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Paper text:
the local dwarf spheroidal galaxies form a sample of small galaxies which , due to their relatively nearby locations and close association with the milky way could in principle furnish crucial observational and theoretical information on a range of astrophysical phenomena . as these systems are eventually disrupted and incorporated into the milky way they illustrate locally one of the mechanisms thought to be responsible for the build up of large galaxies . thus comparing their stars and those now found in our galaxy we can obtain a first estimate of the relevance of late mergers ( unavane et al .
dynamical studies of their stars have yielded valuable constraints on the nature and structure of dark matter halos at the smallest scales ( lin & faber 1983 , gerhard & spergel 1992 ) .
their orbits probe the galactic halo in a range of distances not sampled by any other objects and can thus be used to study the outer galactic halo .
additionally , their small sizes make them in principle the simplest galactic systems , where key processes such as star formation and gas flows can be studied under relatively well defined conditions .
however , this situation is complicated by the fact that we not only lack a theoretical understanding of these systems , but also an observational record of their evolution ; only a present day snapshot of their physical parameters is available , as is the case with most galactic systems . whilst high redshift observations have recently opened up new areas of research as they begin to yield an statistical description of the evolution of bright galaxies , such an approach is likely to remain out of reach for these small systems for some time .
fortunately , their neighboring locations allow the study of their individual stars , which offers the possibility of directly probing their evolutionary histories by inferring star formation rates as a function of time , @xmath2 .
the recent availability of detailed colour - magnitude diagrams for several nearby systems has prompted the development and allowed the application of careful statistical methods aimed at reconstructing the star formation histories of these objects ( e.g. chiosi et al .
( 1989 ) , aparicio et al .
( 1990 ) and mould et al .
( 1997 ) using magellanic and local clusters , and mighell & butcher ( 1992 ) , smecker - hane et al .
( 1994 ) , tolstoy & saha ( 1996 ) , aparicio & gallart ( 1995 ) and mighell ( 1997 ) using local dsph s ) .
although much has been learnt of the complex @xmath2 of these systems , existing studies have lacked two major ingredients : a homogeneous set of observations including several of the dsph galaxies does not exist , and different data sets are generally analyzed using different techniques .
these two points make comparisons between the derived @xmath2 at best risky .
a further difficulty lies in the fact that the available rigorous statistical studies approach the problem parametrically , which is something one should try to avoid when the actual structure of the function one is trying to recover can be crucial , as is the case when the underlying physics is unknown .
an example of this last point is the case of the carina dwarf .
hurley - keller et al . ( 1998 ) solve for the best fitting three discrete bursts solution to the @xmath0 and conclude that star formation has proceeded spasmodically , whilst mighell ( 1997 ) uses a non parametric star count approach , albeit not a fully consistent statistical method , to obtain a more gradual solution for carina s @xmath0 .
in this paper we have attempted to improve on the determination of the star formation histories of local dsph systems by addressing the two points mentioned above .
we use recent hst observations of the resolved populations of a sample of dsph galaxies ( carina , leoi , leoii , ursa minor and draco ) uniformly taken and reduced , to recover the @xmath0 of each , applying a new non - parametric maximum likelihood method .
this allows meaningful comparisons to be made , as any systematics , at any level , will affect all our galaxies equally .
the outline of our paper is as follows : in section 2 we discuss the observations , in section 3 we include a brief outline of our method , which was introduced in our paper i ( hernandez et al . 1999 ) .
the results are presented in section 4 , and in section 5 we summarize our results .
the main requirements of our observations were that they should comprise a homogeneous sample of local dsph galaxies , mostly in terms of the data reduction .
only such an internally consistent data set allows robust comparisons between different galaxies , once uniform data reduction and analysis methods are adopted .
we extracted available archive hst data for the carina , leoi , leoii , ursa minor and draco galaxies , and used standard data reduction methods and standard hst calibration numbers throughout the sample ( e.g. elson et al .
1996 ) .
the currently available data cover only small ( and variable ) sections of the total extent of these systems .
this fact clearly limits the inferences which can be drawn to the small observed fractions , the star formation histories of these regions might not by representative of the average for a whole galaxy .
while this limitation introduces an extra uncertainty to our results , it highlights the interesting possibility of studying spatial variations in the evolutionary histories of dsph galaxies , if comprehensive hst studies where undertaken . in the above sense
, our results for the different galaxies refer in the strictest sense only to the fractions covered by the observed fields .    as we did not require the hr diagrams to extend much fainter than the oldest turnoff points ( @xmath3 ) , or to be complete into the faintest limits ( the faintest stars were in fact excluded from the analysis ) the data reduction was straightforward .
our resulting cmds do not show any systematic difference from comparable published ones for the galaxies we study .
the technical details of the images used appear in the appendix .
in this section we give a summary description of our hr diagram inversion method , which was described extensively in our paper i. in contrast with other statistical methods , we do not need to construct synthetic colour magnitude diagrams ( cmd ) for each of the possible star formation histories being considered .
rather we use a direct approach which solves for the best @xmath0 compatible with the stellar evolutionary models assumed and the observations used .          as the weak metallicity dispersions measured in the galaxies we are studying ( around 0.3 dex ) are comparable to the errors in the metallicity determinations themselves , we have not attempted to introduce any enrichment histories for any of our galaxies .
in fact , as small internal metallicity spreads present in these galaxies would introduce only small differential age offsets in our inferred @xmath0 ( see table i ) , we shall in all cases use single metallicity isochrone sets .
once a metallicity has been selected , we use the latest padova isochrones ( fagotto et al .
1994 , girardi et al .
1996 ) together with a detailed constant phase interpolation scheme using only stars at constant evolutionary phase , to construct an isochrone library having a chosen temporal resolution .    in this case
we implement the method with a resolution of 0.15 gyr , sufficient for our present problem .
it is one of the advantages of the method that this resolution can be increased arbitrarily ( up to the stellar model resolution ) with computation times scaling only linearly with it .
our only other inputs are the positions of , say @xmath5 observed stars in the hr diagram , each having a colour and luminosity , @xmath6 and @xmath7 . starting from a full likelihood model , we first construct the probability that the @xmath5 observed stars resulted from a certain @xmath0 .
this will be given by :        in the above expression @xmath10 is the density of points along the isochrone of age @xmath11 , around the luminosity of star @xmath12 , and is determined by the assumed imf together with the duration of the differential phase around the luminosity of star @xmath12 . @xmath13 and @xmath14 are a maximum and a minimum time needed to be considered , for example 0 and 15 gyr .
@xmath15 is the amplitude of the observational errors in the colour of the stars , which are a function of the luminosity of the stars .
this function is supplied by the particular observational sample one is analyzing .
finally , @xmath16 is the colour the observed star would actually have if it had formed at time @xmath11
. we shall refer to @xmath17 as the likelihood matrix , since each element represents the probability that a given star , @xmath12 , was actually formed at time @xmath11 . since the colour of a star having a given luminosity and age can sometimes be multi - valued function , in practice we check along a given isochrone , to find all possible masses a given observed star might have as a function of time , and add all contributions ( mostly 1 , sometimes 2 and occasionally 3 ) in the same @xmath17 . in this construction we are only considering observational errors in the colour , and not in the luminosity of the stars .
the generalization to a two dimensional error ellipsoid is trivial , however the observational errors in colour dominate the problem to the extent of making this refinement unnecessary .
although the amplitude of luminosity errors is only a factor of @xmath18 smaller than colour errors , as can be inferred from the fact that cmd diagrams typically display a range of luminosities 5 times larger than in colour , in discriminating between isochrones , errors in colour are @xmath19 times as important as errors in luminosity .
the absence of a colour dependence from @xmath10 is a direct consequence of having neglected errors in the luminosity of the stars .
a star of a given observed luminosity and assumed age will thus have a colour determined by the isochrones used .
equation ( 2 ) is essentially the extension from the case of a discrete @xmath0 used by tolstoy & saha ( 1996 ) , to the case of a continuous function ( continuous in time , but obviously discrete with respect to the stars ) in the construction of the likelihood .
the challenge now is to find the optimum @xmath0 without evaluating equation ( 2 ) i.e. without introducing a fixed set of test @xmath0 cases from which one is selected .
the condition that @xmath20 has an extremal can be written as @xmath21 and a variational calculus treatment of the problem applied .
firstly , we develop the product over @xmath12 using the chain rule for the variational derivative , and divide the resulting sum by @xmath22 to obtain :                this in effect has transformed the problem from one of searching for a function which maximizes a product of integrals ( equation 2 ) to one of solving an integro - differential equation ( equation 4 ) .
we solve this equation iteratively , with the boundary condition sfr(15)=0 .
details of the numerical procedure required to ensure convergence to the maximum likelihood sfr(t ) can be found in our paper i , where the method is tested extensively using synthetic hr diagrams .
the main advantages of our method over other maximum likelihood schemes are the totally non parametric approach the variational calculus treatment allows , and the efficient computational procedure , where no time consuming repeated comparisons between synthetic and observational cmd are necessary , as the optimal @xmath0 is solved for directly .
the lower main sequence region of the cmd diagram is totally degenerate with age , and contains the lower brightness stars , where the errors are larger .
we have seen from using synthetic hr diagrams that excluding this region produces a faster and more accurate convergence of the method , and have in analyzing real galaxies excluded stars of magnitudes fainter than @xmath28 .
this last cut together with the fact that our isochrones only extend out to the tip of the red giant branch ( to go further would necessitate combining results from different physical models , which we preferred not to do ) leaves us with a mass range which actually varies as a function of time .
to include also the fraction of the @xmath0 outside this region , we apply a minor correction factor to the result of equation ( 4 ) , which accounts for the fraction of mass outside the sampled range , as a function of time , as given by the imf used .
.age offset in gyr between simulated and recovered populations as a function of input age ( in gyr ) and metallicity ( in solar units ) , for a @xmath29 0.2 dex metallicity mismatch [ cols="<,^,^,^,^,^",options="header " , ]                  the reddenings used are refined iteratively after the fitting of isochrones to the ms region .
note , however , that because the reddenings are small , and the reddenings are restored to the data afterwards , even if the initial guess for the reddening is wrong by a substantial amount , very little effect is seen in the final photometry .
( e.g. a change of the reddening by 0.1@xmath34 changes photometry by less than 0.005@xmath34 . )      as holtzman et al .
( 1995 ) point out , there remain some aspects of the photometric calibration of wfpc2 which are uncertain .
for example , holtzman et al.(1995 ) note discrepancies of @xmath35 0.05 magnitudes between long and short exposures , which are not understood .
several more minor systematic and random effects at the level of a few percent ( corresponding to a few hundredths of a magnitude ) are not well understood .
furthermore , the conversion that has been made here to standard v and i colours , introduces more minor uncertainties .
it must be noted that the systematic effects in zeropoints may be as large as 0.1 magnitudes .
any such uncertainties appear as an offset primarily in adopted distance modulus , and would affect most cmds in the same way .
differentially , any effects should be minimal .
we use recent hst colour - magnitude diagrams of the resolved stellar populations of a sample of local dsph galaxies ( carina , leoi , leoii , ursa minor and draco ) to infer the star formation histories of these systems , @xmath0 . applying a new variational calculus maximum likelihood method which includes a full bayesian analysis and allows 
 a non - parametric estimate of the function one is solving for , we infer the star formation histories of the systems studied . 
 this method has the advantage of yielding an objective answer , as one need not assume _ a priori _ the form of the function one is trying to recover . 
 the results are checked independently using saha s @xmath1 statistic . 
 the total luminosities of the systems are used to normalize the results into physical units and derive sn type ii rates . 
 we derive the luminosity weighted mean star formation history of this sample of galaxies .    
 methods : statistical  stars : formation  galaxies : evolution  local group