THE ASTRONOMICAL JOURNAL, 117 :2244È2267, 1999 May1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

STAR FORMATION HISTORIES FROM HUBBL E SPACE T EL ESCOPE COLOR-MAGNITUDEDIAGRAMS OF SIX FIELDS OF THE LARGE MAGELLANIC CLOUD1

KNUT A. G. OLSEN

Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatories,2 Casilla 603, La Serena, ChileReceived 1998 December 4 ; accepted 1999 January 20

ABSTRACTWe present results on the analysis of background Ðeld stars found in Hubble Space T elescope Wide

Field Planetary Camera 2 observations of six of the old globular clusters of the Large Magellanic Cloud.Treated as contaminants by the globular cluster analysts, we produce (V [I, V) color-magnitude dia-grams (CMDs) of the Ðeld stars and use them to explore the LMCÏs star formation history. The photom-etry approaches V D 26, well below the turno† of an ancient (D14 Gyr) LMC population of stars. TheÐeld star CMDs are generally characterized by an upper main sequence broadened by stellar evolution,an old red giant branch, a prominent red clump, and an unevolved lower main sequence. The CMDsalso contain a few visual di†erences, the most obvious of which is the smeared appearance of the NGC1916 Ðeld caused by heavy di†erential reddening. More subtly, the base of the subgiant branch near theold turno† appears extended in V , and the red giant branch appears broad in V [I in four of the Ðeldsbut not in the NGC 1754 Ðeld. We use a maximum likelihood technique to Ðt model CMDs drawn fromBertelli et al. isochrones to the observed CMDs. We constrain the models by the age-metallicity relationderived from LMC clusters, test four initial mass function (IMF) slopes, and Ðt for the reddening, dis-tance modulus, and star formation rate. We Ðnd that we can just resolve structure in SFR(t) with timesteps of D0.15 in log (age), implying a resolution of D4 ] 108 yr at an age of 1 Gyr. For a SalpeterIMF, our derived star formation history for the NGC 1754 Ðeld is characterized by an enhanced starformation rate over the past 4 Gyr, qualitatively resembling that derived by others for a variety of LMCÐelds. The remaining four Ðelds, however, appear to have had high levels of star formation activity aslong as 5È8 Gyr ago ; these Ðelds lie in the LMC bar whereas the NGC 1754 Ðeld lies in the disk, sug-gesting that the inner regions of the LMC contain signiÐcantly more older stars than the outer regions.Examining the residuals of the models and observations, we Ðnd that the old red giant branches of themodels provide a poor Ðt to the observations, which suggests an error in the model isochrones. Thee†ect of the disagreement appears to be to underestimate the contribution of the old population.Key words : galaxies : evolution È galaxies : stellar content È Magellanic Clouds È stars : formation

1. INTRODUCTION

Because of both its proximity and its unique nature, theLarge Magellanic Cloud is an excellent laboratory forstudying the e†ects of a variety of processes thought toa†ect the formation of stars, e.g., interactions with neigh-boring galaxies, the inÑuence of bar dynamics, stellar feed-back, and metallicity of the environment. Thus, knowledgeof the star formation history of the LMC is a step towardunderstanding the physical origin of stellar populations.This star formation history is recorded in the LMCÏsnumerous clusters and Ðeld stars. Because clusters are rela-tively easy to study, measuring their age and metallicitydistribution is a natural way to study the star formationhistory of the LMC. Researchers have shown that theLMCÏs clusters are predominantly younger than 4 Gyr inage, with only a small number of old (D14 Gyr) globularclusters and one cluster, ESO 121-SC03, with an age of D8Gyr (Mateo, Hodge, & Schommer 1986).

While the cluster results suggest that the LMC formedthe bulk of its stars over the past 4 Gyr, recent observations

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ1 Based on observations made with the NASA/ESA Hubble Space T ele-

scope, obtained at the Space Telescope Science Institute, which is operatedby the Association of Universities for Research in Astronomy, Inc., underNASA contract NAS 5-26555.

2 The National Optical Astronomy Observatories are operated by theAssociation of Universities for Research in Astronomy, Inc., under coo-perative agreement with the National Science Foundation.

of LMC Ðeld stars taken with the Hubble Space T elescope(HST ) imply that stars formed even during the lull in clusterformation, assuming a standard Salpeter initial mass func-tion (Holtzman et al. 1997, hereafter H97 ; Geha et al. 1998).These provocative results open a number of questions, suchas what might cause a switch between cluster and Ðeld starformation modes, and indicate that Ðeld stars need to beaccounted for in a complete description of the LMCÏs starformation history. Moreover, as a complete record of starformation is available in a single pointing and Ðeld starshave an almost continuous age distribution, Ðeld obser-vations appear ideally suited for identifying major star for-mation events and for mapping spatial variations in the starformation history, both keys to understanding the physicalorigin of stellar populations.

The clear disadvantage of using Ðeld stars rather thanclusters to study the star formation history, however, is thatthe information they hold is difficult to extract. This diffi-culty was recognized by M. Schwarzschild, who wrote,““ The random sample represented by the nearby stars cer-tainly includes a large range of masses, possibly somevariety in initial composition, and most likely a greatmixture of stellar ages. A great variety of evolutionarytracks therefore is involved . . . and their disentanglementseems somewhat hopeless ÏÏ (Schwarzschild 1958, p. 2).Luckily, modern tools now allow astronomers to attack theproblem with brute force on a scale that Schwarzschildcould not have done. Observational facilities such as HST

2244

LMC STAR FORMATION HISTORIES 2245

and a variety of excellent ground-based telescopes gatherdata efficiently and accurately, while the stellar evolution-ary models used in comparison with the color-magnitudediagrams (CMDs) have substantially improved (see, e.g.,Schaller et al. 1992 ; Bertelli et al. 1994, hereafter B94 ; Cha-boyer & Kim 1995 ; VandenBerg 1997). Aided by a numberof techniques that take advantage of easily available fastcomputing facilities, work on the star formation histories ofthe Magellanic Clouds and of other Local Group galaxies isproceeding at a rapid pace (e.g., Gallart et al. 1996 ; Apari-cio, Gallart, & Bertelli 1997 ; Dohm-Palmer et al. 1998 ;Tolstoy et al. 1998), often with dramatic results (e.g.,Smecker-Hane 1997).

Our present work seeks to add to the growing body ofknowledge of the star formation history of the LMCthrough observations of six Ðelds taken with HST . Theseobservations were taken with the primary purpose ofproducing CMDs of the LMCÏs inner old globular clusters(Olsen et al. 1998, hereafter Paper I). However, the imagesalso contain signiÐcant numbers of Ðeld stars ; viewed ascontaminants in Paper I, they are the focus of this paper.The locations of the Ðelds are shown in Figure 1 ; Ðve ofthem lie in the bar while the other lies in the disk. As theÐelds are well separated, the observations are nicely situatedfor exploring spatial variations in the star formationhistory. In particular, we will address the question ofwhether the bar and disk have had di†erent star formationhistories, as would be expected if the bar induces star forma-tion or if the event that formed the bar also triggered asigniÐcant star-forming episode. While the majority ofpublished LMC Ðeld star studies have avoided the bar

because of the severe e†ects of photometric crowding, theresolution a†orded by HST enables us to produce CMDsthat include stars well below the oldest main-sequenceturno† even in the bar.

In the following we brieÑy describe the observations andreductions in ° 2, referring the bulk of this discussion toPaper I. Section 3 describes the method used to extract thestar formation histories. The method, which is based on thework of Dolphin (1997), determines the star formationhistory by Ðtting a linear combination of model isochronesto the histogram of points in the CMD (the ““ Hessdiagram ÏÏ), accounting for observational errors. Tests of themethod using simulated CMDs are described in ° 4. Section5 discusses the application of the method to the observedCMDs and evaluates its success. We present our conclu-sions in ° 6.

2. OBSERVATIONS AND REDUCTIONS

As described in Paper I, observations of the six LMCglobular clusters NGC 1754, 1835, 1898, 1916, 2005, and2019 were taken during Cycle 5 of HST using the WideField Planetary Camera 2 (WFPC2) and the F555W andF814W Ðlters. By taking both long and short exposuresthrough each Ðlter, we were ensured reliable photometry ofthe brightest stars in the Ðelds while still reaching V Z 25.We centered the clusters on the Planetary Camera, asshown in the sample image (Fig. 2), leaving the Wide Field(WF) chips containing primarily Ðeld stars.

Paper I fully describes the image reductions, photometry,and production of CMDs. In summary, cosmic-ray rejec-tion, correction for the y-dependent charge transfer effi-

FIG. 1.ÈPositions of the target LMC Ðelds with respect to the parent galaxy. Plate courtesy of Harlow Shapley, Harvard College Observatory, BoydenObservatory photograph. [See the electronic edition of the Journal for a color version of this Ðgure.]

2246 OLSEN Vol. 117

FIG. 2.È““ True color ÏÏ WFPC2 mosaic formed from the F555W and F814W frames of NGC 2019. The Wide Field chips contain mostly Ðeld stars and thePlanetary Camera mostly cluster stars, although there is signiÐcant overlap.

ciency e†ect (Holtzman et al. 1995), geometric distortioncorrection, and bad-pixel Ñagging were applied to theimages before performing photometry. For the photometry,we used version 2.5 of the proÐle-Ðtting photometricpackage DoPHOT (Schechter, Mateo, & Saha 1993), modi-Ðed by E. Deutsch to handle Ñoating-point images. For theWF chips, aperture corrections were applied as a functionof position in the frame using measurements of isolatedbright, unsaturated stars taken in the proÐle-subtractedimages. In agreement with the report of Whitmore & Heyer(1997), we found an o†set in mean magnitudes between the

short- and long-exposure photometry. We corrected for thise†ect by adjusting the short-exposure magnitudes to match,on average, those of the long exposures. Finally, we mergedthe short- and long-exposure lists of photometry, trans-formed the magnitudes from the WFPC2 system toJohnson V /Kron-Cousins I, and produced CMDs bymatching stars with measurements in both V and I by posi-tion. Table 1 contains the photometry.

Figure 3 shows the CMDs obtained from the combinedWF frames of each Ðeld. The main features of the CMDs area lower main sequence seen to V D 25.5, a broad upper

No. 5, 1999 LMC STAR FORMATION HISTORIES 2247

TABLE 1

A. NGC 1754 WIDE FIELD CAMERA PHOTOMETRY

Star X (V ) Y (V ) V pV

a I pIa V Typeb I Typeb Removed?c

0 . . . . . . 214.82 76.38 20.2346 0.0330 20.1296 0.0390 1 11 . . . . . . 184.19 81.18 19.5396 0.0360 19.4586 0.0330 1 12 . . . . . . 273.77 82.37 20.2416 0.0300 20.1716 0.0340 1 13 . . . . . . 96.77 91.67 19.3856 0.0240 18.4896 0.0250 1 14 . . . . . . 506.60 105.65 19.3696 0.0300 18.2548 0.0260 1 105 . . . . . . 513.20 109.20 19.8426 0.0360 18.8726 0.0310 1 1 y6 . . . . . . 296.19 110.51 20.0626 0.0270 19.6656 0.0250 1 17 . . . . . . 119.41 118.82 19.1496 0.0250 19.1116 0.0220 1 18 . . . . . . 247.40 127.18 19.6766 0.0350 19.2936 0.0290 1 19 . . . . . . 238.33 132.13 19.5236 0.0310 18.6766 0.0260 1 1 y

NOTE.ÈTables 1AÈ1F are presented in their entirety in the electronic edition of the Astronomical Journal. A portionis shown here for guidance regarding their form and content.

a Photometric errors reported by DoPHOT.b Where the short-exposure photometry was used, the DoPHOT object type has been multiplied by 10.c Stars removed by cluster-star cleaning procedure are marked with ““ y. ÏÏ

main sequence, a prominent red clump containingintermediate-age horizontal-branch stars, and an old redgiant branch (RGB). The stars falling to the blue of themain-sequence envelope at V D 21 in the NGC 1835, 2005,

TABLE 2

EXPECTED CLUSTER CONTAMINATION

N(cluster stars)/ N(oldest bin)/Field N(cluster stars) N(total) N(total)

NGC 1754 . . . . . . 2031 0.40 0.51NGC 1835 . . . . . . 708 0.07 0.48NGC 1898 . . . . . . 2353 0.23 0.29NGC 2005 . . . . . . 1130 0.10 0.25NGC 2019 . . . . . . 1000 0.08 0.18

and 2019 Ðelds could be blue horizontal-branch stars ;however, most old, metal-poor horizontal-branch stars willbe difficult to distinguish from the broad upper mainsequence. While the old RGB in the NGC 1754 Ðeld forms atight sequence, in the other Ðelds (excluding NGC 1916) thegiant branch and subgiant branch (SGB) appear broader,suggesting a range of turno† ages. The NGC 1916 ÐeldCMD looks remarkably di†erent from the others, havinga much broader lower main sequence and RGB anda stretched-out red clump. As was discussed in Paper I,this appearance is most likely the result of strong di†er-ential reddening, clear evidence of which is seen in ground-based images of the region (A. Walker 1997, privatecommunication). While we cannot rule out a contributionfrom signiÐcant depth in the Ðeld stars local to NGC 1916,the simplest hypothesis is that di†erential reddening

TABLE 3

PARAMETERS OF MODEL STAR FORMATION HISTORIES

E(B[V ) (m[ M)0 Fields Modeled E(B[V ) (m[ M)0 Fields Modeled

IMF x \ 1.5 : IMF x \ 2.35 :0.02 . . . . . . 18.4È18.7 NGC 1835 0.02 . . . . . . 18.4È18.7 NGC 18350.04 . . . . . . 18.3 NGC 2005 0.04 . . . . . . 18.3 NGC 2005

18.4È18.7 All 18.4È18.7 All0.06 . . . . . . 18.3 NGC 2005 0.06 . . . . . . 18.3 NGC 2005

18.4È18.7 All 18.4È18.7 All0.08 . . . . . . 18.3 NGC 2005 0.08 . . . . . . 18.3 NGC 2005

18.4È18.7 All 18.4È18.7 All0.10 . . . . . . 18.3 NGC 2005 0.10 . . . . . . 18.3 NGC 2005

18.4È18.7 All 18.4È18.7 All0.12 . . . . . . 18.3 NGC 2005 0.12 . . . . . . 18.3 NGC 2005

18.4È18.7 All 18.4È18.7 AllIMF x \ 2.0 : IMF x \ 3.0 :

0.02 . . . . . . 18.4È18.7 NGC 1835 0.02 . . . . . . 18.4È18.7 NGC 18350.04 . . . . . . 18.3 NGC 2005 0.04 . . . . . . 18.3 NGC 2005

18.4È18.7 All 18.4È18.7 All0.06 . . . . . . 18.3 NGC 2005 0.06 . . . . . . 18.3 NGC 2005

18.4È18.7 All 18.4È18.7 All0.08 . . . . . . 18.3 NGC 2005 0.08 . . . . . . 18.3 NGC 2005

18.4È18.7 All 18.4È18.7 All0.10 . . . . . . 18.3 NGC 2005 0.10 . . . . . . 18.3 NGC 2005

18.4È18.7 All 18.4È18.7 All0.12 . . . . . . 18.3 NGC 2005 0.12 . . . . . . 18.3 NGC 2005

18.4È18.7 All 18.4È18.7 All

NOTE.ÈAn age-metallicity relation of [Fe/H]\ [0.073[age/(109 yr)][ 0.3 was used for all models.

2248 OLSEN Vol. 117

FIG. 3.ÈColor-magnitude diagrams of the combined WF chips of each Ðeld, restricted to those stars with DoPHOT object type 1 (see discussion in Olsenet al. 1998). Each CMD is roughly characterized by a broad upper main sequence, a lower main sequence extending to V D 25.5, and an old red giant branch.The smeared appearance of the NGC 1916 Ðeld is evidence of strong di†erential reddening.

entirely accounts for the anomalous appearance of theNGC 1916 Ðeld CMD.

Some of the old RGB and lower main-sequence stars areexpected to be globular cluster stars spilling over into theWF chips. From the estimated core and tidal radii of the

clusters (Paper I), we predict the numbers of contaminantslisted in Table 2. In Paper I, we described using these esti-mates to iteratively subtract Ðeld stars from the clusterCMDs, and vice versa. Figure 4 shows the Ðeld star CMDsafter the cluster stars have been removed. The NGC 1754

No. 5, 1999 LMC STAR FORMATION HISTORIES 2249

FIG. 4.ÈField star CMDs after statistical subtraction of cluster stars. The subtraction has removed signiÐcant portions of the old RGBs, particularly inthe NGC 1754 and NGC 1898 Ðelds. The subtraction is not perfect, in some cases carving out regions of the CMDs.

and NGC 1898 Ðeld CMDs have been altered signiÐcantly,with the NGC 1754 Ðeld CMD losing most of its RGB,while the others appear relatively unchanged. The subtrac-tion was not perfect, as some regions of the CMDs have achoppy appearance ; it appears that the large number ofcontaminants present in some of the Ðelds bring out the

faults of the subtraction technique. We did not Ðnd thisproblem when subtracting Ðeld stars from the clusterCMDs, as there are fewer contaminating Ðeld stars in thePlanetary Camera. Moreover, the estimated number of con-taminants in the Ðelds depends strongly on the somewhatuncertain tidal radii of the King model Ðts to the cluster

2250 OLSEN Vol. 117

proÐles. For these reasons, we decided it was easier tomodel the Ðeld star CMDs without correcting for contami-nation from the clusters ; all of the star formation historieswe derive thus include a contribution from the globularclusters.

3. METHOD FOR DETERMINING THE STAR

FORMATION HISTORIES

The study of the star formation history of the LMC usingÐeld star CMDs has a history dating to more than 20 yearsago, the most signiÐcant developments of which are sum-marized by Gallagher et al. (1996). The earliest analyseswere conducted by visual examination of the main featuresof the CMDs (the upper main sequence, RGB, and asymp-totic giant branch) and by comparing the CMDs with Ðdu-cial sequences of clusters of known age and metallicity.Bertelli et al. (1992) developed a more sophisticated tech-nique that involved solving for parameters governing thestar formation history of their LMC Ðelds. The solutionswere made by comparing the average density of stars inbroad regions of the observed CMDs with predictions fromstellar evolutionary models. Similar techniques have beendeveloped and reÐned by Gallart et al. (1996), amongothers, and used to study the star formation histories ofLocal Group dwarf galaxies. Recently, H97 and Geha et al.(1998) used the luminosity functions of three LMC Ðelds toÐt several model star formation histories.

A common feature of the methods described in the pre-vious paragraph is that they do not use all of the informa-tion available in the Ðeld star CMDs; they use eitherselected portions of the CMDs or luminosity functions toextract the star formation histories. By disregarding the fulldistribution of colors and magnitudes, those approachesavoid the problems of modeling the uncertain later phasesof stellar evolution, but at a cost in age resolution. Higherresolution is desirable for addressing questions such as thepossible e†ect of interactions on the LMC star formationhistory and for exploring di†erences among CMDs taken indi†erent Ðelds. The theoretical framework for a moredetailed comparison of models and observations has beenassembled by a number of researchers (e.g., Tolstoy & Saha1996 ; Dolphin 1997 ; Ng 1998) but has been mostly untestedin full capacity against observations. As our CMDs are wellpopulated by stars in most of the phases of evolution seen inthe models, and we wish to address the question of whetherthere are di†erences among the star formation histories theyrepresent, our CMDs provide good motivation and a favor-able environment for exploring the potential of a detailedcomparison with the models.

3.1. Principles of the MethodTo Ðrst approximation, the Ðeld star CMDs are a linear

combination of isochrones of several ages and metallicitiesgoverned by a particular star formation rate as a function oftime [SFR(t)], a universal initial mass function (IMF), anage-metallicity relation [Z(t)], a single value of the red-dening [E(B[V )], and a single distance modulus [(m[

Additional factors that may a†ect the appearanceM)0].of the CMDs are the presence of binary stars, the spread inZ at each age t, variation of the IMF, and the spread in

and E(B[V ) within the Ðelds. For this work we(m[ M)0ignore these additional factors ; the Ðrst three are disre-garded for the sake of simplicity. We consider the assump-tion of uniform distance to be a safe one, as the LMC is

known to be thin compared with its distance (Caldwell &Coulson 1986). Also, from the tight appearance of thecluster sequences in Ðve of the six Ðelds (Paper I), theamount of di†erential reddening must be small. For theNGC 1916 Ðeld, however, di†erential reddening is obvi-

FIG. 5.ÈAn interpolated isochrone (dashed line) with log (age)\ 8.05and [Fe/H]D [0.4, shown with adjacent isochrones of log (age)\ 8.0 and8.1 (solid lines). While the interpolation faithfully reproduces the shape ofthe isochrone over the majority of evolutionary phases, it fails on theasymptotic giant branch.

FIG. 6.ÈA linear Ðt to the LMC cluster age-metallicity relation,adapted from Olszewski, Suntze†, & Mateo (1996). At the metal-poor end,the average abundance of the globular clusters studied in Olsen et al. (1998)has been used.

No. 5, 1999 LMC STAR FORMATION HISTORIES 2251

ously strong ; we thus exclude this Ðeld from the currentanalysis.

Given these assumptions, the problem of determining thestar formation history is reduced to constructing a modelCMD whose SFR(t) best reproduces the distribution ofstars in the observed CMD for a given Z(t), IMF, E(B[V ),and One approach to this problem is to con-(m [ M)0.struct model CMDs from a small number of template starformation histories and to compare them with the observedCMDs (see, e.g., Gallart et al. 1996 ; Aparicio et al. 1997).However, in order to take full advantage of the excellentphotometry of LMC stars possible with HST , the numberof templates we would need to construct would be exceed-ingly large. For example, if we wished to extract the starformation history using 10 age bins spanning the range of10 Myr to 15 Gyr with Ðve possible values for the SFRwithin each bin, and we wished to test a small number ofdi†erent IMF slopes, reddenings, and distance moduli, wewould need to construct more than 107 model star forma-tion histories. As each model CMD needs to contain at leastas many individual stars as the observed CMD, our avail-able computing resources would be greatly overmatched.

A more convenient approach is to treat SFR(t) as a freeparameter for which we Ðnd the best solution for a given setof input parameters. With this approach, we constructmodel CMDs representing a constant SFR in each age binof the desired solution, constrained by the input Z(t) rela-tion, IMF, reddening, and distance modulus. By solving forthe coefficients of the linear combination of models thatbest Ðts the distribution of stars of the CMD, we recoverSFR(t). The number of model star formation histories weneed to construct is then vastly decreased and is simplyequal to the number of combinations of Z(t) relations, IMFslopes, reddenings, and distance moduli we wish to testtimes the number of age bins in the solution. An additionaladvantage of this approach is that a Ðt parameter can beused to test the goodness of Ðt of the models, and errors caneasily be calculated from Monte Carlo simulations. Animplementation of such an approach was Ðrst discussed byDolphin (1997), who used s2 as the Ðt parameter and singu-lar value decomposition (Press et al. 1992) to solve for thebest-Ðt star formation history from a suite of models with anumber of di†erent IMFs, reddenings, distance moduli, andabundances. Dolphin applied the method to simulatedCMDs to test the accuracy with which the input parameterscould be recovered but did not apply the method to actualobservations.

We used the principles of this automated approach toconstruct a method to extract the star formation historiesfrom our Ðeld star CMDs. The method consists of fourbasic steps, which we describe in the following subsections.

3.2. Interpolation of IsochronesThe B94 isochrones are well suited for our purposes

because they model every postÈmain-sequence evolutionaryphase, incorporate the recent OPAL opacities (Rogers &Iglesias 1992), and are calculated for V and I, our observedpassbands. Their main drawback is that they are difficult tointerpolate, as they are not tabulated according to commonevolutionary points. To properly model the CMDs, weneeded to interpolate the isochrones in metallicity as well asin age. We performed this interpolation by breaking theisochrones into segments based on the characteristic evolu-tionary phases listed in Tables 7È12 of B94, an electronic

copy of which was kindly provided by G. Bertelli. Each ofthese segments was then broken into a Ðxed number of massbins. A new isochrone could then be interpolated point bypoint from adjacent isochrones, with the segments ensuringthat the isochrone shape was preserved. We encountered afew problems, however. A problem of a minor nature wasthat we discovered that the V [I and V for the masses ofthe evolutionary phases listed in B94Ïs Tables 7È12 wereoften inconsistent with those tabulated in the isochrones.Instead of using V [I and V from Tables 7È12, we thereforeused the V [I and V from the isochrones for the pointsnearest the masses in the tables. Another small difficultyis that the number of evolutionary points listed in B94ÏsTables 7È12 occasionally changes for isochrones of di†erentages and di†erent metallicities. For isochrones a†ected bythese changes, we used our best judgment in deciding thecorrect mapping. A more serious problem is that largechanges in the curvature of the isochrones do not alwayscorrespond to evolutionary points listed in the tables.Figure 5 illustrates this problem. The solid lines are iso-chrones with values of log (age [yr]) of 8.0 and 8.1 and[Fe/H]D [0.4, while the dashed line is an interpolatedisochrone with log (age) of 8.05. While the interpolated iso-chrone faithfully reproduces the shape of the adjacent onesover most of the isochrone, even on the blue loop, it devi-ates on the asymptotic giant branch. This is because there isno evolutionary point in Table 10 of B94 designating theÐnish of the blue loop and the start of the asymptotic giantbranch. While we found that the interpolated isochroneshave a number of areas where they deviate from the correctshape, the deviations occur in short-lived phases of stellarevolution. We therefore did not attempt to correct the prob-lems, as few of the stars in our observed CMDs will occupythese phases.

3.3. Construction of the Model Star Formation HistoriesWe produced model CMDs using reddenings of

E(B[V ) \ 0.04È0.12 with steps of 0.02, distance moduli of18.4È18.7 with steps of 0.1, and IMF slopes of x \ 1.5, 2.0,2.35, and 3.0, where the IMF is given by the proportionalityN(m) P m~x. The combinations of parameters are sum-marized in Table 3. The ranges of E(B[V ) were chosen tospan the likely range for these Ðelds, the distance moduliinclude both the ““ short ÏÏ and ““ long ÏÏ distance scales of theLMC, and the IMFs were picked to span the range of anumber of derivations of the IMF in the LMC. The age-metallicity relation adopted is linear in age with a slope of[0.073 dex Gyr~1 and a zero point of [0.3 dex. Thismodel, shown in Figure 6, agrees with the age-metallicityrelation suggested by LMC clusters (Olszewski, Suntze†, &Mateo 1996), with ours having a slightly higher metallicityat older ages that reÑects our readjustment of the abun-dances of the inner globular clusters (Paper I) over Olszewskiet al. (1991). However, because of the large gap in theLMC cluster age distribution, our adopted age-metallicityrelation must be regarded as very uncertain over the D5È10Gyr range. Moreover, it is possible that the chemical evolu-tion of the clusters and Ðeld stars is to some degreedecoupled. As discussed by Gallagher et al. (1996) in thecontext of the LMC, the degenerate e†ect of age and metal-licity on stars in the CMD makes the distribution of metalli-cities a critical assumption, particularly for older stars. Lackof knowledge about the Ðeld star metallicities thus necessar-ily weakens our Ðnal conclusions.

-1 0 1 2 3V-I

26

24

22

20

18

16

14

V

a

-1 0 1 2 3V-I

26

24

22

20

18

16

14

V

b

-1 0 1 2 3V-I

26

24

22

20

18

16

14

V

c

-1 0 1 2 3V-I

26

24

22

20

18

16

14

V

d

2252 OLSEN Vol. 117

FIG. 7.È(a) Sample model CMD, showing all 36 age bins of the model at once. The stars have been selected from a Ñat IMF, constrained by theage-metallicity relation of Fig. 6, and given an E(B[V ) of 0.08 and of 18.5. (b) The CMD shown in (a) after application of photometric errors(m [ M)0calculated from artiÐcial-star tests. (c) Hess diagram of (b), scaled logarithmically to bring out the weaker features. Scalings have been applied so that thediagram simulates a Salpeter IMF and a constant star formation rate of 1 yr~1. (d) Same as (c), but scaled linearly. The maximum gray scale is half thatM

_of (c).

We constructed the model CMDs using 36 logarithmicage bins. The bin widths, set to D0.08 in the log, andnumber of bins were chosen to approximately match theresolution with which we could hope to extract the starformation history from the observed CMDs. We chose touse logarithmic age bins to compensate for the fact thatisochrones are spaced more closely together at larger ages,decreasing the resolution with which one can discriminatebetween populations of di†erent ages. Within each age bin,we selected 20,000È30,000 stars with randomly distributedages and masses, with limits on the ages imposed by the binedges and limits on the masses set approximately equal tothe maximum and minimum masses of stars on the iso-chrones within the age bin. The number of stars was chosento greatly exceed the number expected in the observedCMDs so as to limit counting statistics in the models yetbe smaller than the number of artiÐcial stars available forsimulating the errors. In order to increase the sampling of

stars with higher masses, we selected the masses from a ÑatIMF (x \ 0.0) but assigned each star a weight as if it wereselected from the desired IMF of the model. After all starswere selected, we calculated the total weighted mass in starswithin each age bin, adding to this total the integral of theIMF over the range 0.08 but exclud-M

_¹M ¹ 120 M

_ing the mass range occupied by the simulated stars. Thetotal mass represented in the age bin was then used tocalculate the star formation rate within each age bin.

For each model star, we linearly interpolated V [I and Vfrom the nearest isochrones of appropriate age and metalli-city. Photometric errors were applied to these stars throughthe use of the artiÐcial-star tests described in Paper I. Foreach star in the simulated CMD, we randomly selected anearby artiÐcial star. If the artiÐcial star was not recoveredduring photometry, we removed the star from the simulatedCMD. Otherwise, the star was moved according to thephotometric shifts experienced by the artiÐcial star. To

No. 5, 1999 LMC STAR FORMATION HISTORIES 2253

FIG. 8.ÈInput star formation history used in tests of the solutionmethod described in the text.

increase the number of artiÐcial stars from which we couldsimulate the errors, we allowed the program to pick artiÐ-cial stars of a restricted range in magnitude but of any color.Because stars experience systematic photometric shifts thatdepend on color, we corrected for the di†erent systematicshifts before moving the model stars.

We stored the simulated CMDs as Hess diagrams withbins of 0.0625 in V [I and 0.2 in V . At this stage, theweights to bring the CMDs into keeping with the desiredIMF were applied and the histograms were scaled so thateach age bin represents a star formation rate of 1 yr~1.M

_Figures 7aÈ7d show a sample raw simulated CMD, thesimulated CMD with photometric errors, and the binnedCMD, all summed over the 36 age bins.

3.4. Comparison of Model CMDs with Observed CMDsFor deriving SFR(t) and identifying the best-Ðt model, the

s2 parameter tested by Dolphin (1997) is not ideal, as s2 ishighly sensitive to outlying points. We instead chose tominimize the following parameter, which is derived from theLorentzian distribution :

PLor \;i

logG1 ] 1

2CO

i[ M

i(x)

pi

D2H(1)

(Press et al. 1992), where are the observed number ofOistars in the CMD at point i in the (V [I, V ) grid, is theM

i(x)

number of stars predicted at grid point i by the model withparameters x, and is the uncertainty in the number ofp

istars at i, which we calculated from the models assumingPoisson statistics. The Lorentzian, with its large tails, ismuch less sensitive to outliers than is s2. We expect to Ðndoutliers when comparing our observations with the modelsfor several reasons. A primary reason is that the modelisochrones used to generate the model CMDs do not per-fectly describe the process of stellar evolution, especially forthe evolved phases. In addition, we are assuming that asingle reddening, distance modulus, and IMF are appropri-

ate for all of the stars in a single CMD, and that all stars of asingle age have the same metallicity, while in reality therewill be some spread in these quantities. Finally, it is unlikelythat we have exactly reproduced the photometric errorsthrough our artiÐcial-star tests, and there could be signiÐ-cant tails in the photometric distributions that we have notmodeled correctly. However, as long as the bulk of the starsare described by our single parameters and our models ofthe photometric errors apply to most of the stars, theremaining stars may be treated as outliers. For the NGC1916 Ðeld, the assumption of a single reddening is clearlystrongly violated, so that di†erential reddening must beincluded for a viable model Ðt.

3.5. Method for Solution of SFR(t)For each model with di†erent E(B[V ), distance

modulus, and IMF, we can solve for SFR(t) by minimizingequation (1) over all of the model age bins. Because ourmodels are normalized to represent a star formation rate of1 yr~1, the best-Ðt linear combination will produceM

_SFR(t) in units of yr~1 over the area of the Ðeld. AM_partial code for producing the solutions was kindly provid-

ed by A. Dolphin, which we rewrote and amended for ourpurposes. The program minimizes using the NumericalPLorRecipes routine AMOEBA (Press et al. 1992). Given a start-ing guess at the correct values of the Ðtted parameters andthe likely scale of the parameters, AMOEBA searches theN-dimensional parameter space for a minimum in thesurface, stopping when small changes in the parameterschange the value of the function by less than a givenamount, which we set to be slightly larger than the machineprecision. Because AMOEBA stops when a minimum isfound, the minimum may not be the desired globalminimum. To more thoroughly search the surface for aPLorglobal minimum, AMOEBA was restarted several timeswith the new input guess and scale of the parameters setequal to the previous output. After four to Ðve restarts,AMOEBA returned to the same minimum as the previousrun, indicating that no other nearby minimum exists. In thisfashion, we established solutions of SFR(t) for each modelwith di†erent E(B[V ), distance modulus, and IMF. Foreach IMF, the best-Ðt E(B[V ) and distance modulus weredecided by picking the model with the lowest value of PLor.

4. TESTS

Dolphin (1997) has shown, using a method very similar tothe one described in this work, that it is possible to faithfullyextract the star formation history from a simulated CMDrepresenting a typical Local Group dwarf galaxy. An out-standing question, however, is the age resolution with whichthe star formation history may be solved. We explored thisissue, as well as the ability of our method to solve for SFR(t)and identify the correct E(B[V ) and by produc-(m [ M)0,ing a CMD representing a simple star formation historyand running it through the solution procedure. The testcase was constructed by taking a model with a uniform starformation rate and adjusting the star formation rate in eachage bin by a randomly selected amount. To avoid usingpreexisting knowledge of the star formation history to ouradvantage during the solution, the input star formationhistory was kept secret until after the solutions were made.Choosing E(B[V ) \ 0.08, and x \ 2.35(m[ M)0 \ 18.5,for the IMF slope, we simulated a CMD for the test case byselecting 1000 stars in each age bin from the interpolated

2254 OLSEN Vol. 117

surface used to select the best-Ðt parameters of the solution to the star formation history in Fig. 8. The minimum occurs at E(B[V ) \ 0.08FIG. 9.ÈPLorand which were the input values.(m [ M)0 \ 18.5,

B94 isochrones, to which we applied the completeness frac-tions and photometric uncertainties derived for the NGC1754 WF frames, using the artiÐcial-star tests of Paper I.We binned this simulated CMD to the resolution of themodels. We also smoothed both the test CMD and themodel CMDs with a Gaussian having a dispersion of onebin width, as we found that this limited the stochastic e†ectsimposed by the binning. Figure 8 shows the input star for-mation history for the test case.

Making a starting guess at the star formation rate in eachbin, we used four iterations of AMOEBA to Ðnd the best-Ðtcoefficients of the models in each of the 36 age bins and eachvalue of E(B[V ) in the range 0.04[0.12 and in(m[ M)0the range 18.4È18.7, with x \ 2.35. Figure 9 shows the com-puted surface as a function of E(B[V ) andPLor (m [ M)0.The minimum of the surface occurs at E(B[V )\ 0.08 and

in agreement with the input values. The(m[ M)0 \ 18.5,top left panel of Figure 10 shows the star formation historyderived for the minimum compared with the input.PLor,The output contains wild disagreements with the input starformation history, mimicking a ““ bursting ÏÏ-type star forma-

tion history. Guessing that the problem is one of ageresolution, we redid the solutions after grouping several agebins of the models together. The remaining panels of Figure10 show the best-Ðt solutions to the star formation historyafter grouping every two, three, and four models together.These solutions reproduce the input model much better ; thedisagreements are all within the uncertainties calculatedfrom repeated test runs using resampled CMDs, which wedescribe below.

Two factors contribute to the limiting age resolution ofthe star formation history solutions : the number of starsthat fall within each age bin and the spacing of the modelsin the CMD. The Ðrst factor is a function mainly of the totalnumber of stars in the observed CMD, whereas the secondis a function of the age of the models and the photometricerrors. Grouping the models together has the e†ect ofreducing the contribution of both factors ; the number ofstars within each age bin is increased while the spacingbetween the models also increases. To describe the e†ectsthat the number of stars and the model spacing have on theerrors in the solution, we ran repeated tests using the model

No. 5, 1999 LMC STAR FORMATION HISTORIES 2255

FIG. 10.ÈT op left : The thin line shows the star formation history derived from a CMD with stars selected from the proÐle shown by the thick line, using36 logarithmic age bins in the solution. The noisy solution indicates that the achievable age resolution is lower than that which was attempted. T op right,bottom left, bottom right : Solutions using 18, 12, and 9 age bins, respectively, showing the reduction in noise as the number of age bins is decreased. Error barshave been calculated from repeated solutions obtained from resampled CMDs.

star formation history described above with varyingnumbers of stars in the simulated CMDs. Table 4 lists theparameters and number of runs of the tests. For each set ofruns, we calculated the dispersion in the recovered-minus-input star formation rates separately for each age bin of the

solution. The top panel of Figure 11 shows the dispersion intaken as a median over all age bins in theSFRout [ SFRin,solution, plotted against the number of stars per age bin in

the CMDs used in each set of tests. The Ðgure demonstratesthat the median uncertainty in the star formation rates

TABLE 4

PARAMETERS OF TEST STAR FORMATION HISTORY

Stars per Bin afterE(B[V ) (m[ M)0 IMF x Stars per Bin Completeness Correction Runs

0.08 . . . . . . 18.5 2.35 250 D185 10500 D370 10

1000 D740 252000 D1480 103000 D2220 10

0 1000 2000 3000# Stars per age bin

0.0

0.5

1.0

1.5

2.0

Fra

ctio

nal U

ncer

tain

ty

7.0 7.5 8.0 8.5 9.0 9.5 10.0log Age

0.0

0.5

1.0

1.5

Fra

ctio

nal U

ncer

tain

ty2256 OLSEN Vol. 117

FIG. 11.ÈT op : Plot of vs. number of starsp(SFRin [ SFRout)/SFRinper age bin for the test runs described in the text. The error bars indicatethe dispersion about the median which was taken over all age bins ofpSFR,the solution. Bottom : Plot of vs. age, with ap(SFRin [ SFRout)/SFRinseparate line for each run of tests. The dashed, dotted, solid, and dash-dotted lines represent the runs with 3000, 2000, 1000, and 500 stars per agebin, respectively.

increases exponentially with decreasing numbers of starsper age bin, as is expected from Poisson statistics. Thebottom panel of Figure 11 shows that the dispersion aboutthe median over the age bins, indicated by the error bars inthe top panel, is due to the e†ect of isochrone spacing. Atthe youngest and oldest ages, the uncertainty in the recov-ered star formation rate is higher than for intermediate ages,where model isochrones of similar ages are more easily dis-tinguishable. We will bear these e†ects in mind when dis-cussing the age resolution of the star formation historiesderived for the observed CMDs.

5. RESULTS FOR FIELD STAR CMDs AND

ERROR ANALYSIS

5.1. SFR(t), Best-Fit Parameters, and ErrorsThe SFR(t) solutions for the LMC Ðeld star CMDs were

made following the procedure outlined in ° 3. As was done

TABLE 5

BEST-FIT REDDENINGS AND DISTANCES

Field E(B[V ) (m[ M)0NGC 1754 . . . . . . 0.06 ^ 0.01 18.50 ^ 0.05NGC 1835 . . . . . . 0.04 ^ 0.01 18.50 ^ 0.05NGC 1898 . . . . . . 0.06 ^ 0.01 18.40 ^ 0.05NGC 2005 . . . . . . 0.06 ^ 0.01 18.45 ^ 0.05NGC 2019 . . . . . . 0.06 ^ 0.01 18.50 ^ 0.05

for the tests described in ° 4, we binned the observed CMDsto the same resolution as the models and smoothed boththe observed and model color-magnitude grids with aGaussian having a 1 pixel dispersion. We producedseparate solutions for each combination of the parameterslisted in Table 3 and for 18, 12, and 9 age bins. For NGC1835 and NGC 2005, we found that we needed to add addi-tional values of the reddening and distance modulus tobracket the full range of best-Ðt solutions. For each IMF,we found the reddening and distance modulus that bestmatch the observed CMD by choosing the minimum of the

surface, as was done for the test. A sample surface isPLorshown in Figure 12. While the minimum is well deÐned,PLorits value is not as low as those found during the tests, a signthat the model isochrones as input do not agree perfectlywith the observations.

We calculated the errors in SFR(t), E(B[V ), and (m[through Monte Carlo simulations using bootstrappedM)0samples of the observed CMDs. We ran 25 simulations for

each value of the IMF; with each run we resampled theobserved CMDs and produced solutions of SFR(t) for eachvalue of E(B[V ) and exactly as was done for the(m [ M)0,analysis of the true observed CMDs. Again, we chose thebest-Ðt solution from the set by picking the minimum ofthe surface.PLorFigures 13aÈ13c show the derived star formation his-tories and associated errors using 18, 12, and 9 age bins inthe solution and the Salpeter IMF, while Table 5 lists thebest-Ðt values of E(B[V ) and for these solutions.(m[ M)0There are several common features in the derived star for-mation histories. All of the Ðelds show signiÐcant star for-mation in the last 3È4 Gyr with similar amplitude, with theNGC 1754 Ðeld, on account of its lower star density, pro-ceeding at an overall rate D2È3 times lower than the rest.The star formation has the appearance of a burst, peakingat D1 Gyr, in all of the Ðelds. While little can be said aboutthe very young stars because of the small size of the Ðeldsand the close spacing of the young isochrones, there appearsto be a decline in SFR(t) over the last D0.5È1 Gyr. Thisapparent recent decline could simply be evidence that thestars formed in the past 1 Gyr are not thoroughly mixed ;the crossing time in the bar is D80 Myr, implying a mixingtime of a few hundred million years. All of these featuresappear independent of the number of age bins used in thesolution.

The NGC 1754 Ðeld star formation history appears toagree with that derived by H97 for a Ðeld near NGC 1866 ;both show an increase in the recent (\4 Gyr) star forma-tion rate by a factor of D3, although the exact numberdepends on the age resolution of the solution. However, weÐnd that the star formation did not instantly increase but,rather, did so steadily. We do not Ðnd evidence for the short

No. 5, 1999 LMC STAR FORMATION HISTORIES 2257

surface for solutions of the star formation history of the NGC 1754 Ðeld for the Salpeter IMF. The minimum of the surface is atFIG. 12.ÈPLorE(B[V )\ 0.06 and (m[ M)0\ 18.5.

burst of star formation occurring D2 Gyr ago that wasfound by Gallagher et al. (1996).

While the NGC 1754 Ðeld seems to only recently haveformed a large number of stars, star formation in the barÐelds was active as long as D5È8 Gyr ago. The existence ofthis older population in the bar Ðelds naturally explains thedi†erences in RGB and SGB morphology seen in theCMDs, as the bar Ðelds clearly have broader RGBs andSGBs than the NGC 1754 Ðeld. The older population ismost clearly seen in the solution containing nine age bins,but it remains in the solutions with higher age resolution.This result implies that there is a signiÐcant di†erence in thestar formation histories between the inner and outer regionsof the LMC. Because a bar does not conÐne any given starto the bar pattern, we of course cannot directly age-date theLMC bar through its stellar populations. However, theolder population of stars that we see within the bar could bereadily explained by a bar instability triggering star forma-tion as early as 5È8 Gyr ago. This interpretation disagreeswith that of Elson, Gilmore, & Santiago (1997), who havesuggested, on the basis of a WFPC2 CMD of a Ðeld in thebar, that the bar formed D1 Gyr ago, or 1È2 Gyr after theLMC disk. Our results agree with Elson et al. in that we do

Ðnd a peak in the star formation rate occurring D1 Gyr agoin the bar Ðelds ; however, we also see this peak in the diskÐeld. Our interpretation also disagrees with the suggestionof Ardeberg et al. (1997) that the bar may be younger than0.5 Gyr. Again noting that 0.5 Gyr is of the order of themixing time in the bar, it is quite possible that the largeyoung population seen by Ardeberg et al. in their Ðeld rep-resents a local star-forming event. Our results also do notfully agree with those of Hardy et al. (1984), who found thatno signiÐcant star formation had occurred in the bar before3 Gyr ago. However, Hardy et al. may not have been able todetect an older population, as their CMD was not deepenough to reach the turno† and lower main sequence.

An encouraging feature of each of the star formation his-tories of Figures 13aÈ13c is a signiÐcant number of stars inthe last age bin, where we expect the contaminating globu-lar cluster stars to be found. How much of the old stellarpopulations can be attributed to the clusters spilling intothe WF frames? While we found we could not cleanly sta-tistically subtract the cluster stars from the Ðeld star CMDs,we can use the estimates of their numbers derived pre-viously to address the question. In Table 2, we compare thenumber of cluster stars in the WF frames estimated from the

2258 OLSEN Vol. 117

FIG. 13a

FIG. 13.È(a) Star formation histories derived for the HST Ðelds using the method described in the text, adopting 18 age bins in the solution. In each Ðeld,the oldest age bin contains a contribution from the nearby globular cluster. The error bars were calculated from Monte Carlo simulations using bootstrappedsamples of the observed CMDs. The NGC 1754 Ðeld star formation rates are shown on a scale one-third that of the other Ðelds to bring out the detail in itsstar formation history. (b) Star formation histories derived adopting 12 age bins in the solution. (c) Star formation histories derived adopting nine age bins inthe solution. While the loss in resolution over (a) is signiÐcant, most of the salient features remain, and the error bars are smaller.

King model Ðts with the number of stars in the last age binof the star formation histories of Figure 13a. We Ðnd thatthe cluster stars contribute signiÐcantly to each Ðeld, whilein the NGC 1754 and NGC 1898 Ðelds they dominate theoldest bin. This Ðnding leads us to suppose that the narrowRGB of the NGC 1754 Ðeld is produced almost entirely bycluster contamination, whereas the broad RGBs and SGBsof the bar Ðelds truly represent an old Ðeld population.

5.2. Goodness of Fit of the Model Star Formation HistoriesSo far we have implicitly assumed that the best-Ðt model

star formation histories provide good Ðts to the obser-

vations. However, our models make a number of simpliÐca-tions over what is realistically expected ; e.g., we assumeduniform reddening, no binary stars, and a simple linear age-metallicity relation. To obtain an idea of whether thederived star formation histories provide sensible matches tothe data, we overlaid isochrones representing peaks in thederived star formation histories on the Ðeld star CMDs.Figure 14 shows the ages at which the isochrones werechosen and the corresponding matches with the CMDs.The inability to tell by eye whether the chosen peaks in thestar formation rate correspond to increases in the stellardensity in the CMD reinforce the need for an automatedtechnique to derive the star formation histories. We can tell,

No. 5, 1999 LMC STAR FORMATION HISTORIES 2259

FIG. 13b

however, that there is a clear di†erence between the NGC1754 CMD and the CMDs of the bar Ðelds. While the NGC1754 Ðeld appears to have a single old turno†, the bar Ðeldshave many stars D0.5 mag brighter than the oldest turno†,which we interpret as the 4È8 Gyr population in Figures13aÈ13c. Overlaying the isochrones also shows that ouradopted age-metallicity relation does not provide an idealmatch to the CMDs. The most striking disagreement is thatthe RGBs of the older isochrones fall too far to the red tomatch the observed giant branch. While the observed giantbranch could be Ðtted by older isochrones if the distancemodulus were decreased, it would then be difficult to matchthe blue envelope of the upper main sequence, which isreproduced well with the current models. A possible expla-nation is that the metallicity of the model old populations istoo metal-rich. Because the metallicity of the oldest age binof the model was chosen to equal the average abundance ofthe old globular clusters (Paper I), this would imply that theÐeld contains an old population that is considerably more

metal-poor than the clusters. However, because the globu-lar cluster sequences fall quite closely on top of the Ðeld starRGBs, this would mean that the measured globular clusterabundances are too metal-rich by D0.5 dex. As the globularcluster abundances were measured from the slopes of theRGBs, which are empirically calibrated with Milky Wayclusters using high-dispersion spectroscopic abundances,we would be surprised if they were found to be so grosslyincorrect. Another possibility, however, is that the B94 iso-chrones do not correctly predict the color of the RGB forlow metallicities ; our comparison suggests that the modelRGBs are too red by D0.08 mag, given that we have solvedfor the correct reddening and distance modulus of the Ðeldstars.

In Figure 15, we show the residuals produced by themodels of Figure 13a. The residuals are shown as fractionsof the observed star density in the CMDs. The gray scalestretches from [1 to 1 ; white pixels are regions where toomany stars are predicted by the models, black pixels rep-

2260 OLSEN

FIG. 13c

resent an excess of observed stars, and gray pixels representthe regions where the models closely match the obser-vations. The residuals show that over large areas of theCMDs, the models reproduce the observations fairly well.However, as expected from the comparisons with iso-chrones, there are several areas where the Ðts are poor, mostnotably the area of the RGB, the horizontal-branch andred-clump stars, and a small section of the upper mainsequence. These problems are not as apparent if the modeland observed luminosity functions are compared instead ofthe CMDs. These luminosity functions are shown in Figure16 ; the errors in the points include both Poisson countingstatistics and the uncertainties in the derived star formationrates of the models. The use of luminosity functionssmooths over many of the disagreements between themodels and observations ; application of the s2 statisticshows that the models are not strongly excluded, except forthe case of the NGC 1835 Ðeld.

5.3. Exploration of Systematic Errors on the Derived StarFormation Histories

The star formation histories derived in ° 5.1 may beincorrect in detail because of the inconsistent match of themodels and observations. To explore the e†ects that thesystematic disagreements between the models and obser-vations might have on the star formation histories, we setthe reddening and distance modulus to extreme values andexamined the resulting star formation histories. Figure 17ashows the star formation history of the NGC 1835 Ðeld forthe case of 18 age bins in the solution after setting E(B[V )and to extremes. In general, decreasing either(m[ M)0E(B[V ) or increases the relative contribution of(m[ M)0the old stellar populations, while increasing either one hasthe opposite e†ect. This can be understood from Figure 17b,where we plot isochrones having the extreme values ofE(B[V ) and over the CMD. For the low values(m[ M)0

FIG. 14.ÈIn the left column of panels, peaks and features in the star formation histories of Fig. 13a are marked by vertical lines. Isochrones correspondingto the correct ages and metallicities of these features are overlaid on the CMDs in the right column, using the derived reddenings and distance moduli. Whilethe model isochrones reproduce well the envelope of the upper main sequence, they fall too far to the red of the RGBs. [See the electronic edition of theJournal for a color version of this Ðgure.]

2262 OLSEN

FIG. 14.ÈContinued

of E(B[V ) and the old isochrones cover the(m[ M)0,lower main sequence and RGB while the young isochronesdo not Ðt the main sequence as well. For the high values, theold isochrones do not Ðt the turno† and RGB, but theyoung isochrones follow the main-sequence envelope. Wenote that decreasing x, the slope of the IMF, has the samee†ect on the star formation history as decreasing E(B[V )or (see Fig. 18). This is because, with a shallower(m[ M)0model IMF, more evolved stars are needed to reproduce theobserved luminosity function (see also discussion in H97).

Thus, we might expect that the e†ect of the model dis-agreements would be to suppress the old populations in thederived star formation histories. Because the disagreementsare largest for the older stars, we should be able to quantifythis expectation by using our method to solve for the starformation history of one of the globular cluster CMDsshown in Figure 12 of Paper I. By using a cluster CMDwith Ðeld stars statistically removed, we expect to recover astar formation history dominated almost entirely by theoldest bin. In the left panel of Figure 19, we show the starformation history derived for the NGC 2019 Ðeld starÈsubtracted cluster CMD. We have adopted a reddening ofE(B[V )\ 0.06 for the cluster, as derived in Paper I, and adistance modulus of 18.5. Indeed, the star formation historyis characterized by a large peak in the oldest age bin with

almost all of the remaining bins consistent with a star for-mation rate of zero within the errors. In the right panel ofFigure 19, we show the model isochrone having age andmetallicity equal to the average of the oldest bin of thesolution. While the isochrone Ðts the lower main sequenceand turno† luminosity, the model RGB falls too far to thered, in a fashion similar to the Ðts shown in Figure 14, andthe horizontal-branch morphology is not reproduced. WeÐnd that the integral of the model luminosity function overthe range 17 ¹ V ¹ 23 is D50% lower than that of theobserved luminosity function. This result agrees with ourexpectation that the star formation rates in the oldest agebins of Figures 13aÈ13c have been underestimated becauseof the model disagreements. Furthermore, because we donot see signiÐcant star formation in the age range 4È8 Gyrin Figure 19, it appears unlikely that the model disagree-ments could be the cause of the di†erent star formationhistories derived for the NGC 1754 and bar Ðelds.

6. SUMMARY AND CONCLUSIONS

Forty years ago, Martin Schwarzschild remarked on thehopelessness of disentangling a color-magnitude diagram ofstars that contains a wide mix of ages and compositions.Modern computational power, aided by improved astro-nomical facilities, has made the problem no longer unap-

-1 0 1 2 3

26

24

22

20

18

16

14

30

25

20

15

10

V

30

25

20

15

10

-1 0 1 2 3

-1 0 1 2 3

NGC 1754

-1 0 1 2 3

26

24

22

20

18

16

14

30

25

20

15

10

30

25

20

15

10

-1 0 1 2 3

-1 0 1 2 3

NGC 1835

-1 0 1 2 3

26

24

22

20

18

16

14

30

25

20

15

10

V

30

25

20

15

10

-1 0 1 2 3

-1 0 1 2 3

NGC 1898

-1 0 1 2 3

26

24

22

20

18

16

14

30

25

20

15

10

30

25

20

15

10

-1 0 1 2 3V-I

-1 0 1 2 3

NGC 2005

-1 0 1 2 3

26

24

22

20

18

16

14

30

25

20

15

10

V

30

25

20

15

10

-1 0 1 2 3V-I

-1 0 1 2 3

NGC 2019

FIG. 15.ÈResiduals of the best-Ðt model star formation histories of the HST Ðelds and the observed CMDs, adopting a Salpeter IMF. The residuals havebeen divided by the number of observed stars to remove the e†ect of the luminosity function. Bright patches indicate an excess of predicted stars, while darkpatches are regions where there are more stars observed than predicted. Gray areas indicate good agreement between the models and observations. The graylevels have been stretched from [1 to 1.

2264 OLSEN Vol. 117

FIG. 16.ÈComparison of the observed luminosity functions (squares) with the best-Ðt model star formation histories (solid lines), adopting a Salpeter IMF.The error bars on the points are those due to Poisson number statistics and the uncertainties in the derived star formation rates. The Ðgures indicate the s2per degree of freedom for the Ðts and the probability that s2 would exceed the given values by chance.

proachable. In this work, we have applied an automatedtechnique to the task of deriving star formation histories ofÐve Ðelds in the LMC from deep HST CMDs. However,assumptions still need to be made to make the problemtractable. We have assumed throughout that the Ðeld starCMDs can be parameterized by a single reddening, distancemodulus, and IMF; that binary stars may be ignored ; thatthe IMF does not vary in time ; and that a simple linearage-metallicity relation, based on LMC clusters, describesthe enrichment history.

The star formation histories derived for the Ðelds for aSalpeter IMF (Figs. 13aÈ13c) contain a number of featuresthat match our commonsense expectations. All of the Ðeldshave a signiÐcant number of old stars conÐned to the last

age bin. As these Ðelds lie within 1@ of Ðve old globularclusters, contamination from globular cluster stars isexpected ; in two of the Ðelds, the contamination canaccount for the entire population of oldest stars detected inthe Ðeld.

All of the Ðelds show signiÐcant recent Gyr) star([3È4formation, in agreement with a large number of other LMCÐeld star studies and with the LMC cluster age distribution.In particular, the star formation history we derive for theNGC 1754 Ðeld agrees well with that derived by H97 for aÐeld near NGC 1866, once the contribution due to contami-nating globular cluster stars is ignored. The decline in thestar formation rate between the ages of 0.5 and 1 Gyr seenin each of the Ðelds is naturally explained by inadequate

No. 5, 1999 LMC STAR FORMATION HISTORIES 2265

FIG. 17a

FIG. 17.È(a) In these four plots, E(B[ V ) and have been set to extreme values and the e†ect on the NGC 1835 Ðeld star formation history(m [ M)0shown. In general, decreasing E(B[ V ) or increases the contribution of older stars, while increasing these parameters has the opposite e†ect. (b)(m [ M)0Comparison of selected model isochrones with the NGC 1835 Ðeld CMD for the combinations of E(B[ V ) and shown in (a) [see the electronic(m [ M)0edition of the Journal for a color version of this Ðgure].

mixing of stars formed most recently ; the crossing time inthe bar is D80 Myr, implying a mixing time of a fewhundred million years.

We Ðnd that the four bar Ðelds (NGC 1835, 1898, 2005,and 2019) experienced signiÐcantly more star formation inthe age range 4È8 Gyr than did the NGC 1754 Ðeld, whichlies in the disk ; this is most clearly seen in the solutioncontaining nine age bins (Fig. 13c). The existence of thispopulation of stars unique to the bar naturally explains thebroader giant branches and the existence of the stars 0.5mag brighter than the turno† in each of the bar Ðelds butnot in the NGC 1754 Ðeld. This result implies that theLMCÏs inner region is older than has been suggested, bye.g., Elson et al. (1997) and Ardeberg et al. (1997).

However, we also Ðnd disagreements between the best-Ðtmodels and observation ; the biggest disagreement is in theregion of the RGB, where the model sequences fall D0.08mag to the red of the observed sequence. To explore theconsequences of this disagreement, we set the reddeningsand distances of the models to extreme values and examinedthe resulting star formation histories. In addition, we exam-ined the success of our method in deriving the star forma-tion rate of the NGC 2019 cluster CMD. Both explorationssuggest that the model disagreements cause us to underesti-mate the contribution of the old populations. Thus, weexpect that the star formation histories of Figures 13aÈ13care roughly correct within the assumptions made about theIMF, the e†ect of binary stars, and the age-metallicity rela-

2266 OLSEN Vol. 117

FIG. 17b

tion. Of these assumptions, the age-metallicity relation is onthe weakest footing because of poor knowledge of LMCmetallicities in the D5È10 Gyr range.

While we think the method used here to derive star for-mation histories has potential, it can clearly be considerablyimproved. Helpful next steps would be to include binarystars in the models, explore the e†ect of di†erent models ofchemical evolution, and test additional sets of isochrones.Future analyses will hopefully beneÐt from future improvedknowledge of LMC metallicities and from better under-standing of the later phases of stellar evolution.

I would like to thank Paul Hodge for his guidancethroughout this project. I also thank Bob Schommer, NickSuntze†, Alistair Walker, and Chris Smith for helpful dis-cussions and for thorough readings of the original manu-script. I thank the anonymous referee for several excellentcomments that improved the manuscript. Andy Dolphinprovided the routines around which I built the program toextract the star formation histories. This work was sup-ported by STScI grant GO-5916 to Nick Suntze†.

No. 5, 1999 LMC STAR FORMATION HISTORIES 2267

FIG. 18.ÈPlots showing the e†ect of setting the slope of the IMF, x, to extremes on the star formation history of the NGC 1835 Ðeld. Lowering x increasesthe relative contribution of the old populations, while raising x emphasizes the young stars.

FIG. 19.ÈStar formation history derived by using the method employed for the Ðeld star CMDs on the CMD of the NGC 2019 cluster. As expected, thestar formation history is dominated by the oldest age bin, while most of the remaining age bins have star formation rates consistent with zero within theuncertainties. [See the electronic edition of the Journal for a color version of this Ðgure.]

REFERENCESAparicio, A., Gallart, C., & Bertelli, G. 1997, AJ, 114, 669Ardeberg, A., Gustafsson, B., Linde, P., Nissen, & P.-E. 1997, A&A, 322,

L13Bertelli, G., Bressan, A., Chiosi, C., Fagotto, F., & Nasi, E. 1994, A&AS,

106, 275 (B94)Bertelli, G., Mateo, M., Chiosi, C., & Bressan, A. 1992, ApJ, 388, 400Caldwell, J. A. R., & Coulson, I. M. 1986, MNRAS, 218, 223Chaboyer, B., & Kim, Y.-C. 1995, ApJ, 454, 767Dohm-Palmer, R. C., et al. 1998, AJ, 116, 1227Dolphin, A. 1997, NewA, 2, 397Elson, R. A. W., Gilmore, G. F., & Santiago, B. X. 1997, MNRAS, 289, 157Gallagher, J. S., et al. 1996, ApJ, 466, 732Gallart, C., Aparicio, A., Bertelli, G., & Chiosi, C. 1996, AJ, 112, 1950Geha, M. C., et al. 1998, AJ, 115, 1045Hardy, E., Buonanno, R., Corsi, C. E., Janes, K. A., & Schommer, R. A.

1984, ApJ, 278, 592Holtzman, J. A., et al. 1995, PASP, 107, 156ÈÈÈ. 1997, AJ, 113, 656 (H97)Mateo, M., Hodge, P., & Schommer, R. A. 1986, ApJ, 311, 113Ng, Y. K. 1998, A&AS, 132, 133Olsen, K. A. G., Hodge, P. W., Mateo, M., Olszewski, E. W., Schommer,

R. A., Suntze†, N. B., & Walker, A. R. 1998, MNRAS, 300, 665 (Paper I)

Olszewski, E. W., Schommer, R. A., Suntze†, N. B., & Harris, H. C. 1991,AJ, 101, 515

Olszewski, E. W., Suntze†, N. B., & Mateo, M. 1996, ARA&A, 34, 511Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992,

Numerical Recipes in C (2d ed. ; Cambridge : Cambridge Univ. Press)Rogers, F. J., & Iglesias, C. A. 1992, ApJS, 79, 507Schaller, G., Schaerer, D., Meynet, G., & Maeder, A. 1992, A&AS, 96, 269Schechter, P. L., Mateo, M., & Saha, A. 1993, PASP, 105, 1342Schwarzschild, M. 1958, Structure and Evolution of the Stars (New York :

Dover)Smecker-Hane, T. 1997, in AIP Conf. Proc. 393, Star Formation Near and

Far, ed. S. S. Holt & L. G. Mundy (Woodbury, NY: AIP), 571Tolstoy, E., et al. 1998, AJ, 116, 1244Tolstoy, E., & Saha, A. 1996, ApJ, 462, 672VandenBerg, D. A. 1997, in IAU Symp. 189, Fundamental Stellar Proper-

ties, ed. T. R. Bedding, A. J. Booth, & J. Davis (Dordrecht : Kluwer), 439Whitmore, B., & Heyer, I. 1997, New Results on Charge Transfer Effi-

ciency and Constraints on Flat-Field Accuracy (WFPC2 Instrum. Sci.Rep. 97-08) (Baltimore : STScI)