubas et al. 2005) and OGLE 2003-BLG-235 (Bond et al.
2004).
1.3. Detection efficiency in physical units
Here, we present groundbased photometric data of the event OGLE 2007-BLG-050which we use, for the first time, to constrain
both the presence of planets and the mass of the lens.
This is also the first event for which parallax and xallarap (source orbital motion) are analyzed simultaneously. However, we
find that the apparent xallarap signal is probably due to minor remaining systematic effects in the photometry.
Access to the physical propertiesof the lens allows us to compute the planetarydetection efficiencyin physical units (r⊥,mp),
where r⊥is the projected separation in AU between the planet and the lens and mpis the planet mass in Earth mass units.
OGLE-2007-BLG-050 had a high sensitivity to planetary companions of the lens, with a substantial efficiency to Neptune-
mass planets and even Earth-mass planets.
2. Observational data
The microlensing event OGLE-2007-BLG-050 was identified by the OGLE III Early Warning System (EWS ; Udalski 2003)
(α = 17h58m19.39s, δ = −28o38′59′′(J2000.0) and l = +1.67o, b = −2.25o) on 2 Mar 2007, from observations carried
out with the 1.3 m Warsaw Telescope at the Las Campanas Observatory (Chile). The peak of the event occured on HJD′≡
HJD − 2,450,000 = 4221.904 (2007 May 2 at 9:36 UT).
The event was monitored over the peak by the Microlensing FollowUp Network (µFUN ; Yoo et al. 2004a) from Chile (1.3m
SMARTS telescope at the Cerro Tololo InterAmerican Observatory), South Africa (0.35 m telescope at Bronberg observatory),
Arizona (2.4 m telescope at MDM observatory, 1.0 m telescope at the Mt Lemmon Observatory), New Zealand (0.40 m and
0.35 m telescopes at Auckland Observatory and Farm Cove observatory respectively) and on the wings from the Vintage Lane
(Marlborough, New Zealand), Wise (Mitzpe Ramon, Israel) and Palomar 60-in (Mt Palomar California, USA) observatories.
However, the last three were not included in the final analysis because they do not significantly improve the constraints on
planetary companions. Data from all the three sites are consistent with single-lens model.
It was also monitored by Microlensing Observations in Astrophysics (MOA) with the 1.8 MOA-II telescope at Mt John
University Observatory (New Zealand), and Probing Lensing Anomalies Network (PLANET ; Albrow et al. 1998) from 5 differ-
ent telescopes : the Danish 1.54 m at ESO La Silla (Chile), the Canopus 1 m at Hobart (Tasmania), the Elizabeth 1 m at the South
African Astronomical Observatory (SAAO) at Sutherland, the Rockefeller 1.5 m of the Boyden Observatory at Bloemfontein
(South Africa) and the 60 cm of Perth Observatory (Australia). The RoboNet collaboration also followed the event with their
three 2m robotic telescopes : the Faulkes Telescopes North (FTN) and South (FTS) in Hawaii and Australia (Siding Springs
Observatory) respectively, and the Liverpool Telescope (LT) on La Palma (Canary Islands).
In this analysis, we use 601 OGLE data points in I band, 104 µFUN data points in I band, 77 µFUN data points close to R
band, 121 PLANET data points in I band, 55 RoboNet data points in R band and 239 MOA-Red data points (wide band covering
R and I bands).
3. Event modelling
OGLE-2007-BLG-050is a very high magnification event (A ≃ 432) due to its small impact parameter u0. Because they are quite
obvious on the observed light curve, finite-source effects must be incorporated in the modeling. Moreover, the long timescale of
the event implies that parallax effects are likely to be detectable.
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4 V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency
3.1. Finite-source effects
When observing a microlensing event, the resulting flux for each observatory-filter i can be expressed as,
Fi(t) = Fs,iA[u(t)] + Fb,i,
where Fs,iis the flux of the unmagnified source, Fb,iis the backgroundflux and u(t) is the source-lens projected separation in the
lens plane.
When the source can be approximated as a point, the magnification of a single-lens event is given by (Einstein 1936;
Paczy´ nski 1986)
A(u) =
u2+ 2
√u2+ 4
u
However, in our case the source cannot be considered as a point (u ? ρ∗) and the variation in brightness of the source star across
its disk must be considered using the formalism of Yoo et al. (2004a). When limb-darkening of the source profile are neglected
(uniform source), the magnification can be expressed as (Gould 1994a; Witt & Mao 1994; Yoo et al. 2004a),
Auni(u/ρ∗) ≃ A(u)B0(u/ρ∗),
B0(z) ≡4
πzE(k,z)
where E is the elliptic integral of the second kind and k = min(z−1,1). Separating the u and z = u/ρ∗parameters allows fast
computation of extended-sourceeffects.
To include the limb-darkening, we parameterize the source brightness S by,
S(θ)
S0
= 1 − Γ
?
1 −3
2(1 − cosθ)
?
,
where θ is the angle between the normal to the stellar surface and the line of sight. The new magnification is then expressed by
adding the B1(z) function of Yoo et al. (2004a) related to the linear limb-darkening law,
Ald(u/ρ∗) = A(u)[B0(z) − ΓB1(z)].
The limb-darkening coefficients Γ have been taken equal to 0.49 for the I filter and 0.60 for the R filter, which are results from a
single-lens fit. From Claret (2000) and Afonso et al. (2000) models, considering a subgiant similar to our source (log g = 4, T =
5250K),we find 0.44 and 0.53,respectivelyfor I and R filters. These values are close to those of our model andlead to essentially
the same parameter values as shown in TABLE 1.
In Fig.1, we present the OGLE-2007-BLG-050 light curve modeled with extended-source effects (black curve) and without
these effects (red curve). Finite-source effects are clearly noticeable by a characteristic flattening and broadening of the light
curve at the peak.
For each data set, the errors were rescaled to make χ2per degree of freedom for the best-fit extended-source point-lens
(ESPL) model close to unity. We then eliminated the largest outlier and repeated the process until there were no 3 σ outliers.
None of the outliers constitute systematic deviations that could be potentially due to planets.
3.2. Source properties from color-magnitude diagram and measurement of θE
To determine the dereddened color and magnitude of the microlensed source, we put the best fit color and magnitude of the
source on an (I,V − I). calibrated color magnitude diagram (CMD) (cf. Fig.2). We use calibrated OGLE-III data. The magnitude
and color of the target are I = 18.21 ± 0.03 and (V − I) = 2.32 ± 0.01. The mean position of the red clump is represented by an
open circle at (I,V − I)RC= (15.95,2.37),with an error of 0.05 for both quantities. The shift in position of our target relative to
the red clump is then ∆I = 2.26 ± 0.05 and ∆(V − I) = −0.05 ± 0.05.
For the absolute clump magnitude, we adopt the Hipparcos clump magnitude MI,RC = −0.23 ± 0.03 (Stanek & Garvanich
(1998)).ThemeanHipparcosclumpcolorof(V−I)0,RC= 1.05±0.05isadopted(JenniferJohnson,2008,privatecommunication).
Assuming that the source is situated in the bulge and a Galactic center distance of 8kpc, µGC= 14.52 ± 0.10 (Einsenhauer et al.
2005).
The magnitude of the clump is given by I0,RC = MI,RC+ µGC = 14.29 ± 0.10. We derive (I,V − I)0,RC = (14.29,1.05) ±
(0.10,0.05). Hence, the dereddened source color and magnitude are given by : (I,V − I)0 = ∆(I,V − I) + (I,V − I)0,RC =
(16.55,1.00)± (0.12,0.08).
From (V − I)0, we derive (V − K)0using the Bessel & Brett (1988) diagram for giants, supergiants and dwarfs : (V − K)0=
2.31 ± 0.13. The measured values of I0and (V − I)0then lead to K0= 15.24 ± 0.09.
For completeness, we also derive an extinction estimate [AI,E(V − I)] = (1.66,1.32), which leads to an estimate RVI =
AV/E(V − I) = 2.02.
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V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency5
OGLE-2007-BLG-050
OGLE
Auckland
Farm Cove
MDM
Bronberg
CTIO
Mt Lemmon
MOA
SAAO
UTas
Danish
Perth
Boyden
FTN
FTS
Liverpool
Palomar
Wise
VLO
42154220 42254230
16
15
14
13
12
11
4221.842224222.2
12.4
12.2
12
11.8
11.6
11.4
4221.84222 4222.2
0.02
0
-0.02
Fig.1. Top: Light curve of OGLE-2007-BLG-050near its peak on 2007 May 1. Middle: Zoom onto the peak showing the finite-
source effects. Bottom : Magnitude residuals. They correspond to the real residuals and are not exactly equal to the difference
between data and model of the light curve shown above, because the model is given in I band and the R band data points have
been linearly converted into the I OGLE system. We show the model with finite source and parallax effects. As a comparison, a
model without finite source effects is shown in red.
The colordeterminesthe relationbetween dereddenedsourceflux andangularsourceradius.We use the followingexpression
given by Kervella et al. (2004) for giants between A0 and K0 :
log2θ∗= 0.5170− 0.2K0+ 0.0755(V − K)0,
giving θ∗= 2.20± 0.06µas.
With the angular size of the source given by the extendedsource point lens (ESPL) fit, ρ∗= 0.00458±0.00003,we derive the
angular Einstein radius θE: θE= θ∗/ρ∗= 0.48 ± 0.01mas, where the error is determined by : (σθE/θE)2= (σθ∗/θ∗)2+ (σρ∗/ρ∗)2.
This first fit takes into account finite source effects only. The values of ρ∗and θEwill not change significantly when adding new
effects (see parallax effects later) but the induced modifications will be included in the final results.
Then, combined with the fitted timescale of the event tE= 66.9 ± 0.6days, gives the geocentric relative lens-source proper
motion : µ = θE/tE= 2.63± 0.08mas/yr,with the same method for calculating the error.
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6V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency
(V−I)
I
11.522.53
20
19
18
17
16
15
Fig.2. Calibrated color-magnitudediagramof the field around OGLE-2007-BLG-050.The clump centroidis shown by an empty
open circle, while the OGLE-III I and V − I measurements of the source are shown by an open circle surrounding1σ error bars.
3.3. Parallax effects
3.3.1. Orbital parallax effects
The source-lens projected separation in the lens plane, u(t) of Eq.3.1, can be expressed as a combination of two components, τ(t)
and β(t), its projections along the direction of lens-source motion and perpendicular to it, respectively :
u(t) =
?
τ2(t) + β2(t).
If the motion of the source, lens and observer can all considered rectilinear, the two components of u(t) are given by,
τ(t) =t − t0
tE
In the case of a simple point-source point-lens model, only five parameters are fitted : the source flux Fs, the blending flux Fb
(both duplicated if more than one observatory), the time of the closest approach t0, the impact parameter u0and the timescale of
the event tE.
However, for long events, like OGLE-2007-BLG-050 (where tE≥ yr/2π), the motion of the Earth cannot be approximated
as rectilinear and generates asymmetries in the light curve. Parallax effects then have to be taken into account. To introduce these
effects, we use the geocentric formalism (An et al. 2002 and Gould 2004) which ensures that the three standard microlensing
parameters (t0, tE, u0) are nearly the same as for the no-parallax fit. Now two more parameters are fitted. These are the two
components of the parallax vector, πE, whose magnitude gives the projected Einstein radius, ˜ rE= AU/πEand whose direction is
that of lens-source relative motion.
The parallax effects imply additional terms in the Eq.3.3.1
τ(t) =t − t0
tE
where
(δτ(t),δβ(t)) = πE∆s = (πE.∆s,πE× ∆s)
;
β(t) = u0.
+ δτ(t);
β(t) = u0+ δβ(t)
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V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency7
0.40.20 -0.2-0.4
-0.4
-0.2
0
0.2
0.4
Fig.3. Likelihood contours as a function of the parallax vector πE(1, 2, 3, 4σ). The best fit is πE= (0.099,−0.072).There is a
hard 3σ lower limit πE> 0.086 which implies M < 0.67M⊙and a 3σ upper limit πE< 0.23 which implies M > 0.25M⊙.
and ∆s is the apparent position of the Sun relative to what it would have been assuming a rectilinear motion of the Earth.
The Extended-Source Point-Lens (ESPL) fit yields a determination of the components (πE,N,πE,E) of the parallax vector πE
projected on the sky in North and East celestial coordinates. This is done by mapping a grid over the πEplane and searching for
the minimum of χ2(cf. Fig.3). In addition to the best ESPL fit presented in Section 3.4, this grid search was done to probe the
likelihood contours as a function of πE, holding each trial parameter pair πE= (πE,N,πE,E) fixed while allowing all remaining
parameters to vary. The best fit is πE = (0.099,−0.072). There is a hard 3σ lower limit πE > 0.086 and a 3σ upper limit
πE < 0.23. The error of πEis calculated from the 1σ contour : πE = 0.12 ± 0.03. The likelihood contours in the πEplane
are slightly elongated along the North-South axis. This tendency, which is weak here due to the long timescale, is explained in
Gould et al. (1994) by the fact that for short events the Earth’s acceleration vector is nearly constant during the event.
The Fig.4 shows the modeling improvement when we include the orbital parallax effects in the fit. These plots only show the
OGLE and MOA residuals because these data mostly constrain the parallax since they cover a long time range.
As discussed by Smith et al. (2003a), there is a u0↔ −u0degeneracy.For a low magnification event with |u0| ∼ 1, the u0> 0
and u0< 0 solutions will behave differently, but for a high magnification event with |u0| ≪ 1 like OGLE-2007-BLG-050, the
u0↔ −u0transformation can be considered as a symmetry and there is no possibility to distinguish one solution from orbital
motionalone.Inprinciple,these can bedistinguishedfromso-called“terrestrialparallax”effects caused bythe differentpositions
of the telescopes on the surface of the Earth.
3.3.2. Terrestrial parallax effects
We investigate terrestrial parallax in order to check if it is consistent with the vector parallax determined from orbital parallax
effects and to distinguish the u0 > 0 and u0 < 0 solutions. The resulting χ2of the orbital+terrestrial parallax model does not
show any improvement and is actually worse than orbital parallax alone (∆χ2= 4, χ2
explanationfor this discrepancyis that the much stronger (∆χ2= 235, χ2
the true parallax and the small terrestrial parallax “signal” is actually just due to low-level systematic errors.
orbital parallax= 1760.5). The most likely
without parallax= 1995.4)signal from orbital effects reflects
3.3.3. Xallarap effects
We also consider the possibility that the orbital parallax signal is actually due to xallarap (orbital motion of the source) rather
than to real parallax. Of course an orbital motion of the source, in case of a binary orbit that fortuitouslymimics that of the Earth,
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8V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency
Fig.4. OGLE (stars) and MOA (hexagons) residuals (magnitude) for models with (upper panel) and without (lower panel)
parallax effects. The residuals have been binned for clarity.
can reproduce the same light curve as the orbital parallax effects but here we are looking for orbital motion that is inconsistent
with the Earth-motion explanation.
We thereforesearch forxallarap solutions (orbitalmotionof the source)by introducing5 new parameters in the model related
to the orbital motion of the source : P the period of the source’s orbit, ξE,Nand ξE,Ethe xallarap vector which is analogous to the
πEvector, and α2and δ2, the phase and inclination of the binary orbit which function as analogs of the celestial coordinates of
the source in case of parallax. The rather long timescale does not justify removingparallax effects to search for xallarap only and
moreover, searching for a model including only xallarap effects does not provide significant improvements. For these reasons,
we search for a solution that takes into account both orbital + terrestrial parallax and xallarap effects with a Markov Chain Monte
Carlo algorithm (MCMC). We explore a large range of periods, from 0 to 700 days, and find a χ2improvement (χ2= 1717.7,
∆χ2= −43)for periods above250 days in comparisonwith the orbital parallaxeffects only.The χ2is essentially flat in the period
range [250 - 500] days with a very shallow minimum around P = 290 days.
The P = 290 days solution gives : ξE= (0.958,−0.273),and thus a source orbital radius : as= DSθEξE= 3.74AU.
Kepler’s third law (expressed in solar-system units),
a3= P2M ; M ≡ Ms+ Mc
and Newton’s third law,
Msas= Mcac ⇒ a ≡ ac+ as=
?
1 +Ms
Mc
?
as
imply
M3
M3
c
a3
s= P2M ⇒a3
s
P2=
Mc
[1 + (Ms/Mc)]2.
From the position of the source relative to the red clump on the CMD diagram (Fig.2), we conclude that the source is a sub-giant
situated in the bulge and, because the bulge is an old population, we infer that the source mass Msis close to a solar mass with
an upper limit of 1.2M⊙. This mass limit and the long orbital period require a companion with Mc> 70M⊙, thus a black hole,
which has an extremely low a priori probability. And if the companion is neither a black hole nor a neutron star, its mass has to
be less or equal than the source mass since the source is an evolved star and a slightly heavier companion would therefore be
much brighter. To explore these other possible star companions, we add a new constraint on the magnitude of the xallarap vector
Page 9
V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency9
100150 200 250 300
1720
1730
1740
1750
1760
Fig.5. χ2as a function of the period of the source’orbit from a MCMC algorithm with parallax and xallarap effects. The dashed
line is the case without any constraint on the companion mass and leads to a black hole solution. The solid line is for a constraint
(ξE<0.31/3
3.7P2/3= 0.18(P/yr)2/3) and leads to a solar mass companion with a minimum of χ2at P = 170 days.
in the MCMC program, assuming that Ms< 1.2M⊙and Mc/Ms≤ 1, which can be expressed as :
ξE<0.31/3
3.7
P2/3= 0.18(P/yr)2/3.
The minimum of χ2(χ2= 1730) is obtained for a source orbital period equal to 170 days as shown in the Fig.5. When we put
the correspondingparameters(P, α2, δ2) in a differential-methodprogramto reacha moreaccuratesolution,we find χ2= 1728.1.
The xallarap vector of this solution (ξE,N, ξE,E)=(−0.0142, 0.0940) implies a source orbital radius as= DSθEξE= 0.40AU
and a companion mass close to 1M⊙. The MCMC algorithm permits us to explore an 11-dimensional space (t0, tE, u0, ρ∗, πE,N,
πE,E, ξE,N, ξE,E, P, α2, δ2). We plot the 1σ and 3σ limits of the |πE| = πEas given in the Fig.6. The resulting parallax is then
πE= 0.94 ± 0.10.
3.4. Characteristics of the extended-source models with parallax and xallarap effects
Consideringthe finite-sourceeffects andparallax+ xallarapeffects, andthe 16observatoriesinvolvedinthe eventmonitoring,we
haveto fit 43 parameters(the3 standardparameters,1 forthe angularsize ofthe source,2 for parallax,5 forxallarapand2x16for
the fluxes Fsand Fbof the different telescopes). The best ESPL fit model including parallax and xallarap effects (χ2= 1717.7)
corresponds to a binary system in which the source companion is a black hole (see §.3.3.3). One more reasonable solution could
be the solar mass companion obtained when using a constraint on the xallarap (see §.3.3.3). This solution has χ2= 1728.1 for
1745data points and43 fit parameters,to giveχ2/d.o.f = 1.01,while the best ESPL fit with parallaxeffects onlyhas χ2= 1760.5
and the one without any parallax nor xallarap effects gives χ2= 1995.4, a difference of ∆χ2= 267.3. The corresponding best-fit
parameters and their errors as determined from the light curve of three different models are shown in Table 1 and Table 2 (see
also Fig.1).
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10V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency
0.40.6 0.81 1.21.4
0
2
4
6
8
10
Fig.6.
(0.31/3/3.7)P2/3= 0.18(P/yr)2/3on the companion mass. πE= 0.94± 0.10.
χ2as a function of the magnitude of the parallax vector πE from MCMC runs including the constraint ξE <
TABLE 1
OGLE-2007-BLG-050 Fit Parameters
Extended-Source Point-Lens with Parallax
ParametersWithout
xallarap
Xallarap
Black hole
P = 290 days
1717.7
Xallarap
Solar mass
P = 170 days
1728.1
χ2
1760.5
t0(days)
σt0(days)
u0
σu0
tE(days)
σtE(days)
ρ∗
σρ∗
πE
σπE
ξE
σξE
4221.9726
0.0001
0.00204
0.00002
68.09
4221.9725
0.0001
0.00215
0.00003
64.96
4221.9725
0.0001
0.00214
0.00003
65.11
0.66 0.750.75
0.00450
0.00004
0.12
0.03
0.00473
0.00006
2.33
0.07
1.00
0.06
0.00471
0.00006
0.94
0.10
0.17
0.07
/
/
Table 1. Fit parameters for three different models : 1/ with orbital parallax effects only, 2/ with orbital parallax + xallarap
(black-hole source companion), 3/ with orbital parallax + xallarap (solar-mass source companion).
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V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency 11
TABLE 2
OGLE-2007-BLG-050 Flux Parameters
Extended-Source Point-Lens
Orbital Parallax + Xallarap (solar mass companion)
Observatory
Fs
σFs
Fb
σFb
OGLE I
MOA I
µFUN R New-Zealand (Auckland)
µFUN R New-Zealand (Farm Cove)
µFUN I Arizona (MDM)
µFUN R South Africa (Bronberg)
µFUN I Chile (CTIO SMARTS)
µFUN I Arizona (Mt Lemmon)
PLANET I South Africa (SAAO)
PLANET I Australia, Tasmania (UTas)
PLANET I Chile (Danish)
PLANET I Australia (Perth)
PLANET I South Africa (Boyden)
Robonet R Faulkes North (Hawaii)
Robonet R Faulkes South (Australia)
Robonet R Liverpool (Canaries Island)
0.96
0.96
1.08
0.91
0.99
0.232
6.24
3.94
5.60
2.74
14.45
0.793
9.59
0.115
0.129
2.75
0.01
0.01
0.01
0.009
0.17
0.002
0.06
0.04
0.05
0.03
0.15
0.008
0.09
0.001
0.001
0.03
0.28
0.29
3.06
-27.0
242.3
0.28
-3.9
21.8
-1.9
163.8
13.0
6.5
447.3
-0.26
-0.229
17.7
0.01
0.01
0.05
0.2
75.0
0.04
0.2
0.1
1.6
1.2
0.4
0.1
0.6
0.02
0.009
0.3
Table 2. Source flux and blending for telescopes that observed OGLE 07-BLG-050. The given values corresponds to the model
with parallax and xallarap, in a case of a solar mass companion for the source. They do not change significantly for the other
models.
3.5. Lens mass and distance estimates
Gould (1992) showed that if both θEand πEcould be measured, then the mass M and the lens-source relative parallax πrelcould
be determined as given in Eq.1.2 and then the lens distance could be deduced from :
πrel= 1AU
?1
DL
−
1
DS
?
The resulting characteristics of the lens are given in Table 3 for each model that we have presented : parallax only, parallax +
xallarap (black-hole companion) and parallax + xallarap (solar-mass companion). Due to the high parallax magnitude obtained
with the “black hole” model (see Table 1), the lens mass is a brown dwarf (M = 0.025M⊙) in the extreme foreground (DL =
824pc). Moreover, the extreme black hole mass (Ms> 70M⊙), by itself, virtually rules out this model. We take this as evidence
for unrecognized systematic errors at the ∆χ2∼ 40 level, and hence do not believe inferences based on ∆χ2at this level are
robust. Systematic errors at this level are not uncommon for microlensing events.
The model with a solar-mass companion is suspect as well, still with a brown-dwarf lens in the foreground, meaning that it
results from the same systematics. We therefore conclude that the xallarap “signal” is probably spurious and we present these
two models only for completeness. We expect that the presence of these systematics will corrupt the parallax measurements by
of order
?
estimates. However, this systematic error is too small to qualitatively impact the conclusions of this paper.
For the model with parallax effects only, the lens star is a M-dwarf (Table 3) and situated in the disk, lying 5.5 kpc from the
observer. With the added uncertainties due to systematics, the parallax becomes πE= 0.12±0.03±0.01, the lens mass estimates
M = 0.50±0.14 M⊙( 28%) and the relative parallax πrel= 57.9±14.5µas. For the rest of the analysis, we will only consider this
model when the physical parameters of the lens are needed.
As discussed by Ghosh et al. (2004), future high-resolution astrometry could allow the direct measurement of the magnitude
and direction of the lens-source relative proper motion µ and substantially reduce the parallax uncertainty and thus the stellar
mass uncertainty. But according to our initial estimate of the relative proper motion (µ = 2.63 ± 0.08mas/yr), it would take at
least a 20 years to clearly detect the lens (especially since the source is very bright), but hopefully, within a decade, either ELT,
GMT or TMT (giant telescopes) will be built, in which case the lens could be observed thereafter.
∆χ2
xallarap+parallax/∆χ2
parallax∼
√(235 ± 43/235) − 1 ∼ 9%, which will impact the lens mass and relative parallax
4. Planet Detection efficiency
4.1. Introduction and previous analyses
To provide reliable abundance limits of Jupiter- to Earth-mass planets in our Galaxy, it is essential to evaluate the apparent
non-planetary events, especially the well-covered high magnification events. A necessary step is to evaluate the confidence with
which one can exclude potential planetary companions for each event.
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12V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency
TABLE 3
OGLE-2007-BLG-050 Lens mass and distance
Extended-Source Point-Lens with Parallax
ParametersWithout
xallarap
Xallarap
Black hole
P = 290 days
Xallarap
Solar mass
P = 170 days
θE
M (M⊙)
πrel (µas)
DL (kpc)
0.48 ± 0.01
0.50 ± 0.13
58 ± 15
5.47 ± 0.45
0.47 ± 0.041
0.025 ± 0.001
1088 ± 46
0.82 ± 0.07
0.47 ± 0.01
0.0618 ± 0.0007
440 ± 58
1.77 ± 0.20
Table 3. Lens mass and distance for three different models : 1/ with orbital parallax effects only, 2/ with orbital parallax +
xallarap (black-hole source companion), 3/ with orbital parallax + xallarap (solar-mass source companion).
Since OGLE-2007-BLG-050 presents strong finite-source effects, one may wonder whether a given planetary perturbation
would have been so washed out by these effects as to become undetectable. Using many such efficiency calculations the aim is
to determine the selection function to the underlying population of planets.
Gaudi & Sackett (2000) developed the first method to calculate detection efficiency for a single planet, which was extended
to multiple planets detection efficiency by Gaudi et al. (2002), who analyzed 43 microlensing events from the 1995-1999 obser-
vational seasons. Three of them were high magnification events [OGLE-1998-BLG-15 (Amax∼ 170), MACHO-1998-BLG-35
(Amax∼ 100) and OGLE-1999-BLG-35 (Amax∼ 125)]. This 5-year analysis provided the first significant upper abundance limit
of Jupiter- and Saturn-mass planets around M-dwarfs. Tsapras et al. (2003) and Snodgrass et al. (2004) derived constraints on
Jovian planet abundance based on OGLE survey data of 1998-2000 and 2002 seasons respectively.
Computing detection efficiency for individual events is thus required to estimate the frequency of planetary signatures in
microlensing light curves, and a couple of complex events have indeed been analyzed separately. For example the high mag-
nification event OGLE-2003-BLG-423 (Amax∼ 256) by Yoo et al. (2004b) who found that the event was not as sensitive as it
should have been if better monitored over the peak. Another high magnification (Amax∼ 525) example is MOA-2003-BLG-32 /
OGLE-2003-BLG-219was analyzed by Abe et al. (2004) and Dong et al. (2006) (Appendix B). This well-covered event showed
the best sensitivity to low-mass planets to date. Finally, the highest magnification event ever analyzed, OGLE-2004-BLG-343,
was unfortunately poorly monitored over its peak, and Dong et al. (2006) showed that it otherwise would have been extremely
sensitive to low-mass planets.
4.2. Planet detection efficiency in Einstein Units
To characterizetheplanetarydetectionefficiencyofOGLE-2007-BLG-050,we followthe Gaudi & Sackett(2000) methodwhich
consists of fitting binary models with the 3 binary parameters (d,q,α) held fixed and the single lens parameters allowed to vary.
Here d is the planet-star separation in units of θE, q the planet-lens mass ratio, and α the angle of the source trajectory relative
to the binary axis. In Gaudi & Sackett (2000), the single lens parameters, u0, t0and tE, are related to a PSPL fit. In this analysis,
we also fit the radius of the source ρ∗(scaled to the Einstein radius) and compare the binary lens fits to the best ESPL fit for this
event.
From the resulting fitted binary lens χ2
compared with a threshold value χ2
χ2
systems.
For each (d,q), the fraction of angles 0 < α < 2π that was excluded is called the ”sensitivity” for that system. Indeed, the
detection efficiency ǫ(d,q) can be expressed as :
(d,q,α), we calculate the χ2improvement : ∆χ2
C. If ∆χ2
(d,q,α)= χ2
(d,q,α)− χ2
ESPL, and ∆χ2
(d,q,α)is
(d,q,α)>
(d,q,α)< −χ2
C, the (d,q,α) planetary (or binary) system is detected, while if ∆χ2
C, it is excluded. Gaudi et al. (2002) argued that a threshold of 60 is high enough to be confident in excluding binary lens
ǫ(d,q) =
1
2π
?2π
0
Θ(∆χ2(d,q,α) − χ2
C)dα
where θ is the step function. To perform the calculations of binary light curves, we use a binary-lens finite-source algorithm
developed by Dong et al. (2006) (Appendix A). The resulting grids of χ2as a function of d and α are shown in Fig.7 for some
values of q. The complete computation has been done for every possible combinations between the following values of d, q and
α :
– q : 19 values with a constant logarithmic step over the range [10−6, 10−2].
– d : 40 values with a constant logarithmic step over the range [0.1, 10].
Page 13
V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency 13
– α : 121 values linearly spaced from 0 to 360o.
The resulting detection efficiency diagram for OGLE-2007-BLG-050 is shown in Fig.8. The first observation is that no
planet is detected since there is no configuration that gives ∆χ2< −60. This event is very sensitive to the presence of planets,
especially in the [0.8 - 1.2] separation range in Einstein units, where the detection efficiency reaches 100% for Jupiter mass ratios
(q = 9x10−4), 75% for Neptune mass ratios (q = 5x10−5) and 10% for Earth mass ratios (3x10−6). In larger separation ranges, as
[0.4 - 2.7] RE, we exclude Jupiter mass ratios with 95% confidence.
In future statistical analyses of microlensing planetary detection efficiency,one will likely be forced to use a higher exclusion
threshold than 60 because, while planets can sometimes be reliably excluded at this threshold (as in the present case), it is
unlikely that they can be reliably detected at this level, particularly in high-magnification events. Because we cannot predict the
exact threshold that will be adopted by future studies, we show both our exclusion level (∆χ2> 60) and a somewhat arbitrarily
chosenvalue,∆χ2> 250.Theimportantpointis thatthedetectionefficiencydiagramsinthetwocases (Fig.8andwitha threshold
equal to 250 in Fig.9) are very similar.
4.3. Planet detection efficiency in physical units
Having an estimate of the angular Einstein radius θE, the distance DLof the lens from the observer and the lens mass M, we
derive estimates of the physical parameters(r⊥,mp) for the tested planetary models, where r⊥is the projectedseparation between
the planet and its host star and mpthe planet mass, and calculate the associated detection efficiency.
r⊥(AU) = d DL(kpc)θE(mas)
mp= qM
To simplify the translation between efficiency diagrams in Einstein units and physical units, we have considered the values
of M, DLand θEas perfectly known. A proper analysis would involve a convolution of the detection efficiency map in terms of
native parameters d,q over the probability density distribution of the primary lens parameters (e.g. Yoo et al. (2004b)). While
our procedure of keeping M, DLand θEfixed is an approximation, considering the logarithmic scale of the efficiency maps, the
uncertainties in the primary lens parameters will not have an important effect on the shape of the resulting efficiency diagrams.
We take the parameters related to the fit with extended source and parallax effects, where M = 0.50 ± 0.14 M⊙, DL =
5.47 ± 0.45kpc and θE= 0.48 ± 0.01mas. The resulting detection efficiency diagram in physical units is shown in Fig.8 as well,
but the corresponding axes are those on the top and the right of the graphic. This demonstrates that OGLE-2007-BLG-050 is
sensitive to Neptune-mass planets as well as some Earth-mass configurations. Indeed, for a [1.8 - 3.1] AU projected separation
range between the planet and the lens star, Jupiter, Neptune and Earth-like planets are excluded with a 100%, 95% and 10%
confidence respectively. For a range of [1.4 - 4] AU, the detection efficiency reaches 100% for Jupiter mass planets and 75% for
Neptune mass planets, and for a much bigger range of [0.6 - 10] AU, Jupiter-like planets are excluded with a 75% confidence.
4.4. Planet detection efficiency as a function of central caustic size
Chung et al. (2005) analyzed the properties of central caustics in planetary microlensing events in order to estimate the pertur-
bation that they induce. They gave an expression for the central-caustic size as a function of the planet-star separation and the
planet/star mass ratio. Several authors have considered the size and shape of the central caustic as a function of the parameters
of the planet for high-magnification events (Griest & Safizadeh 1998; Dominik 1999; Dong et al. 2009b). In the analysis of the
cool Jovian-mass planet MOA-2007-BLG-400Lb, Dong et al. (2009b) conducted the initial parameter space search over a grid
of (w,q) rather than (d,q) where w is the “width” of the central caustic. For MOA-2007-BLG-400,the angular size of the central
caustic is smaller than that of the source (w/ρ ∼ 0.4), and w can be directly estimated by inspecting the light curve features.
Dong et al. (2009b) find the (w,q) parametrizationis more regularly defined and more efficient in searching parameter space than
(d,q).
The source size of OGLE-2007-BLG-050 is ρ = 0.0045 which is relatively big, and since finite-source effects smear out
the sharp magnification pattern produced by the central caustics, one way to present the planetary detection efficiency results
is to estimate the ratio w/ρ that is reached at the detection/exclusion limits. Assuming that detectable planets should produce
signals ≥ 5%, Han & Kim (2009) estimated the ratio w/ρ must be at least equal to 0.25. Here we present the detection efficiency
diagram in (d,w/ρ) space in Fig.10, still considering ∆χ2> 60 as the criterion of exclusion. This diagram shows a clear frontier
in red at w/ρ values between 0.1 and 0.3 above which the detection efficiency is greater than 50%, which also corresponds to
the 50% detection’s contours in Fig.8. On this frontier, the value of w/ρ goes down to 0.1 for d ∼ 1 and increases to 0.3 for
d >> 1 or d << 1. Our realistic estimate of detection efficiency is in general agreement with the simple criterion in Han & Kim
(2009). Given the high photometric precision and dense sampling, our data allow detections below the 5% threshold adopted
by Han & Kim (2009). We also note that the w/ρ threshold is weakly dependant on d, which is a result of the enhancement in
detection efficiency of the resonant caustics at small mass ratios.
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14V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency
We have presented a new way of visualizing the detection efficiency in (d,w/ρ) space. It offers a physically straightforward
way to understand the planetary sensitivity in events with pronounced finite-source effects. We find that the data obtained by
current observation campaigns can probe planetary central caustics as small as ∼ 20% of the source size for high-magnification
events.
5. Conclusion
OGLE-2007-BLG-050is a rarecase ofahighmagnificationeventwithwell measuredfinite sourceeffectsanddetectableparallax
effects. This leads to an estimate of the angular Einstein radius θE= 0.48 ± 0.01 mas, the parallax πE= 0.12 ± 0.03, the mass
M = 0.50 ± 0.14 M⊙and distance DL= 5.5 ± 0.4 kpc of the lens star. This is only the second reasonably precise mass estimate
(to within 28%) for an unseen single object using any method.
When computingplanet detection efficiency,we did not find anyplanetarysignature and the resulting maps in (d,q,α),where
d is the planet-star separation in Einstein units, q the planet-lens mass ratio, and α the angle of the source trajectory relative to the
binary axis, reveal a good sensitivity to low mass ratios q, with a 75% and 10% efficiencies for Neptune- and Earth-mass ratios
respectively in the range [0.8 - 1.2] RE, and a 100% detection efficiency for Jupiter-mass ratio in [0.4 - 2.7] RE.
It also permits the calculation of efficiency maps in physical space (r⊥,mp), where r⊥is the projected planet/star separation
and mpis the planet mass. Here we show that this microlensing event is very sensitive to Neptune-mass planets and has (10%)
sensitivity to Earth-mass planets within a [1.8 - 3.1] AU projected separation range.
Acknowledgements. We thank Thomas Prado and Arnaud Tribolet for their careful reading of the manuscript. VB thanks Ohio State
University for its hospitality during a six week visit, during which this study was initiated. We acknowledge the following support:
Grants HOLMES ANR-06-BLAN-0416 Dave Warren for the Mt Canopus Observatory; NSF AST-0757888 (AG,SD); NASA NNG04GL51G
(DD,AG,RP); Polish MNiSW N20303032/4275 (AU); HST-GO-11311 (KS); NSF AST-0206189 and AST-0708890, NASA NAF5-13042
and NNX07AL71G (DPB); Korea Science and Engineering Foundation grant 2009-008561 (CH); Korea Research Foundation grant 2006-311-
C00072 (B-GP);Korea Astronomy and SpaceScience Institute(KASI);Deutsche Forschungsgemeinschaft (CSB);PPARC/STFC,EU FP6pro-
gramme “ANGLES” (ŁW,NJR); PPARC/STFC (RoboNet); Dill Faulkes Educational Trust (Faulkes Telescope North); Grants JSPS18253002,
JSPS20340052 and JSPS19340058 (MOA); Marsden Fund of NZ(IAB, PCMY); Foundation for Research Science and Technology of NZ;
Creative Research Initiative program (2009-008561) (CH); Grants MEXT19015005 and JSPS18749004 (TS). This work was supported in part
by an allocation of computing time from the Ohio Supercomputer Center.
Page 15
V. Batista et al.: OGLE-2007-BLG-050 : Planetary detection efficiency15
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Fig.7. Binary-lens finite-source grids of χ2as a function of (x,y) where x = dcosα and y = dsinα for different values of q. The
value appearing in the upper part of each diagram corresponds to the value of logq. The color scale shows the variations of the
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