THE ACS LCID PROJECT. V. THE STAR FORMATION HISTORY OF THE DWARF GALAXY LGS-3: CLUES TO COSMIC REIONIZATION AND FEEDBACK^*

Our analysis of the SFH of LGS-3 and the rest of LCID galaxies is done upon two fundamental requirements: (1) the consistency of the method, in order to obtain SFHs that are readily comparable and (2) the stability and uniqueness of the solution. To accomplish (1), we have adopted the same criterion on the selection of input parameters to derive the SFH for all galaxies of the LCID project. For (2), we have further developed our method and created new tests and control procedures. Details are given in the following and Appendix A.

The IAC method to solve the SFH is based in three main codes: (1) IAC-star (Aparicio & Gallart 2004) is used for the computation of synthetic stellar populations and CMDs; (2) IAC-pop (Aparicio & Hidalgo 2009) is the core algorithm for the computation of SFH solutions, and (3) MinnIAC (introduced in Aparicio & Hidalgo 2009, but explained in more detail in this paper) is a suite of routines created to manage the process of sampling the parameter space, creating input data, and averaging solutions. As a summary, Figure 3 shows a data flow diagram for the SFH computation process.

Zoom In Zoom Out Reset image size

Considering that time and metallicity are the most important variables in the problem (see Aparicio & Hidalgo 2009), we define the SFH as an explicit function of both. We will use two definitions, one in terms of the number of formed stars and another in terms of gas mass converted into stars. Formally, ψ_N(t, Z)dtdZ is the number of stars formed within the time interval [t, t + dt] and within the metallicity interval [Z, Z + dZ], while ψ(t, Z)dtdZ is the mass converted into stars within the same intervals. The relation between ψ_N(t, Z) and ψ(t, Z) is the average stellar mass at birth. Both can be identified with the usual definitions of the SFR, but as a function of time and metallicity.

Several criteria can be used to grid the CMD. The simplest is to use a uniform grid and count stars within each box. However, not all the regions of the CMD and stellar evolutionary phases contain information about the same quality for the purpose of the SFH derivation. An alternative is to give higher weights to regions with more certain stellar evolutionary phases (such as the main-sequence or the sub-giant branch), regions which are more heavily populated or regions in which the photometry is more accurate. To do so, we have introduced what we call bundles (see Aparicio & Hidalgo 2009, and Figure 4), that is, macro-regions on the CMD that can be subdivided into boxes using different appropriate samplings. The number and sizes of boxes in each bundle determine the weight of that region in the derived SFH. This is an efficient and flexible approach for two reasons. First, it allows a finer sampling in those regions of the CMD where the models are less affected by uncertainties in the input physics. Second, the box size can be increased, and the impact on the solution decreased, in those regions where the small number of observed stars could introduce noise due to small number statistics or where we are less confident of the stellar evolution predictions.

Zoom In Zoom Out Reset image size

IAC-pop provides a solution for the input set of data and parameters. It can also provide several solutions for the same set by changing the initial random number generator seed. Alternatively, it can deliver several solutions by randomly changing the input data O^j (the number of stars in the original, oCMD, boxes) according to a Poisson statistic. However all of these solutions are eventually sensitive to several other input functions and parameters. To better analyze their effects, we can divide them into four groups. The first group consists of sampling parameters; the main ones are the number and boundaries of parent, pCMDs, and the criteria to define the bundles and grid arrays for sampling the CMDs. The second group consists of external parameters; these are the photometric zero points, distance modulus, and reddening. The third group consists of model functions, which are those functions defining the sCMD; the main ones are the initial mass function (IMF), ϕ(m), and the function accounting for the frequency and relative mass distribution of binary stars, β(f, q) (see Aparicio & Hidalgo 2009). Finally, we have the external libraries, which consist of stellar evolution and bolometric correction libraries. Limiting and measuring the effects of all these parameters in the SFH solution requires several thousand runs of IAC-pop using different input sets. MinnIAC is used to test for the effects of the four groups are treated separately. In general, the solutions obtained with varying sampling parameters are averaged; their rms deviation is also a good estimate of the solution internal error. As for the external parameters and model functions, the set with the best χ²_ν is usually adopted. Regarding the external libraries, the solution with the best χ²_ν may be adopted, but all the solutions are usually conserved; their differences can provide an indication of the external uncertainties associated with such libraries.

MinnIAC (from Minnesota and IAC) is a suite of routines developed specifically to manage the process of selection of sampling parameters, creating input data sets, and averaging solutions. More in detail, regarding the parameter selection and data input, MinnIAC is used for two tasks. First, to divide the sCMD in the corresponding pCMD according to the selected age and metallicity bins. Second, to define the bundles and box grids in the CMDs and counting stars in them. With this information, MinnIAC creates the input files for each IAC-pop run. Several selections of both simple populations and bundle grids are used within each SFH solution process. MinnIAC does this task automatically, requiring only the input of bundles and corresponding grid sizes, age and metallicity bin sizes, and the steps by which the starting points of the CMD grid as well as age and metallicity bins should be modified from one IAC-pop run to another. A second module of MinnIAC is used to average single solutions found by IAC-pop as required.

In this section, we will describe with some details the specific application of the IAC method to the case of LGS-3, detailing the different effects mentioned in Section 3: sampling parameters, external parameters, model functions, and external libraries. Our choices are summarized in Table 2. For each item, Columns 2, 3, and 4 give minimum (or choice 1), maximum (or choice 2), and the step used (if relevant), respectively. Column 5 gives the final adopted value (if relevant). The SFH of LGS-3 has been derived independently for the DOLPHOT and DAOPHOT photometry sets following the same procedure as described below.

Table 2. Functions and Parameters Tested and Adopted for the SFH

Function/Parameter	Min. Value	Max. Value	Step	Adopted
Stellar evolution library	BaSTI	Girardi	⋅⋅⋅	BaSTI
Binary fraction	0.0	1.0	0.2	0.4
IMF (0.1 ⩽ m ⩽ 0.5 M☉)a	0.0	4.0	0.1	1.3
IMF (0.5 < m ⩽ 100 M☉)	0.0	5.0	0.1	2.3
ΔZF475Wb	−0.21	0.21	0.105	0.13/0.08
ΔZF814Wb	−0.15	0.15	0.075	0.11/0.11
CMD griddingc,d	0.01 ×  0.2	0.01 ×  0.2	⋅⋅⋅	⋅⋅⋅
Age sampling ($\rm {\le} 1\, Gyr$)d	0.2 Gyr	0.5 Gyr	⋅⋅⋅	⋅⋅⋅
Age sampling ($\rm {>} 1\, Gyr$)	1.0 Gyr	1.0 Gyr	⋅⋅⋅	⋅⋅⋅
Metallicity sampling (Z ⩽ 0.001)d	0.0002	0.0002	⋅⋅⋅	⋅⋅⋅
Metallicity sampling (Z>0.001)	0.0005	0.0005	⋅⋅⋅	⋅⋅⋅

Notes. ^aValues are the exponent x in the IMF defined as ϕ(m) = Am^−x, where m is the mass and A is a normalization constant. ^bZero-point offsets are relative to Sirianni et al. (2005). The adopted values were selected by interpolation. Results for DOLPHOT/DAOPHOT are given in the last column. ^cValues are the box size (color × magnitude) for the CMD sampling. ^dSeveral bin sets were tested using the same bin size but applying different offsets to the starting point of the bin set.

Download table as:  ASCIITypeset image

Using IAC-star, several sCMDs were computed, one with each of the IMFs and binarity functions listed in Table 2. Each sCMD was built with 8 × 10⁶ stars with a constant SFR in the range of age 0–13.5 Gyr and a metallicity range 0.0001–0.005. The latter value has been assumed after a preliminary test with isochrones which showed that stars with metallicity larger than 0.005 do not exist in LGS-3. Using a large number of synthetic stars in the sCMD is important in order to minimize the stochastic effects of the Monte Carlo simulations used in IAC-star, to have a statistically significant number of stars in each pCMD (>6000 stars), and to reduce the uncertainties in the stellar composition of simple populations in the pCMDs.

The bundles and one of the grid sets used to bin the CMDs are shown in Figure 4. In total, 72 combinations of IMF and binarity (model functions) have been tested. For each of them, 24 solutions have been obtained by varying the CMD binning and the simple stellar population sampling (sampling parameters). The average $\bar{\psi }$ of these 24 solutions is adopted as the solution for the specific IMF and binarity. These averages have been obtained by using a boxcar of width 1 Gyr and step 0.1 Gyr for t and width 0.0006 and step 0.0001 for Z. A $\bar{\chi ^2_\nu }$ is also obtained as the average of the 24 single χ²_ν values. The average solution with the smallest $\bar{\chi ^2_\nu }$ is assumed as the best one. The IMF and the binary fraction corresponding to this solution are assumed here as the ones best representing our photometry data, although we do not intend to reach any strong conclusions concerning their true values.

To minimize the effects of external parameters (distance, reddening and photometry zero point), the SFH is derived for different offsets in color and magnitude applied to the oCMD. The offset giving the minimum $\bar{\chi ^2_\nu }$ is assumed as the best. These are obtained when the oCMD is shifted by [Δ(F475W−F814W), Δ(F814W)] = [0.02, 0.11] and [ − 0.03, 0.11] for the DOLPHOT and DAOPHOT photometry sets, respectively. These tests also provide a way to estimate how distance, reddening, and zero-point uncertainties translate into the solution of the SFH. Defining $\bar{\chi ^2}_{\nu,\rm min}$ as the $\bar{\chi ^2_\nu }$ value of the best solution, if $\sigma _{\bar{\chi ^2}_{\nu,\rm min}}$ is its corresponding rms deviation, all the solutions with $\bar{\chi ^2}_\nu \le \bar{\chi ^2}_{\nu,\rm min}+\sigma _{\bar{\chi ^2}_{\nu,\rm min}}$ should be considered valid. We can assume that the rms deviation of all the corresponding single solutions is a proxy of the error of the adopted solution, σ_P.

The former error, σ_P, includes the effects of sampling and external parameters. However, there is one more source of uncertainty that must be taken into account, namely, the effect of statistical sampling in the oCMD. To estimate this, the solution is again obtained after varying the numbers of stars in the oCMD grid according to a Poisson statistic. This is done 20 times and the rms deviation of the solutions, σ_I, is obtained. The final adopted error is the sum in quadrature σ_SFH = (σ²_I + σ²_P)^1/2.

Finally, solutions have been obtained using both the BaSTI (Pietrinferni et al. 2004) and the Girardi et al. (2000) stellar evolutionary libraries (external libraries). In both cases, the bolometric corrections by Bedin et al. (2005) were used. Both produced qualitatively similar results, and a comparison between them together with the results obtained with other methods is presented in Section 4.3. For detailed analysis presented in the following sections, for simplicity, we use only the BaSTI stellar evolutionary libraries.

Figure 5 shows plots of the SFH of LGS-3, ψ(t, Z), as a function of both t and Z derived from the two different photometry sets. ψ(t) and ψ(Z) are also shown in the ψ − t and ψ − Z planes, respectively. The projection of ψ(t, Z) on the age–metallicity plane shows the AMR, including the metallicity dispersion.

Zoom In Zoom Out Reset image size

In Figure 6 we show ψ(t), the AMR, and the cumulative mass fraction of LGS-3 as a function of time with their associated errors. Three vertical dashed lines show the ages of the 10th, 50th, and 95th percentiles of the integral of ψ(t). Table 3 gives a summary of the integrated and mean quantities for the observed area and estimates for the whole galaxy.

Zoom In Zoom Out Reset image size

The main features of the SFH of LGS-3 can be observed in Figure 6. The bulk of the star formation of LGS-3 occurred at old epochs, with a main peak located ∼11.5 Gyr ago. The initial episode of star formation lasted until roughly 9 Gyr ago. Approximately 85% of the stars were formed in this initial episode. Since then, the SFR has been very low and slowly declining until the present. The SFR at the measured main peak is a factor of 15 larger than the mean SFR between 0 and 9 Gyr ago (because of limited time resolution, this is a lower limit, see Section 4.2). It is worth noting that the age at which the rate of the AMR starts to increase more steeply matches up with the age at which the SFR decreases to a low level. This may indicate a scenario in which the stars were formed at a higher rate at old epochs from fresh, low-metallicity gas. When the initial episode of star formation was over, the gas reservoir would have been nearly exhausted. As a consequence, subsequent star formation would occur at a much lower rate and from an ISM including a high fraction of processed gas, hence significantly increasing the metallicity enrichment rate.

It is interesting to test whether the number of SNe produced in LGS-3 during its lifetime is sufficient to produce the chemical enrichment we have derived. Focusing on the main episode, 1.7 × 10⁶ M_☉ of gas was converted into stars from 13.5 to 8.5 Gyr ago. Assuming a minimum progenitor mass for core collapse SNe of 6.5 M_☉ (Salaris & Cassisi 2005) we obtain a total of 2.9 × 10⁴ SNe produced in the old episode. Accounting for stellar mass loss and assuming a minimum initial mass of 10 M_☉ for SNe progenitors, the total number of SNe decreases to 1.6 × 10⁴. This can be compared to an estimate of the total number of SNe necessary to produce the observed chemical enrichment. Assuming the solar abundance distribution from Grevesse & Noels (1993), that only Type II supernovae (SNe II) are involved, and using the iron mass production rate from Nomoto et al. (1997) (see Monelli et al. 2010c, for more details), we estimate that ∼80 SNe II are enough to produce an increment of metallicity from 0 to 0.0007, which is the upper value obtained for LGS-3 8.5 Gyr ago. This estimate was derived assuming the iron mass production of a 13 M_☉ star (which may vary by a factor of 2–3 with different assumptions). Although relatively uncertain, our calculation shows that the galaxy needs to retain only roughly 1% of the material released by SNe in order to produce the observed metallicity increase. The same calculation can be made for the chemical evolution from 8.5 Gyr ago to the present. In this case, the mass of gas converted into stars is 2.4 × 10⁵ M_☉ and the corresponding numbers of SNe are 4.1 × 10³ and 2.4 × 10³. In this time, the metallicity increases from Z = 0.0007 to a current upper value of Z = 0.003, requiring 290 SNe. This is less than 15% of the total, a much larger fraction than calculated for the initial star formation episode. This implies that a significantly larger fraction of the newly synthesized material was captured at later times at lower SFRs. Note that only SNe II have been considered for this estimate; if Type Ia supernovae (SNe Ia) were also included, even lower fractions of captured newly synthesized material would result. Note also that much of the latter enrichment could be achieved by SNe Ia from the initial star formation episode.

The SFH shown in Figure 6 corresponds to the observed area, which covers the center and a major fraction of the galaxy. The total mass of stars created in this area over the lifetime of the galaxy is 2.0 × 10⁶ M_☉. Assuming an exponential density distribution of scale length 112 pc (S. Hidalgo et al. 2011, in preparation), we estimate that the total mass of stars ever formed in LGS-3 is 1.1 × 10⁶ M_☉. Adding in the gas mass (Young & Lo 1997, see Section 1), the total baryonic mass of LGS-3 is 1.5 × 10⁶ M_☉. These values correspond to the best solution with the adopted IMF (x = 1.3 for the interval 0.1 ⩽ m ⩽ 0.5 M_☉; x = 2.3 for the interval 0.5 ⩽ m ⩽ 100 M_☉; see Table 2). If a Salpeter IMF with an exponent of 2.35 for the range 0.1–100 M_☉ is used, then the result would be larger by a factor of ∼1.5.

Figure 7 shows the observed CMD, the best-fit CMD, and the corresponding residuals. The calculated CMD has been built using IAC-star with the solution SFH of LGS-3 as input. Residuals are given in units of Poisson uncertainties, obtained as $(n^o_i-n^c_i)/\sqrt{n^c_i}$, where n^o_i and n^c_i are the numbers of stars in bin i of a uniform grid defined on the observed and calculated CMDs, respectively.

Zoom In Zoom Out Reset image size

Observational uncertainties as well as the uncertainties inherent to the SFH computation procedure result in a smoothing of the derived SFH and a decrease of the age resolution (see Appendix A.3). Having an accurate estimate of this age resolution is fundamental in order to obtain reliable information of the actual properties of the galaxy. We have calculated the age resolution as a function of time by recovering the SFH of a set of mock stellar populations corresponding to very narrow bursts occurring at different times. Five mock populations have been computed, each one with a single burst at ages of 1, 4, 8, 11.5, and 13 Gyr. We used IAC-star to create the CMDs associated with each mock stellar population. For the mock populations at ages of 11.5 and 13 Gyr, five simulations have been done with different random number seeds in order to better characterize them. The assumed metallicities and SFRs correspond to those obtained for LGS-3. Observational uncertainties were simulated in the CMDs by using the results of the DOLPHOT completeness tests. The SFHs of the mock stellar populations were then obtained using the identical procedure as for the real data. In all cases, the recovered SFH is well fitted by a Gaussian profile with σ depending on the burst age. Figure 8 shows the input bursts and the corresponding Gaussian profile fits to the solutions, including their peak age and σ. From these results, we estimate an intrinsic age resolution of ∼1.1 Gyr at the oldest ages which decreases to ∼0.5 Gyr at an age of 1 Gyr. Also, a slight shift is observed between the input and solution ages. For the oldest trials, the shift is toward younger ages (up to 0.4 Gyr for the oldest trial) and for the youngest trials it is toward older ages (about 0.2 Gyr).

Zoom In Zoom Out Reset image size

It is particularly important to determine the oldest age at which star formation occurred in LGS-3 and the real duration of the first, main star formation episode. Our SFH solution (Figure 6) is well fitted by a Gaussian profile with a peak, μ_obs = 11.5 Gyr and σ_obs = 1.2 Gyr in the age range 9.5–13.5 Gyr (using the DOLPHOT photometry). The shifts shown in Figure 8 indicate that the actual star formation peak would have occurred at about 11.7 Gyr ago. A comparison of the solution with our estimated resolution indicates that the initial episode of star formation is just barely resolved in our observations. A quadratic subtraction of σ ∼ 1.1 that we have obtained for a sharp burst at this age from the derived σ_obs = 1.2 provides a first estimate of the actual, intrinsic dispersion of σ_real ∼ 0.5.

Given the importance of this measurement, we have pursued a more detailed analysis. We have extended the former tests to a set of mock stellar populations with SFHs defined as Gaussian profile shaped bursts of different σ and centered at 11.5 Gyr. Observational uncertainties were simulated in the associated CMDs and the SFHs were recovered following the same process as for the galaxy. In all cases, the recovered SFHs are well fitted by Gaussian profiles in the age range 9.5–13.5 Gyr. Figure 9 shows a quadratic fit to the standard deviation of the solutions, σ^mock_rec, as a function of the input ones, σ^mock_in. If the actual LGS-3 SFH were well described as a Gaussian profile, its σ could be inferred by interpolating in this fit, which gives σ^obs_in = 0.6 Gyr. We will adopt this value for the forthcoming discussion.

Zoom In Zoom Out Reset image size

As for the rest of the LCID galaxies, we have obtained the SFH of LGS-3 using the Match (Dolphin 2002) and Cole (Skillman et al. 2003) methods in addition to the IAC method. These two methods use Girardi et al. (2000) for the stellar evolution library, and we have adopted the DOLPHOT photometry to be used with them for this comparison (for a description of the main features of these methods and the particulars of their application to LCID galaxies, see Monelli et al. 2010b). For a proper comparison, we have also obtained the SFH with IAC-star/MinnIAC/IAC-pop using the same library and photometry. This also allows us to compare the solutions obtained with the IAC method using two different stellar evolution libraries.

Figure 10 shows the cumulative mass fraction of LGS-3 as obtained with the three tested methods. The agreement between the three methods resulting in the same cumulative mass fraction (0.8) at 8 Gyr is very good. This strongly supports the conclusion that the majority of the star formation in LGS-3 occurred during this initial episode of star formation. Also, all three methods find star formation continuing at much lower rates, until the present. The similar behavior of the three solutions allows us to be confident that our scientific conclusions are independent of the SFH solution method. In particular, the fact that the galaxy has formed most of the stars after the reionization epoch is also confirmed and even reinforced.

Zoom In Zoom Out Reset image size

Figure 10 also shows the effect of using different stellar models by comparing the IAC-star/MinnIAC/IAC-pop results using two different stellar libraries. Here, we see that the SFH derived using the BaSTI stellar evolution library has a slightly faster rise compared to the SFH derived using the Girardi et al. (2000) library. However, otherwise, the two are in very good agreement, and the main features of the SFH of LGS-3 are found in both derived models at roughly equivalent levels.

An analysis of the actual resolution provided by the Match and Cole methods of the kind of that done for IAC-star/MinnIAC/IAC-pop in Section 4.2 would be necessary before going deeper in this comparison, which is beyond our objectives here. In the following, we will use only the results obtained with IAC-star/MinnIAC/IAC-pop and the BaSTI stellar evolution library.

Dwarf galaxies are the focus of a major cosmological problem affecting the CDM scenario; the number of dark matter subhalos around Milky Way-type galaxies predicted by CDM simulations is an order of magnitude larger than observed (Kauffmann et al. 1993; Klypin et al. 1999; Moore 1999). Several explanations have been proposed to solve this problem but most of them use two main processes that can dramatically affect the formation and evolution of dwarf-sized halos: heating from the ultraviolet (UV) radiation arising from cosmic reionization and internal SN feedback. Both are, in principle, capable of stopping the star formation in a dwarf halo and even to fully remove its gas. The detailed information about the SFH at early and intermediate ages, such as we have obtained here, may provide fundamental insight into the problem and help to discriminate between the different scenarios proposed so far. Detailed discussions about this complex problem can be found in Mac Low & Ferrara (1999), Bullock et al. (2001), Stoehr et al. (2002), Kravtsov et al. (2004), Ricotti & Gnedin (2005), Strigari et al. (2008), and Sawala et al. (2010), among others. Here, we will give a short summary just to place our forthcoming discussion in context.

Feedback from SNe can suppress star formation or even blow out the gas completely from the smallest systems. Mac Low & Ferrara (1999) presented detailed models showing that the mass ejection efficiency is low in galaxies with baryonic mass above ∼10⁷ M_☉ and that only galaxies with baryonic mass below ∼10⁶ M_☉ could have their gas completely blown away almost independently of the SNe mechanical luminosity. It is worth mentioning that, according to the assumptions of Mac Low & Ferrara (1999), a baryonic mass of ∼10⁶ M_☉ would correspond to a total mass of 6.8 × 10⁷ M_☉, while a baryonic mass of ∼10⁷ M_☉ would correspond to 3.5 × 10⁸ M_☉.

There is a consensus that UV background heating establishes a characteristic time-dependent minimum mass (a filtering mass) for halos that can accrete gas (e.g., Gnedin 2000; Kravtsov et al. 2004). The UV background will heat the gas in low-mass halos, preventing star formation, and eventually photoevaporate them. As a result, halos not massive enough to have accreted gas before the reionization epoch would fail to do so later, unless they accrete mass faster than the rate at which the filtering mass increases. The redshift of the reionization epoch has been estimated from polarization observations of the CMB to be z = 10.9 ± 1.4 (WMAP 5 year results; Komatsu et al. 2009). However, the presence of the Gunn–Peterson trough in quasars at z ∼ 6 shows that the universe was not yet fully reionized at earlier epochs (Loeb & Barkana 2001; Becker et al. 2001).

According to models (see, e.g., Mac Low & Ferrara 1999), the minimum circular velocity for a dwarf halo to accrete and cool gas in order to produce star formation is v_c ∼ 30 km s⁻¹, which corresponds to a total mass of ∼10⁹ M_☉. However, most dwarf galaxies in the Local Group, including all the Milky Way satellites and five out of the six of the LCID sample, show circular velocities well below that value. Nevertheless, there are currently several proposed ways to overcome this apparent contradiction. Bullock et al. (2001) have proposed that dSph and the rest of the smallest halos would have been formed during the prereionization era. The processes of photoionization feedback resulted in the suppression of the star formation below an observable level in 90% of them. This was supported by Ricotti & Gnedin (2005), who found that if positive feedback is considered, a fraction of low-mass halos (total mass below 10⁸ M_☉) would have been able to form stars before the reionization era. The star formation in these galaxies would have been halted by internal mechanisms, like photodissociation of H₂ or SN feedback in advance of the reionization era. Another possibility is that masses of dark matter halos of the dSphs galaxies would be much larger than those measured at the optical limit of the galaxies and larger than the limit required by the reionization scenario (Stoehr et al. 2002). In these scenarios, dark matter halos of lower masses would have failed to form stars and remain dark. Alternatively, dark matter halos could have been much larger in the reionization epoch, with total masses above the mass required for star formation, but they lost a large fraction of their mass afterward due to tidal harassment (Kravtsov et al. 2004). It is also possible that a self-shielding mechanism would be present, protecting the gas in the inner regions or in regions denser than some limit from being heated by the UV background (Susa & Umemura 2004). Finally, it has been suggested that inhomogeneous reionization can lead to large variations in the masses of halos that survive and produce stars (Busha et al. 2010).

In summary, it is useful to note that the first scenario (Bullock et al. 2001; Ricotti & Gnedin 2005) and also the last ones (Susa & Umemura 2004; Busha et al. 2010) allow star formation in low-mass halos (total mass below 10⁸–10⁹ M_☉), while the second and third are compatible with star formation in more massive halos only (total mass above 10⁹ M_☉). Additionally, the first scenario would imply that star formation would have been produced before reionization and would have been halted afterward in the smallest galaxies.

Recently, Sawala et al. (2010) have presented a set of high-resolution hydrodynamical simulations of the formation and evolution of isolated dwarf galaxies including the most relevant physical effects, namely, metal-dependent cooling, star formation, feedback from Type II and Ia SNe, and UV background radiation. Their results are very useful for a direct comparison with observations. They study halos with present-day total masses between 2.3 × 10⁸ M_☉ and 1.1 × 10⁹ M_☉ and reach an interesting set of conclusions that include many of the aforementioned ones. First, halos that are not massive enough to accrete sufficient gas to form stars before z = 6 lose their gas subsequently due to UV background heating. In this sense, reionization sets a lower mass limit of the halo for star formation. Second, in halos that form stars, feedback is the main process driving the evolution of the galaxy and regulating its star formation. It alone can blow out all the gas of a dwarf system before z = 0, while UV background alone has almost no effect. However, if feedback has previously made the gas diffuse and reduced its radiative cooling efficiency, then the UV background has a strong effect, producing a sharp cutoff in the star formation at the epoch of reionization. Finally, self-shielding is effective only in the inner regions of the halos at the more massive end.

All the aforementioned mechanisms are founded on solid and extensive theoretical modeling. However to decide to what extent one or another should be preferred, detailed observational evidence is necessary. One such observable is whether the star formation in the smallest galaxies was halted at reionization. To answer this question, a precision of ∼1 Gyr is required in the determination of the SFH at 12.5–13 Gyr ago (Ricotti & Gnedin 2005), which is indeed what we have reached in our work. In the following, we will discuss the different possibilities in light of our results for LGS-3 paying attention first to SN feedback effects and second to UV background effects.

The SFH obtained for LGS-3 allows us to make a direct estimate of the importance of feedback in LGS-3. To do so, the mechanical luminosity of the SNe produced in the main star formation episode and the mass of the galaxy can be compared with the results of Mac Low & Ferrara (1999). They used models to calculate the conditions for blow-away, partial mass loss, and no mass loss of dwarf galaxies as a function of the baryonic mass of the galaxy and the mechanical luminosity, L_w, of the SNe produced during a central, instantaneous star formation episode.

The kinetic energy released to the interstellar medium by the SNe produced in the old, main star formation episode in LGS-3 can be computed as follows. For a continuous SFR of 1 M_☉ yr⁻¹, the STARBURST99 models (Leitherer et al. 1999) predict a mechanical luminosity of log L_w = 41.9. This is the equilibrium value reached after a few tens of million years from the first star formation. The SFR is for a mass range 1–100 M_☉ and is computed using a classical IMF with exponent −2.35. For direct comparison, we convert this rate to the same stellar mass interval and IMF that we have used and obtain 1.7 M_☉ yr⁻¹. Scaling these values to the SFR obtained for LGS-3, the total mass converted into stars in the initial episode is 1.7 × 10⁶ M_☉. Assuming a Gaussian profile of σ = 0.6 Gyr, or a duration (FWHM) of 1.4 Gyr, results in an average SFR of 1.3 × 10⁻³ M_☉ yr⁻¹. Scaling STARBURST99 models to this average SFR results in a mechanical luminosity of L_w = 5.3 × 10³⁸ erg s⁻¹.

Alternatively, we can compute the mechanical luminosity from core collapse SNe only (note that STARBURST99 includes all sources of mechanical luminosity). From Section 3, the total number of SNe formed during the old, main episode was 2.9 × 10⁴ or 1.6 × 10⁴ for minimum core collapse progenitor masses of 6.5 M_☉ or 10 M_☉, respectively (Salaris & Cassisi 2005). Assuming an energy release per SN of 10⁵¹ erg (Leitherer et al. 1999), we obtain a total mechanical luminosity released during the old, main episode of L_w = 6.8 × 10³⁸ erg s⁻¹ or L_w = 3.8 × 10³⁸ erg s⁻¹, respectively. Since these values are compatible with the STARBURST99 estimate, we will adopt L_w = 5.3 × 10³⁸ erg s⁻¹. Combining this with a baryonic mass of 1.5 × 10⁶ M_☉ (Section 4.1), the model results of Mac Low & Ferrara (1999) shown in their Figure 1 place LGS-3 in the regime of mass loss, but fairly close to the blow-away regime.

However, Equation (1) from Mac Low & Ferrara (1999) gives a relation between the baryonic and dark matter masses of dwarfs which is an extrapolation to low-mass galaxies of the relation derived by Persic et al. (1996) for more massive galaxies (total masses above 10¹⁰ M_☉). From this relation, the total mass for LGS-3 is 9 × 10⁷ M_☉. Alternately, Sawala et al. (2010) give the final (z = 0) stellar, gaseous, and total masses for their model runs. The relations between these parameters depend on the model ingredients, but for a baryonic mass like that of LGS-3, the total mass is in the range from 2 × 10⁸ M_☉ to 4 × 10⁸ M_☉. Using this estimate instead of the Persic et al. (1996) extrapolation places LGS-3 well within the regime of mass loss due to feedback and far from the blow-away regime.

A flat Einstein–de Sitter universe with $H_0=70.5\ \rm km s^{-1} Mpc^{-1}$, Ω_m = 0.273 (as derived from the WMAP 5 year data; Komatsu et al. 2009), has an age ∼13.7 Gyr. The WMAP 5 year data indicate the epoch of reionization occurred at z ∼ 10.9 corresponding to a look-back time of ∼13.3 Gyr. Evidence from quasar spectra indicates that the epoch of reionization is over at a redshift of z ∼ 6 or a look-back time of ∼12.7 Gyr (Becker et al. 2001). We will focus on the latter since it is more conservative from the point of view of the forthcoming discussion.

Did LGS-3 form most of its stars before or after the epoch of reionization? Using the cumulative mass fraction from Figure 6, we find that more than 80% of the stellar mass of LGS-3 has been formed later than ∼12.5 Gyr ago (z ∼ 5). Thus, the observations indicate that the majority of the stars in LGS-3 were formed after the epoch of reionization.

We can use synthetic data to better assess this constraint. In Figure 12, we show the results of two synthetic SFHs, one in which all stars are formed between 13.4 and 12.4 Gyr ago (before the epoch of reionization) and another where all of the stars are formed between 11.9 and 10.9 Gyr ago. Figure 12 shows that the model where all of the stars formed before the epoch of reionization is clearly inconsistent with the SFH of LGS-3. On the other hand, the model with all stars forming after the epoch of reionization is a good match to our SFH for LGS-3, indicating that it is highly unlikely that LGS-3 formed a large fraction of its stars before reionization had concluded.

Zoom In Zoom Out Reset image size

However, the absolute ages in our comparisons are intrinsically dependent on the stellar evolution models which have intrinsic uncertainties (e.g., Chaboyer 1995; Chaboyer et al. 1998). Modern stellar evolution models have addressed many of the shortcomings in earlier generations of models (e.g., see discussion in Dotter et al. 2007), and the results of the relative ages derived from main-sequence stars are in excellent agreement (better than 0.5 Gyr) as demonstrated by Marín-Franch et al. (2009). The accuracy of the absolute ages is also increasing. For example, the average age of the oldest globular clusters obtained from the age analysis by Marín-Franch et al. (2009), using the BaSTI library, is 12.3 Gyr with an age dispersion of ±0.4 Gyr (A. Marín-Franch 2011, private communication). In this regard, the complete inconsistency of our SFH for LGS-3 with the model in which all stars are formed before reionization is persuasive.

Nonetheless, as argued in Monelli et al. (2010c), the strongest constraints come from relative ages. Because all of the LCID galaxies were observed and analyzed in an identical manner, we can make the following argument. Since we find an age difference of >1.5 Gyr between the peak of star formation in LGS-3 and that of Tucana, and since that difference is larger than the difference between the age of the universe and the epoch of reionization, it is highly unlikely that a significant fraction of the stars in LGS-3 have been formed before the epoch of reionization. An identical conclusion was found for the Cetus dSph (Monelli et al. 2010b). In summary, Figure 5 provides strong evidence that most, if not all, of the star formation in LGS-3 occurred after the end of reionization.

Although LGS-3 is a low-mass galaxy, do theoretical models expect it to form its star before reionization? Before putting the former result in the context of current theoretical model predictions, some estimates of integrated physical quantities of LGS-3 are necessary. The velocity dispersion of the gas in LGS-3 is 8 km s⁻¹ at a radius of 470 pc (Young & Lo 1997) which corresponds to a total mass within that radius of 2.1 × 10⁷ M_☉. According to Stoehr et al. (2002), this value should not be considered indicative of the total mass of the galaxy, but only of the virial mass at the measured radius. Following the results of Sawala et al. (2010), the total mass of LGS-3 would be expected to be in the interval ∼2 × 10⁸ M_☉ to ∼4 × 10⁸ M_☉. Equation (16) of Mac Low & Ferrara (1999) gives a relation between circular velocity and total mass in a density distribution given by a modified isothermal sphere. Using h = 0.7, it can be written as v_c(r) = 10.5 M^1/3_h,7, where M_h,7 is the total mass in units of 10⁷ M_☉ and v_c is given in km s⁻¹. Thus, for a total mass in the range of 2 × 10⁸ M_☉ to ∼4 × 10⁸ M_☉, the resulting circular velocity for LGS-3 is between 28 km s⁻¹ and 36 km s⁻¹. These values are close to or somewhat above the limit of 30 km s⁻¹ below which the UV background is expected to prevent any subsequent star formation.

In summary, it is possible that LGS-3 was massive enough that its star formation was not halted by UV background heating and it was also able to preserve at least part of its gas against feedback from SNe. However, models by Sawala et al. (2010) indicate that, for its estimated mass, the combination of the UV background together with SN feedback should have halted the star formation at z ∼ 6 (or ∼12.7 Gyr ago). Since this is evidently not the case, a mechanism such as self-shielding seems necessary to allow the galaxy to continue forming stars, at least in its inner regions, as suggested by Sawala et al. (2010). If the total mass of LGS-3 would be of the order of 10⁸ M_☉ or lower (as obtained from the relation used by Mac Low & Ferrara 1999), the circular velocity would drop to values of the order of 20 km s⁻¹. For this total mass estimate, our observation that most of the star formation in LGS-3 has taken place after z ∼ 6 would be difficult to reconcile with models. However, one possibility is that a mechanism like mass loss due to tidal harassment (as proposed by Kravtsov et al. 2004) would be at play. On the other hand, the Mac Low & Ferrara (1999) relation requires a large extrapolation for the mass of LGS-3, and this may simply imply that the Sawala et al. (2010) mass estimates are to be preferred.

This condition is fulfilled if combinations of simple populations which are within the error bars of the adopted solution produce CMDs which are indistinguishable from the best-fit CMD. It is also required that combinations of simple populations significantly different from the adopted solution produce calculated CMDs significantly different from the solution one. Simulations done during the tests for the IAC-star/MinnIAC/IAC-pop method (Aparicio & Hidalgo 2009; Hidalgo et al. 2009, and this work), consisting in obtaining the SFH of mock stellar population of known SFH, show that the aforementioned conditions are fulfilled in general.

By self-consistency we refer to the capability of the method to reproduce synthetic input. When a mock population is computed according to a given SFH and analyzed in an identical manner as real data, the resulting SFH should be equal to the input one, within errors and resolution limits. Such tests have been executed in Aparicio & Hidalgo (2009) and repeated here using the best-fit SFH given in Figure 5 to compute the mock population to be analyzed. For simplicity, we will show here only the tests for the DOLPHOT photometry but the result is equivalent for the DAOPHOT photometry. See also tests presented in Section 4.2.

To minimize the dependency of the test results on the associated statistical fluctuations inherent to the method (present in the observational uncertainties simulation and in the sCMD modeling), we have obtained the SFHs for 10 CMDs calculated from the SFH of Figure 5 changing only the random number generation seed. An example is shown in Figure 7. Figure 13 shows the average of all of the new SFHs and the AMRs obtained. The input values, i.e., the SFH and AMR of LGS-3, are also shown for comparison. The main conclusion of the self-consistency test is that the SFH recovering process introduces no significant bias in the location of the main peak nor in the AMR, and that the SFH features are properly recovered, although it introduces some degree of degradation in time resolution. This has to be taken into account and corrected for when quantities such as the maximum intensity of a burst or its duration are relevant (see Section 4.2).

Zoom In Zoom Out Reset image size

Stability is related to uniqueness, but the question we want to address here is what is the maximum excursion of the model functions and external parameters producing an SFH within the error bars of the solution? We have tested the effects of changing the fraction of binary stars, the IMF, and the photometric zero points. The latter are discussed in Section 4. The dependency of the solution on the fraction of binary stars and the IMF has been analyzed in detail in Monelli et al. (2010b) and E. Skillman et al. (2011, in preparation), respectively. In short, any fraction of binary stars larger than 0.4 gives an SFH within the error bars. In the case of the IMF, for the interval of stellar masses 0.5 < m ⩽ 100 M_☉, the range of values producing solutions within the errors is 2.2 ⩽ x ⩽ 2.4, where x is the mass exponent index in ϕ(m) = Am^−x. The solution is independent of x for the interval 0.1 ⩽ m ⩽ 0.5 M_☉, although the total stellar mass is not. As an indication, the total mass converted into stars in LGS-3 would be increased by a factor ∼1.5 if the IMF exponent changes from x = −1.3 to x = −2.3 for the interval 0.1 ⩽ m ⩽ 0.5 M_☉. The same factor of ∼1.5 exists between the corresponding current masses in existing stars and stellar remnants. As a summary, Table 4 shows the range of variation for the binary fraction, IMF, and photometric offsets which give solutions within the error bars for the adopted SFH of LGS-3.

Table 4. Model Function and External Parameter Best Fit Range

Function/Parameter	Min./Max. Value
Binary fraction	0.4/0.8
IMF (0.1 ⩽ m ⩽ 0.5 M☉)a	−4.0/0.0
IMF (${\rm 0.5<m}\le 100\, \rm M_\odot$)	−3.0/−2.0
ΔZF475W	−0.04/0.02
ΔZF814W	−0.06/0.04

Note. ^aValues are the exponent x in the IMF defined as ϕ(m) = Am^−x, where m is the mass and A is a normalization constant.

Download table as:  ASCIITypeset image

Since the number of real stars is limited, the observational sampling could be insufficient and could introduce undesirable biases in the solution. To check this point, we have obtained the SFH of five different random subsets of the observed stars. Each one contains 50% of the total, so that each star may have been included in several subsamples. Figure 14 shows the SFH of LGS-3 and the average of the SFHs of the five subsamples. For the comparison to be meaningful, the test results have been normalized by a factor. Error bars show the dispersion of the five subsample SFHs. The agreement is very good and no bias is apparent, which indicates that our solution does not suffer from undersampling effects.

Zoom In Zoom Out Reset image size