Besançon Galactic model analysis of MOA-II microlensing: evidence for a mass deficit in the inner bulge
MMon. Not. R. Astron. Soc. , 1–17 (2002) Printed 22 October 2015 (MN L A TEX style file v2.2)
Besan¸con Galactic model analysis of MOA-II microlensing:evidence for a mass deficit in the inner bulge (cid:63)
S. Awiphan † , E. Kerins and A. C. Robin Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK Institut Utinam, CNRS UMR6213, Universit´e de Franche-Comt´e, OSU THETA Franche-Comt´e-Bourgogne, BP 1615, Besan¸con 25010, France
ABSTRACT
Galactic bulge microlensing surveys provide a probe of Galactic structure. We presentthe first field-by-field comparison between microlensing observations and the Besan¸conpopulation synthesis Galactic model. Using an updated version of the model we pro-vide maps of optical depth, average event duration and event rate for resolved sourcepopulations and for difference imaging (DIA) events. We also compare the predictedevent timescale distribution to that observed. The simulation follows the selectioncriteria of the MOA-II survey (Sumi et al. 2013). We modify the Besan¸con model toinclude M dwarfs and brown dwarfs. Our best fit model requires a brown dwarf massfunction slope of − .
4. The model provides good agreement with the observed aver-age duration, and respectable consistency with the shape of the timescale distribution(reduced χ (cid:39) . . × and 0 . × the number observed, respectively).We perform a 2-dimensional fit to the event spatial distribution to predict the opticaldepth and event rate across the Galactic bulge. The most serious difficulty for themodel is that it provides only ∼
50% of the measured optical depth and event rateper star at low Galactic latitude around the inner bulge ( | b | < ◦ ). This discrepancymost likely is associated with known under-estimated extinction and star counts inthe innermost regions and therefore provides additional support for a missing innerstellar population. Key words: gravitational lensing: micro - stars: statistics - Galaxy: bulge - Galaxy:structure
The microlensing surveys toward the Galactic bulge haveprovided useful information for the search for exoplanets andfor the study of Galactic structure (Paczynski 1996; Gaudi2012). Several microlensing surveys have monitored a largenumber of stars and detected thousands of events over thebulge [e.g. OGLE (Udalski et al. 1994; Sumi et al. 2006;Wyrzykowski et al. 2015), MOA (Sumi et al. 2003, 2013),MACHO (Alcock et al. 1997, 2000; Popowski et al. 2005)and EROS (Afonso et al. 2003; Hamadache et al. 2006)].The microlensing optical depth, τ , measures the fraction ofthe sky covered by the Einstein rings of the lenses for agiven line of sight. As the optical depth is directly relatedto the mass density of the lens population, it can be usedto determine the mass distribution of the bulge. However, a (cid:63) This work uses the Manchester–Besan¸con Microlensing Sim-ulator - MaB µ lS, which is publicly available online at † E-mail: [email protected] difficulty in measuring the microlensing optical depth stemsfrom the fact that it is sensitive to the individual contribu-tions of long duration events. Another measurable propertyfrom the surveys is microlensing event rate, Γ, which hasthe advantage that it is not dominated by a small numberof long duration events but the disadvantage that it is sen-sitive to Galactic kinematics and the stellar mass function,as well as the mass distribution.A number of measurements of the bulge optical depthhave been made by the survey teams, often under differ-ent sample definitions. We loosely categorize these as: re-solved source measurements, difference image analysis (DIA)source measurements and red clump giant (RCG) sourcemeasurements. The resolved source method includes allsources which are brighter than magnitude limit, whilst theDIA method includes fainter sources which may only bedetectable during lensing. The DIA method has the ben-efit that it is less sensitive to blending systematics withincrowded fields and potentially provides a better S/N ratiomeasurement due to the larger available sample size. At theother extreme the RCG method uses samples of events which c (cid:13) a r X i v : . [ a s t r o - ph . E P ] O c t S. Awiphan, E. Kerins and A. C. Robin involve only bright sources which are assumed to be well re-solved and therefore should exhibit a minimal blending bias.In recent studies, DIA optical depth measurements tend tobe about 25% higher that those derived from RCG samples(Sumi et al. 2013).The MOA-II survey (Sumi et al. 2013) determinedthe optical depth from a study of 474 events with sourcesbrighter than 20 th magnitude in the I -band toward thebulge. They determined a value of τ DIA = [2 . ± . e [0 . ± . −| b | ) × − . For the average opticaldepth, they find that τ DIA , = 3 . +0 . − . × − at ( l =0 . ◦ , b = − . ◦ ). These results are broadly consistentwith previous measurements from MOA-I (Sumi et al. 2003),OGLE (Sumi et al. 2006), MACHO (Popowski et al. 2005)and EROS Hamadache et al. (2006) (See Table 3).Over recent years, more detailed theoretical modelshave been developed in order to predict the microlensingoptical depth values (Han & Gould 2003; Wood & Mao2005; Kerins et al. 2009). Kerins et al. (2009) presented syn-thetic maps of optical depth and event rate over the Galacticbulge using catalogues generated from the Besan¸con galacticmodel developed by Robin et al. (2003) with 3D extinctionmaps from Marshall et al. (2006). The observational resulttends to agree with the theoretical models. However, the re-cent MOA-II surveys provide optical depth of RCG 30-40%below the prediction of Kerins et al. (2009) which might bethe result of lacking long crossing time events in observa-tional data (Sumi et al. 2013).One important issue which we do not explicitly addressis source blending. In principle this can be examined withinthe context of a population synthesis model through con-struction of artificial images. However, this is beyond thescope of the present work. Instead, we choose to model onlythe two idealized cases described above (resolved and DIAsources). If the model is a good representation of reality,these cases should provide reasonable upper and lower lim-its on the potential number of events within specific MOA-IIsources sub-sample. Our DIA estimate should always pro-vide a firm upper limit to the observed microlensing rateper star. On the other hand, our resolved source calculationsshould provide a firm lower limit to the number of events perstar. MOA-defined sub-samples such as clump giants shouldyield a rate somewhere intermediate to these regimes as theRCG sources are resolved but known to be confined to thebulge, whereas in our simulation a non-negligible number ofour sources will be closer to the observer. We note that, inorder for the model to accurately compute the microlensingrate per unit sky area it would also be necessary to accu-rately mimic the source colour-magnitude cuts of the survey.In this paper, the microlensing optical depth and eventrate maps are presented by using a recent version of Be-san¸con galactic model (Robin et al. 2014) and compared tothe MOA-II result. The updated model includes an innerbar component. In Section 2, the Besan¸con Galactic modelis summarised. In order to simulate the MOA-II microlens-ing event sample from the Besan¸con model, the selectioncriteria is discussed in Section 3. The calculation method ofmicrolensing parameters and their maps are shown in Sec- The subscripts of optical depths indicate the method of analysisand the long duration cutoff in days. tion 4. In Section 5, the results from the Besan¸con simulationare compared with the observational results of the MOA-IIsurvey. The model parametrisation of simulation results areprovided in Section 6. Finally, in Section 7, the conclusionsof this work are presented.
The Besan¸con model, a Galactic population synthesis model,is designed to describe the observable properties of theGalactic stellar population by relating them to models ofGalactic formation and evolution, stellar formation and evo-lution and stellar atmospheres, using constraints from obser-vation data (Robin et al. 2003, 2012, 2014). In the Besan¸conmodel, stars are created from gas following an initial massfunction (IMF) and star formation rate (SFR), and evolvedaccording to theoretical stellar evolutionary tracks. For eachsimulated star, the photometry, kinematics and metallicityare computed. In order to simulate the Galaxy, four mainpopulations are assumed: a thin disc; thick disc; bulge/bar;and stellar halo.The model also includes a 3D extinction map (Marshallet al. 2006). An interstellar extinction distribution in threedimensions from 2MASS survey (Cutri et al. 2003) towardsthe inner Galaxy ( | l | (cid:54) ◦ and | b | (cid:54) ◦ ), with 15 (cid:48) resolu-tion is used. Marshall et al. (2006) calculated the extinctionas a function of distance along each line of sight by compar-ing observed reddened stars to unreddened simulated starsfrom the Besan¸con model. This distribution can be used todetermine the observed colours and magnitudes of the simu-lated stars. In the following work, a later version of Besan¸conmodel (Robin et al. 2014) has been used. The thin disc is a major component in the Galactic cen-tral region. It is assumed to have an age of 10 Gyr. A con-stant SFR over the past 10 Gyr is assumed, along with anIMF with two slopes, d N/ d m ∝ M − . for M < M (cid:12) andd N/ d m ∝ M − . for M > M (cid:12) . The total mass of thethin disk is 9 . × M (cid:12) . The luminosity function de-termined from Hipparcos observations is adopted (Haywoodet al. 1997a,b; Robin et al. 2003), whilst the underlying den-sity law follows the Einasto (1979) density profile. The discis modelled with a central hole and so the maximum densityof the thin disc is located at about 2.5 kpc from the Galac-tic Centre. The kinematics follow the
Hipparcos empiricalestimates of Gomez et al. (1997). The populations of thindisc are divided into 7 distinct components with differentdistribution in age, scale height and velocities (Robin et al.2012).
The thick disc is of much lower density than the thin disclocally but becomes important at Galactic latitudes aboveabout 8-10 ◦ . In the model it is assumed a separate popula-tion from the thin disc, with distinct star formation history.Recent constraints from SDSS and 2MASS data lead to re-visions of the scale length and scale heights (Robin et al.2014). We here make use of the single thick disc episode of c (cid:13) , 1–17 esan¸con model analysis of MOA-II microlensing formation presented in Robin et al. (2014), modelled by a12 Gyr isochrone of metallicity -0.78 dex, with a density lawfollowing a modified exponential (parabola up to z = 658pc, followed by an exponential with a scale height of 533pc), which is roughly equivalent to a sech function of scaleheight 450 pc. The radial density follows an exponential witha scale length of 2.355 kpc. Its kinematics follow the resultof Ojha et al. (1996). A new model of the bulge of the Besan¸con model has beenproposed by Robin et al. (2012), as the sum of two ellip-soids: a standard boxy bulge (bar), the most massive com-ponent which dominates the stellar content of latitudes be-low about 5 ◦ , and another ellipsoid (thick bulge) with longerand thicker structure which can be observe at higher lati-tudes where the bar starts to be less prominent. However,in Robin et al. (2014), we showed that the “thick bulge”population was in fact the inner part of the thick disc whichshort scale length makes a large contribution in the bulgeregion. Hence, in this new version, the populations in thebulge region are: the thin disc, the bar and the thick disc.The angle of the bar to the Sun-Galactic Centre directionis 13 ◦ . The bar kinematics are taken from the model of Fux(1999) and the bulge kinematics are established to repro-duce the BRAVA survey data (Rich et al. 2007). The stel-lar density and luminosity function are assumed from theresult of Picaud & Robin (2004) with a single burst pop-ulation of 10 Gyr age. The IMF below and above 0.7 M (cid:12) are assumed to be d N/ d m ∝ m − . and a Salpeter slope,d N/ d m ∝ m − . , respectively (Picaud & Robin 2004). Thetotal bar mass is 5 . × M (cid:12) . The model mass to light ratiois 2.0 at the Sagittarius Window Eclipsing Extrasolar PlanetSearch (SWEEPS) field ( l = 1 . ◦ , b = − . ◦ ) in Johnson-I band which is compatible with result of Calamida et al.(2015) in F814W filter (wide I ). The stellar halo is older than the thick disc (14 Gyr) andmetal poor ([Fe / H] = -1.78). A single burst population withan IMF, d N/ d m ∝ m − . , and total mass of 4 . × M (cid:12) are assumed (Robin et al. 2003). The density law has beenrevised in the study of SDSS+2MASS star counts (Robinet al. 2014). It is now modelled with a power law densitywith an exponent of 3.39 and an axis ratio of 0.768. Its kine-matics is modelled with Gaussian distributions of velocitiesof dispersion (131, 106, 85) in km/s in the (U,V,W) plane,and no rotation. In following work, we simulate the MOA-II survey datataken from the 2006 and 2007 observing seasons (Sumi et al.2011, 2013). In order to obtain enough samples in each mag-nitude range, we produce lens/source star catalogues span-ning four H -band magnitude ranges. H -band selection en-sures that we adequately samples all relevant stellar types, Table 1.
Solid angles used for the simulated catalogues. Ω sim isused to compute the spatial maps presented in the paper, whilstΩ sample is used to compute the global time scale distribution (Seetext).
Magnitude range Ω sim (deg ) Ω sample (deg ) − (cid:54) H <
15 2 . × − . × − (cid:54) H <
19 8 . × − . × − (cid:54) H <
23 9 . × − . × − H >
23 1 . × − . × − though we stress that our calculations are performed usingthe corresponding R and I -band magnitudes of the sourcessince these are the relevant filters for MOA-II. Our rangescorrespond to: − (cid:54) H <
15, 15 (cid:54)
H <
19, 19 (cid:54)
H <
H >
23. The latter ranges are dominated by starswhich are too faint to act as sources but which do actas lenses. The solid angle in each catalogue, Ω sim , is cho-sen to contain ∼ l = 1 ◦ , b = − ◦ ) (Table 1). The first catalogue( − (cid:54) H <
15) has a solid angle of 0.026 deg , corre-sponding to the size of the MOA-II sub-fields. The simula-tion catalogues stars out to a distance of 15 kpc and hasthe same overall areal coverage as the MOA-II survey. Ourfinal results are appropriately inverse weighted with Ω sim inorder to recover the relevant microlensing observables.For each line of sight, the microlensing optical depth,average time scale and event rate toward the Galactic bulgeare calculated using all combinations of source and lens pairsfrom the four catalogues. We compute microlensing quanti-ties obtained from all resolved sources above a specific mag-nitude threshold and also from all difference imaging anal-ysis (DIA) sources which have a magnified peak above thesame threshold (Alard 2000; Wozniak 2000; Bramich 2008).Therefore, the baseline magnitude of the DIA sources canbe fainter than the limit. For unresolved sources, the instan-taneous fraction of events with impact parameter u smallenough to be detectable scales as u , though over time therate of detectable events scales as u . Therefore, we weight theoptical depth by min(1 , u ) and the rate-weighted averageduration by min(1 , u ), respectively. The impact parametermoments of Equation 6 are rate-weighted as explained inSection 4.1 in order to reflect the fact that observables arenecessarily obtained from rate-biased samples.The finite source effect is also taken into account in ourcalculations. The events which involve a source star withangular radius larger than the angular Einstein radius arenot used to calculate the microlensing parameters. However,they are accounted for in the source number normalisation.In practice this modification alters our results only at the1% level (See Equation 5). The time scale distribution of the MOA-II observationaldata, excluding the gb21-R-8-53601 event, which is locatedoutside the Besan¸con extinction map, and the Besan¸con sim-ulated data are shown in Figure 1. The histogram of the Be-san¸con data is generated from the sample catalogues usingthe same criteria as Section 3.1 but with smaller solid an- c (cid:13)000
15) has a solid angle of 0.026 deg , corre-sponding to the size of the MOA-II sub-fields. The simula-tion catalogues stars out to a distance of 15 kpc and hasthe same overall areal coverage as the MOA-II survey. Ourfinal results are appropriately inverse weighted with Ω sim inorder to recover the relevant microlensing observables.For each line of sight, the microlensing optical depth,average time scale and event rate toward the Galactic bulgeare calculated using all combinations of source and lens pairsfrom the four catalogues. We compute microlensing quanti-ties obtained from all resolved sources above a specific mag-nitude threshold and also from all difference imaging anal-ysis (DIA) sources which have a magnified peak above thesame threshold (Alard 2000; Wozniak 2000; Bramich 2008).Therefore, the baseline magnitude of the DIA sources canbe fainter than the limit. For unresolved sources, the instan-taneous fraction of events with impact parameter u smallenough to be detectable scales as u , though over time therate of detectable events scales as u . Therefore, we weight theoptical depth by min(1 , u ) and the rate-weighted averageduration by min(1 , u ), respectively. The impact parametermoments of Equation 6 are rate-weighted as explained inSection 4.1 in order to reflect the fact that observables arenecessarily obtained from rate-biased samples.The finite source effect is also taken into account in ourcalculations. The events which involve a source star withangular radius larger than the angular Einstein radius arenot used to calculate the microlensing parameters. However,they are accounted for in the source number normalisation.In practice this modification alters our results only at the1% level (See Equation 5). The time scale distribution of the MOA-II observationaldata, excluding the gb21-R-8-53601 event, which is locatedoutside the Besan¸con extinction map, and the Besan¸con sim-ulated data are shown in Figure 1. The histogram of the Be-san¸con data is generated from the sample catalogues usingthe same criteria as Section 3.1 but with smaller solid an- c (cid:13)000 , 1–17 S. Awiphan, E. Kerins and A. C. Robin
Table 2.
The mass function of the simulated low-mass star pop-ulation
Component Mass range MF slope
Thick disk 0 . M (cid:12) - 0 . M (cid:12) -0.50Bulge 0 . M (cid:12) - 0 . M (cid:12) -1.50Halo 0 . M (cid:12) - 0 . M (cid:12) -0.50 gles, Ω sample , which contain ∼ N (cid:48) Besancon − N MOA ) isshown, where N (cid:48) Besancon is Besan¸con event rate scaled tonumber of MOA-II events per year ( (cid:80) N MOA ). The Be-san¸con data shows a deficit of short time scale events ( < α ∝ log(d N/ d M ), to the H-burning limit of 0 . M (cid:12) (Table 2). We also add in a browndwarf mass function slope, α BD , normalised to the stellarmass function at the H-burning limit and extended down to0 . M (cid:12) . The added populations use the same kinetic pa-rameters as the original catalogue and are used for the lensstars only.In order to find the best value of α BD , the Besan¸condata from sub-field 7-4 of 20 separate fields (Fields gb1-gb20) are normalised by the MOA-II event per year andare used to calculate the timescale distributions. In Fig-ure 2, the reduced chi-squares of the predicted versus ob-served timescale distributions as a function of α BD for val-ues of α BD between -0.9 and 1.0 are shown, along with re-duced chi-squares of the original simulation (without addinglow-mass stars and brown dwarfs) and a simulation addingonly low-mass stars. The result shows that adding low-massstars and brown dwarfs provides a better match to theMOA-II time scale distribution. Sumi et al. (2011) find afavoured mass function index in the brown dwarf regime,0 . M (cid:12) (cid:54) M (cid:54) . M (cid:12) , for the 2006-2007 MOA-II datais α BD = − .
49. From our simulation, an MF slope of α BD = − . α bd > . − . Figure 1.
The Einstein radius crossing time distribution of theMOA-II survey and the Besan¸con data (top) and the scaled resid-ual between the MOA-II survey and the Besan¸con data with theMOA-II distribution error (bottom). The blue shaded area (bluethin line) represents the efficiency corrected time scale distribu-tion for the MOA-II DIA sources, exculding event gb21-R-8-53601(See text). The crossing time distribution of the Besan¸con re-solved sources (red thick line) and DIA sources (blue thick dashedline) are also presented. The error bars of Besan¸con distributionsare shown at 100 times their true size. For the residual, the redline and blue dashed line represent the residual of the Besan¸conresolved sources and DIA sources, respectively. from the MOA-II survey ( N MOA ) is between the numberof events from the Besan¸con resolved sources (0 . N MOA )and DIA sources (2 . N MOA ). In the absence of significantblending effects, we should expect our resolved and DIA pre-dictions to bracket the true result; the fact that it does israther reassuring. However the effects of blending are com-plex and a more detailed comparison would require model-ing both the source selection criteria and the source blendcharacteristics of the MOA-II image data. This is beyondthe scope of the current paper. In the case that all resolvedsource events are detected, we might be tempted to con-clude that 12% of faint stars which can only be detected bythe DIA method are observed. However, differences in theassumed filter response can equally be a factor. c (cid:13) , 1–17 esan¸con model analysis of MOA-II microlensing Figure 2.
The reduced chi-square values of the model timescaledistribution with respect to the MOA-II data for 20 fields is pre-sented as a function of brown dwarf mass function slope. Red cir-cle dots and lines present the Besan¸con resolved source data. Bluesquare dots and dashed lines present the Besan¸con DIA sourcedata. Thick lines and thin line show the original data and thedata with adding only low-mass stars, respectively.
The mean crossing times are shorter for both resolvedsources and DIA sources, at 20.3 and 20.9 days, respectively.This is close to the MOA-II RCG timescale (19.2 days), buta little lower than their mean timescale for all sources (24.0days). These mean crossing times also compatible with meancrossing time of OGLE-III survey of resolved sources arebrighter than
I <
19 mag and the relative errors on cross-ing time are less than 100% with log-normal model in allthree regions: positive longitude ( l > ◦ , 22.0 days), central( − ◦ < l < ◦ , 20.5 days) and negative longitude ( l < − ◦ ,24.2 days) (Figure 3) (Wyrzykowski et al. 2015).The residuals of the distribution (model − data) withadding low-mass stars show a slight deficit of events withshort crossing time between 0.3 and 2 days and very longcrossing time between 30 and 200 days. Moreover, the modeltends to over-predict the number of events with durationbetween 2 and 30 days, though there is not a high statis-tical significance to any of these discrepancies. Overall, ourbest-fit brown dwarf slope provides a match to the MOA-IItimescale distribution with a reduced χ (cid:39) . . × M (cid:12) , thick disk 2 . × M (cid:12) ,halo 4 . × M (cid:12) and bulge 5 . × M (cid:12) .To analyse the structure of the residual histogram, weshow the contributions of each lens component separatelyin Figure 4 for DIA sources. The histogram is calculated byassuming that the proportion of each component to the ob-served rate scales with their proportionate rate within themodel. We find that the bulge lens component dominatesboth the over-predicted and under-predicted regimes, sug-gesting a mismatch in bulge kinematics, or spatial distribu-tion, as the principal source of the problem. Figure 3.
The Einstein radius crossing time distribution of theMOA-II survey, OGLE-III events in − ◦ < l < ◦ fields andthe Besan¸con data with added low-mass stars and brown dwarfs(top) and the scaled residual between the MOA-II survey andthe Besan¸con rates (bottom). The descriptions are the same as inFigure 1. To compare the model optical depth, rate and average du-ration to the MOA-II survey we must ensure that we matchthe timescale selection. Accordingly, the maximum Einsteincrossing time ( t E, max ) of this work is set at 200 days tomatch the MOA-II maximum Einstein crossing time (Sumiet al. 2013). Furthermore, for the minimum Einstein cross-ing time ( t E, min ), events with duration below 40 minutesin fields gb5 and gb9 and 200 minutes in other field con-tribute negligibly (Sumi et al. 2011, 2013). Therefore, theoptical depth of all events ( τ all ) and histograms of Einsteincrossing time in each field are used to calculate optical depth( τ select ), average Einstein crossing time ( (cid:104) t E (cid:105) select ) and eventrate (Γ select ) of the events which meet the criteria. Thus τ select = τ all (cid:80) t E, max i = t E, min t E,i N i (cid:80) ∞ i =0 t E,i N i , (1) c (cid:13)000
The Einstein radius crossing time distribution of theMOA-II survey, OGLE-III events in − ◦ < l < ◦ fields andthe Besan¸con data with added low-mass stars and brown dwarfs(top) and the scaled residual between the MOA-II survey andthe Besan¸con rates (bottom). The descriptions are the same as inFigure 1. To compare the model optical depth, rate and average du-ration to the MOA-II survey we must ensure that we matchthe timescale selection. Accordingly, the maximum Einsteincrossing time ( t E, max ) of this work is set at 200 days tomatch the MOA-II maximum Einstein crossing time (Sumiet al. 2013). Furthermore, for the minimum Einstein cross-ing time ( t E, min ), events with duration below 40 minutesin fields gb5 and gb9 and 200 minutes in other field con-tribute negligibly (Sumi et al. 2011, 2013). Therefore, theoptical depth of all events ( τ all ) and histograms of Einsteincrossing time in each field are used to calculate optical depth( τ select ), average Einstein crossing time ( (cid:104) t E (cid:105) select ) and eventrate (Γ select ) of the events which meet the criteria. Thus τ select = τ all (cid:80) t E, max i = t E, min t E,i N i (cid:80) ∞ i =0 t E,i N i , (1) c (cid:13)000 , 1–17 S. Awiphan, E. Kerins and A. C. Robin
Figure 4.
The Einstein radius crossing time residual histogramof the Besan¸con DIA sources for each Galactic lens component.The residual distribution of all lens components is shown in redline. The blue thin dashed, green thick dashed, cyan dash-dottedand magenta dashed-dot-dotted lines with shaded areas representthin disk, thick disk, halo and bulge lenses, respectively. (cid:104) t E (cid:105) select = (cid:80) t E, max i = t E, min t E,i N i (cid:80) t E, max i = t E, min t E,i N i , (2)and Γ select = t E, max (cid:88) i = t E, min N i , (3)where t E,i and N i are the crossing time and the num-ber of microlensing events associated with the logarithmic timescale bin i , respectively. Figure 5(a) shows optical depth maps for both resolved andDIA source samples for a survey limit M lim = 20. The mapsare computed for the Johnson R and I filter bands, whichare comparable to the Cousins R and I bands of the MOA-IIsurvey. The total optical depth of all source and lens pairsis calculated by averaging the optical depth of all sourcesalong the line of sight, τ = (cid:80) N s s =1 (cid:80) N l ( D s >D l ,M s
Figure 5.
The optical depth (a), average time scale (b), microlensing event rate per square degree (c) and microlensing event rate perstar (d) for resolved sources (left) and DIA sources (right) from the Besan¸con Galactic model. The maps are smoothed by the samekernel function as Sumi et al. (2013). The sub-fields with l > ◦ are excluded. Kerins et al. (2009) and Penny et al. (2013), due to the ad-dition of low-mass star and brown dwarf lenses. There is nomajor difference between the average time scale of resolvedsources and DIA sources. The negative longitudes provideslightly longer time scales than positive longitudes due to the bar geometry resulting in typically larger Einstein radiiat negative longitudes.In Figure 7, we show the average timescale maps indi-vidually for the thin disk, thick disk, stellar halo lens andbulge lens populations. The maps show a reasonably sym- c (cid:13) , 1–17 S. Awiphan, E. Kerins and A. C. Robin (a) Thin disk (b) Thick disk(c) Halo (d) Bulge
Figure 6.
The optical depth maps of each Galactic lens component: (a) thin disk lenses, (b) thick disk lenses, (c) stellar halo lenses and(d) bulge lenses. The maps have same description as the map in Figure 5 metric spatial distribution in the average event duration,with bulge lenses exhibiting typically shorter time scalescompared to the other lens components. Since bulge lensesdominate the event rate in the inner Galaxy (Figure 8) theoverall map of event duration shown in Figure 5(b) closelyresembles that of the bulge lens population. We also con-firm from Figure 7 that the long duration region at longitude l > . ◦ evident in Figure 5(b) arises from the disk lens pop-ulation as the density of bulge lenses become sub-dominantaway from the Galactic Centre. The total event rate is obtained simply by dividing the op-tical depth maps by their corresponding average time scalemaps: Γ = 2 π τ (cid:104) t E (cid:105) . (8)Figure 5(c) and Figure 5(d) show maps of microlensingevent rate per square degree (Γ deg ) and event rate per star(Γ star ), respectively. Γ deg is obtained by integrating the rateover the effective number of sources: N = (cid:80) N s ( M s >M lim ) s =1 Ω Ω s Resolved , (cid:80) N s s =1 (cid:104) u p (cid:105) w Ω Ω s DIA . (9)In Figure 5(c) we see that Γ deg for DIA sources ishigher than for resolved sources, as expected. The area in-tegrated microlensing event rate in the simulated maps forresolved sources and DIA sources is 1,250 and 3,250 events per year, respectively. The maps of Γ star in Figure 5(d) forresolved sources and DIA sources do not show a major differ-ence indicating that, overall, they probe sources and lensesat similar distances with similar kinematics.In Figure 8, the maps of Γ star are shown separately foreach lens population. The strong dominance of bulge lensesover most of the MOA-II region is evident. The optical depth, microlensing event rate per square de-gree and microlensing event rate per star from the Besan¸conGalactic model and survey observations at different Galac-tic latitudes are presented in Figure 9. The results are cal-culated from the optical depth and microlensing event ratebetween l = − ◦ and l = 5 ◦ . The stars in each simulatedsub-field are binned to 0.5 ◦ in Galactic latitude, in simi-lar fashion to the MOA-II survey Sumi et al. (2013). Theresults from the previous measurements, as well as the sim-ulated models of the present paper, are listed in Table 3and 4, respectively. The shaded areas in Figure 9 show 68%confidence interval of the data. The shaded 68% confidenceintervals are obtained by generating random deviate distri-butions of each exponential fit model assuming that errorson the best-fit parameters are Gaussian distributed.From the MOA-II data and the Besan¸con simulateddata, the optical depth at b > . ◦ agrees very well with anexponential fit. For b < . ◦ , the optical depths decrease dueto the high column density of dust in that area. Over lower c (cid:13) , 1–17 esan¸con model analysis of MOA-II microlensing (a) Thin disk (b) Thick disk(c) Halo (d) Bulge Figure 7.
The average time scale maps of each Galactic lens component for DIA sources: (a) thin disk, (b) thick disk, (c) halo and (d)bulge. The maps have same description as the map in Figure 5(a) Thin disk (b) Thick disk(c) Halo (d) Bulge
Figure 8.
The microlensing event rate per star maps of each Galactic lens component for DIA sources: (a) thin disk, (b) thick disk, (c)halo and (d) bulge. The maps have same description as the map in Figure 5c (cid:13) , 1–17 S. Awiphan, E. Kerins and A. C. Robin
Figure 9.
The optical depth (top), microlensing event rate per square degree (middle) and microlensing event rate per star (bottom)as a function of Galactic latitude. The measurements are averaged over Galactic longitudes − ◦ < l < ◦ . Different markers representdifferent survey measurements (See Table 3): OGLE (pentagon), MACHO (circle), MOA (triangle), EROS (square) and simulated datafrom the Besan¸con galactic model (star) (See Table 4). Results of resolved sources, DIA sources and RCG source are presented withunfilled, filled and half-filled makers. The error bars of the Besan¸con simulation results are shown at 100 times their true size. The thindashed, dash-dotted and dotted lines represent fits to the MOA-II all-source sample, EROS RCG sample and MOA-II RCG sample,respectively (Hamadache et al. 2006; Sumi et al. 2013). The thick solid and dashed lines are fits to the resolved source and DIA sourcesimulations of this paper. The shaded areas represent the 68% confidence interval of EROS, MOA-II and Besan¸con fits, respectively (Seetext). c (cid:13) , 1–17 esan¸con model analysis of MOA-II microlensing latitude regions ( b < ◦ ), the Besan¸con DIA optical depth islower than the MOA-II all-source optical depth by a factorof 2, a factor similar to that found by Penny et al. (2013).The exponential models of the Besan¸con optical depth are, τ Res = (1 . ± . e (0 . ± . −| b | ) × − ,τ DIA = (1 . ± . e (0 . ± . −| b | ) × − . (10)The event rate per square degree for the Besan¸con re-solved sources is compatible with the MOA-II all-sourceevent rate, however for DIA sources it is 3 times higherthan the MOA-II result. This might be a consequence ofthe blending effect as discussed in Section 3.2. The resultsfrom both also show the same turning point at l = 1 . ◦ asthe optical depth. The exponential fits for the event rate persquare degree give,Γ deg , Res = (43 ± e . ± . −| b | ) , Γ deg , DIA = (119 ± e (0 . ± . −| b | ) . (11)For the microlensing event rate per star, the results aresimilar to the optical depth result, which is expected giventhe general agreement of the average event duration. Theexponential models of the simulated event rate per star canbe written as,Γ star , Res = (13 . ± . e (0 . ± . −| b | ) × − , Γ star , DIA = (14 . ± . e (0 . ± . −| b | ) × − . (12)Whilst the model average event duration is in reason-able agreement with the MOA-II observations the factor2 discrepancy with both the optical depth and rate sug-gests that the model bulge mass is too low to accommo-date the microlensing results. The bulge mass would needto be increased by a factor 2.6 in order to match the over-all optical depth distribution. However, we note that such achange could not be made in a self-consistent manner with-out also altering the lens and source kinematics and theevent timescale distribution. In Section 4, the model microlensing maps of the MOA-IIfield, filter and time scale cut are shown. In order to com-pare the result with the MOA-II observational data, resid-ual maps are produced ( x Besancon − x MOA ). In Figure 10(a),the Besan¸con model underestimates the optical depth com-pared with the MOA-II data at closer to the Galactic Centre( b < ◦ ), predicting only around 50% of the MOA-II mea-sured optical depth. However, moving away from the Galac-tic Centre, the Besan¸con optical depth is generally in goodagreement with the MOA-II measurement, suggesting thatthe Besan¸con disk model provides a good description of themicrolensing data.The residual average time scale maps in Figure 10(b),show that the Besan¸con model provide a reasonably goodrepresentation of the MOA-II timescales across pretty muchthe whole map. Most of the structures in the map are pro-duced by individual very long time scale MOA-II events ormany short time scale event in observed specific sub-fields. We therefore conclude that the average microlensing kine-matics within both the disk and bulge are consistent withmicrolensing data.In Figures 10(c) and (d), which show the residual eventrate per unit area and per source, respectively, indicate asimilar deficit of the model with respect to the data withinthe inner bulge region.These maps confirm the view that, whilst the modelbulge kinematics provide a good average description of theevent timescale, the mass is insufficient by a factor 2 toexplain the observed number of events. χ of τ and Γ star maps From Section 5.2, the simulated results under-predict theoptical depth and microlensing event rate per star comparedwith the MOA-II observational data and show the structureat low Galactic latitude. The significance of this result canbe assessed by a straightforward reduced χ statistic: χ r = 1 N fld N fld (cid:88) i =1 (cid:18) x Bes ,i − x MOA ,i σ MOA ,i (cid:19) , (13)where ( x Bes , x
MOA ) refer to the (model, observed) microlens-ing quantity. σ is the observational uncertainty within thefield, and N fld is number of fields. The observational un-certainty within the field is calculated using formula fromHan & Gould (1995). The reduced χ contribution of eachMOA-II field are shown in Table 5 and Figure 11. The gb21field is excluded due to the limit of the Besan¸con extinctionmaps. The model optical depth is in agreement with MOA-II data within 3 σ MOA for most fields. The reduced χ ofresolved source and DIA sources optical depth are 2.4 and2.0, respectively.The event rates show higher reduced chi-squared con-tribution than the optical depths. The Besan¸con resolvedsources and DIA source results have χ r values of 2.6 and2.2, respectively. The low Galactic latitude area ( b < ◦ ) ofboth parameters provide the bulk of the disagreement (SeeSection 5.1).In field gb1, there is a long time scale event (gb1-3-1, t E = 157 . χ test, the reduced χ of field gb1 without gb1-3-1, gb1 Cut , is calculated. Theresult in Table 5 shows that the gb1
Cut field provide a betterreduced χ r than original gb1 field.Finally, we cut the events which have crossing timelonger than 100 days which locate in 5 field; gb1, gb9, gb10,gb13 and gb 14. The new reduced chi-squared, χ r, Cut , of op-tical depth (2.3 for resolved sources and 1.8 for DIA sources)and event rate per star (2.2 for resolved sources and 2.0for DIA sources) is reduced. However, fields which locatearound inner bulge except field gb6 still show high reducedchi-squared compare to high Galactic latitude field. There-fore, high reduced chi-squared region around inner bulge donot effect by long time scale event, but show the mismatch ofoptical depth and event rate per star between the Besan¸conmodel and the MOA-II data at low Galactic latitude area. c (cid:13)000
Cut field provide a betterreduced χ r than original gb1 field.Finally, we cut the events which have crossing timelonger than 100 days which locate in 5 field; gb1, gb9, gb10,gb13 and gb 14. The new reduced chi-squared, χ r, Cut , of op-tical depth (2.3 for resolved sources and 1.8 for DIA sources)and event rate per star (2.2 for resolved sources and 2.0for DIA sources) is reduced. However, fields which locatearound inner bulge except field gb6 still show high reducedchi-squared compare to high Galactic latitude field. There-fore, high reduced chi-squared region around inner bulge donot effect by long time scale event, but show the mismatch ofoptical depth and event rate per star between the Besan¸conmodel and the MOA-II data at low Galactic latitude area. c (cid:13)000 , 1–17 S. Awiphan, E. Kerins and A. C. Robin
Table 3.
Observed microlensing optical depth and rate measurements towards the Galactic Centre.
Project Field Method t E,max N event l, b ∗ τ Γ deg Γ star deg days events × yr − deg − × − yr − star − OGLE(1994) [1] ± ◦ ,-3.5 ◦ ± [2] ◦ ,3.64 ◦ +1 . − . - -MACHO(2000) [3] ◦ ,-3.35 ◦ +0 . − . - -EROS(2003) [4] ◦ ,-4.0 ◦ ± [5] ◦ ,-3.8 ◦ +0 . − . - -MACHO(2005) [6] ◦ ,-2.68 ◦ +0 . − . - -OGLE(2006) [7] ◦ ,-2.75 ◦ +0 . − . - -EROS(2006) [8] ◦ -10.00 ◦ ), 1.75 ◦ ± ◦ ,10.00 ◦ ), 2.26 ◦ ± ◦ ,10.00 ◦ ), 2.76 ◦ ± ◦ ,10.00 ◦ ), 3.23 ◦ ± ◦ ,10.00 ◦ ), 4.45 ◦ ± [9] ◦ ,5.00 ◦ ), -1.40 ◦ +1 . − . +22 . − . +25 . − .
52 (-5.00 ◦ ,5.00 ◦ ), -1.77 ◦ +1 . − . +14 . − . +7 . − .
70 (-5.00 ◦ ,5.00 ◦ ), -2.26 ◦ +0 . − . +13 . − . +6 . − .
75 (-5.00 ◦ ,5.00 ◦ ), -2.76 ◦ +0 . − . +9 . − . +3 . − .
67 (-5.00 ◦ ,5.00 ◦ ), -3.25 ◦ +0 . − . +7 . − . +2 . − .
58 (-5.00 ◦ ,5.00 ◦ ), -3.75 ◦ +0 . − . +5 . − . +2 . − .
43 (-5.00 ◦ ,5.00 ◦ ), -4.25 ◦ +0 . − . +4 . − . +2 . − .
22 (-5.00 ◦ ,5.00 ◦ ), -4.74 ◦ +0 . − . +3 . − . +1 . − .
16 (-5.00 ◦ ,5.00 ◦ ), -5.23 ◦ +0 . − . +4 . − . +2 . − . ◦ ,5.00 ◦ ), -5.72 ◦ +0 . − . +5 . − . +3 . − . ◦ ,5.00 ◦ ), -6.23 ◦ +0 . − . +19 . − . +11 . − . RCG 200 16 (-5.00 ◦ ,5.00 ◦ ), -1.69 ◦ +1 . − . +2 . − . +13 . − .
16 (-5.00 ◦ ,5.00 ◦ ), -2.26 ◦ +1 . − . +2 . − . +11 . − .
11 (-5.00 ◦ ,5.00 ◦ ), -2.76 ◦ +0 . − . +1 . − . +7 . − .
14 (-5.00 ◦ ,5.00 ◦ ), -3.25 ◦ +0 . − . +1 . − . +7 . − .
11 (-5.00 ◦ ,5.00 ◦ ), -3.75 ◦ +0 . − . +1 . − . +7 . − . ◦ ,5.00 ◦ ), -4.25 ◦ +0 . − . +0 . − . +5 . − . ◦ ,5.00 ◦ ), -5.15 ◦ +0 . − . +0 . − . +3 . − . ∗ The values of Galactic latitude ( l ) and Galactic longitude ( b ) shown in Table 3 are average position of the map or average Galacticlatitude of field in each Galactic longitude bin. Note [1]: (Udalski et al. 1994), [2]: (Alcock et al. 1997), [3]: (Alcock et al. 2000), [4]: (Afonso et al. 2003), [5]: (Sumi et al. 2003), [6]:(Popowski et al. 2005), [7]: (Sumi et al. 2006), [8]: (Hamadache et al. 2006), [9]: (Sumi et al. 2013)
The MOA-II team parameterise the observed spatial mi-crolensing distribution using a polynomial function. We cando likewise for our simulated maps. We model the structureof the optical depth, average time scale and event rate mapsshown in Figure 5 using a 10-parameter cubic polynomial fitin l and b . The model function can be written as, x = a + a l + a b + a l + a lb + a b + a l + a l b + a lb + a b , (14)where x is the microlensing observable (rate, time-scale oroptical depth). The best-fit models are shown in Figure 12and the model parameters are provided in Table 6. The bestfit models agree to within 20% of the exact model value for | b | < ◦ . A new version of the Besan¸con Galactic model is used to sim-ulated microlensing optical depth, average timescales andmicrolensing event rate maps towards the Galactic bulge. The new model incorporates a refined two-component bulge(Robin et al. 2012). We perform a detailed comparison ofthe model with the recent optical depth study by MOA-II(Sumi et al. 2013) based on 474 events. The MOA-II ob-servational filter, time scale cut and Gaussian kernel areapplied to the maps. This is the first detailed field-by-fieldcomparison between a theoretical microlensing model and alarge-scale microlensing dataset.In its original form the model overestimates the averagetime scale compared to the survey because the model lackslow-mass stars. Allowing for an extension of the model stel-lar mass function into the low mass star and brown dwarfregime, we find that the model correctly produces the ob-served average event timescale provided the mass function isessentially cut off at the hydrogen burning limit. The shapeof the observed timescale distribution shows weak evidencesof an excess of short (0 . < t E < < t E <
200 days) duration events and a deposit ofmoderate duration events (2 < t E <
30 days). However, themodel provides satisfactory match with MOA-II distribution(reduced χ (cid:39) . c (cid:13) , 1–17 esan¸con model analysis of MOA-II microlensing Table 4.
The Besan¸con model microlensing optical depth and event rate within | l | < ◦ using 0 . ◦ bins in b . (cid:104) b (cid:105) N event τ Γ deg Γstarevents × yr − deg − × − yr − star − Resolved source -1.40 ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± DIA source -1.40 ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± ◦ ± ± ± Table 6.
The best fit model parameter of the Besan¸con microlensing parameters. τ Res τ DIA (cid:104) t (cid:105) Res (cid:104) t (cid:105) DIA Γ deg , Res Γ deg , DIA Γ star , Res Γ star , DIA × − × − days days yr − deg − yr − deg − × − yr − star − × − yr − star − a
256 321 21.7 21.5 101 477 305 378 a -14.8 -16.9 -0.151 -0.260 -11.0 -38.9 -17.2 -19.0 a a -4.21 -4.52 0.121 0.126 -3.99 -10.3 -5.63 -5.89 a -7.85 -8.70 0.012 -0.043 -7.49 -22.9 -9.42 -10.1 a -3.84 0.464 0.817 0.631 -12.9 -1.02 -6.93 -0.191 a a -0.720 -0.771 0.016 0.016 -0.735 -1.88 -0.965 -1.00 a -1.02 -1.12 0.004 -0.004 -1.08 -3.13 -1.25 -1.32 a -0.647 -0.408 0.056 0.042 -1.31 -1.10 -1.00 -0.570 event rate is found to lie between the predicted number ofevents from the Besan¸con model for pure resolved sourcesand DIA sources. The number of Besan¸con microlensingevents with resolved sources and DIA sources are 0.83 and2.17 times number of MOA-II detected events. Given thatthe model analysis does not include a correction for blend-ing in the number of available sources, and some expecteddifferences due to differences in the assumed bandpass, thisis a reasonable level of agreement.For the optical depth the residual maps between themodel predictions and MOA-II observations show that thereis generally good agreement over most of the MOA-II sur-vey area and that the disagreement is confined to the re-gions closest to the Galactic Centre ( b < ◦ ). The Besan¸conmodel predicts only 50% of the observed optical depth inthis region. Maps of the event rate per star maps also showa similar disagreement. The fact that there is reasonable agreement in the maps of average duration but disagreementwith the rate and optical depth argues for a mass deficit inthe current bulge model.The bulge mass employed in the current Besan¸conmodel is somewhat lower than inferred in some recent stud-ies such as Portail et al. (2015), who argue that a bar witha mass in the range (1 . − . × M (cid:12) is compatiblewith recent radial velocity and proper motion studies. Sucha massive bar is likely required to solve the optical depth dis-crepancy reported here. It remains to be seen whether such amodel can be straightforwardly accommodated within a fullpopulation synthesis code. However, Robin et al. (2012) ar-gue that the dust map model is likely to be under-estimatedin the innermost regions due to incompleteness of 2MASSstar counts below K (cid:39)
12. They also identify a missing pop-ulation within the inner ∼ ◦ in their model based on starcount residuals. The additional population, along with in- c (cid:13) , 1–17 S. Awiphan, E. Kerins and A. C. Robin
Resolved sources DIA sources (a) Optical depth(b) Average time scale(c) Microlensing event rate per square degree(d) Microlensing event rate per star
Figure 10.
The optical depth (a), average time scale (b), microlensing event rate per square degree (c) and microlensing event rate perstar (d) residual maps of resolved sources (left) and DIA sources (right) from the Besan¸con Galactic model and the MOA-II survey data.Contour level shows zero residual value. creased extinction in this region should permit an increasedoptical depth without violating star count limits. Furtherwork, perhaps using VISTA data to extend the range of thedust map in the innermost regions of the bulge, should en-able us to refine the Besan¸con model. We are now in an era of large-scale microlensingdatasets which will play a pivotal role in our understandingof the inner structure of the Galaxy. There is no shortageof pieces to be included within the Galactic jigsaw, but it c (cid:13) , 1–17 esan¸con model analysis of MOA-II microlensing Resolved sources DIA sources (a) Optical depth(b) Microlensing event rate per star
Figure 11.
The reduced χ contribution of optical depth (a) and microlensing event rate per star (b) of resolved sources (left) and DIAsources (right). Table 5.
Field-by-field contributions to the reduced χ ( χ r ) be-tween the Besan¸con model and the MOA-II data for the opticaldepth and microlensing event rate per star.Field τ Res τ DIA Γ star , Res Γ star , DIA gb1 1.40 1.22 3.67 3.00gb2 1.34 0.91 2.47 2.09gb3 1.01 0.45 0.02 0.06gb4 4.31 3.24 5.91 4.78gb5 7.31 5.47 9.41 7.94gb6 0.04 0.01 0.15 0.02gb7 0.28 0.08 0.67 0.44gb8 0.26 0.77 0.84 1.44gb9 7.34 6.35 7.99 7.08gb10 5.14 4.56 5.76 4.91gb11 10 − − gb13 2.77 2.50 2.03 1.76gb14 1.72 1.39 1.74 1.34gb15 3.37 2.75 3.12 2.60gb16 3.98 3.73 2.36 2.13gb17 1.80 1.53 0.45 0.32gb18 0.64 0.32 0.27 0.12gb19 2.12 1.77 1.28 1.02gb20 2.56 2.02 2.91 2.49 χ r Cut
Cut
Cut
Cut
Cut χ r, Cut remains a testing puzzle to assemble them into a fully syn-thesized view of our Galaxy.
ACKNOWLEDGMENTS
The authors acknowledge the anonymous referees for theirvaluable suggestions that have helped to improve this paper.The authors are also grateful to Takahiro Sumi for provid-ing the MOA-II survey data. SA gratefully acknowledges thesupport from the Thai Government Scholarship and the Uni-versity of Manchester President’s Doctoral Scholar Award.
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