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(Corbelli & Schneider, 1997 ; Corbelli, 2003 In the next section, we outline the chemical evolution equations, which form the backbone of our models. We describe in § how we link these equations with the synthetic CMD fitting method to build a self-consistent model CMD. In § , we present the results of fitting closed b... |
Chemical Evolution Equations To model the SFH of M33 in a self-consistent way we use the chemical evolution equations, which follow the variation of the gas and stellar masses and the abundances of various elements. We use the instantaneous recycling approximation (IRA) in which stellar lifetimes are treated as negligi... |
Pagel & Tautvaišiené ( 1995 , hereafter PT95) In the DPA, there is a delay time, [MATH] , between the birth of a stellar generation and the resulting SNe Ia explosions. |
Our models track the elements O, Mg, Si, Ca, Ti, and Fe. To minimize the effect of uncertainties in the yield of any one particular [MATH] -element, we focus on their sum, which we refer to as [MATH] Unless otherwise noted, [MATH] /Fe] = [(O+Mg+Si+Ca+Ti)/(5 Fe)]. We do not follow carbon and nitrogen because they have a... |
We define the total mass surface density as [MATH] where [MATH] and [MATH] are the stellar and gaseous surface densities, respectively, and [MATH] is the elapsed time of the model. We also define [MATH] as the gas mass surface density in the form of element [MATH] Combining the formalism of Tinsley ( 1980 |
and PT95 , the equations of chemical evolution under the IRA and DPA can be expressed as [EQUATION] [EQUATION] [EQUATION] [EQUATION] |
where [MATH] is the SFR, [MATH] is the inflow rate, [MATH] is the outflow rate, and [MATH] is the fraction of gas in the form of element [MATH] The net stellar yield, [MATH] , is the mass of newly synthesized element [MATH] instantaneously returned to the ISM by a stellar generation per unit mass locked up into stellar... |
(for the case of a variable mass loss efficiency) and Portinari et al. ( 1998 We find that, averaged over all metallicities, [MATH] for a Kroupa et al. ( 1993 initial mass function (IMF). We note in passing that the precise value of [MATH] is inconsequential because the factor [MATH] can be absorbed into the star forma... |
Equs. represent conservation of total, gaseous, stellar, and elemental mass. The first term in Equ. represents the mass of element [MATH] that is originally present in the gas and lost to star formation, plus what is instantaneously returned by the winds and explosive deaths of massive stars. The second term in Equ. re... |
represent, respectively, the inflow and outflow rate of element [MATH] . The chemical composition of the inflowing gas is assumed primordial. The initial heavy element abundance and stellar mass are zero and, except for closed box models, the initial gas mass is zero, also. |
Motivated by the successes of chemical evolution models applied to the MW’s satellites (e.g., Lanfranchi & Matteucci, 2004 we set the outflow rate proportional to the SFR by a constant factor, [MATH] , called the outflow efficiency. The composition of the outflowing gas is the same as the ISM, which is assumed to be ho... |
(Lanfranchi & Matteucci, 2004 ; Carigi et al., 2006 so we restrict [MATH] to be less than 10. The idea behind this approach is that SNe inject kinetic energy into the ISM, possibly causing some gas to leave the system entirely. In reality, the outflow efficiency could depend on factors like the depth of the gravitation... |
As is commonly done in chemical evolution models, we couple the SFR to the gas mass through the so-called Kennicutt-Schmidt (KS) relation, |
[EQUATION] where [MATH] is the star formation efficiency. In the spirit of the pioneering work of Schmidt ( 1959 who found that star formation rate traced gas density in the MW, Kennicutt ( 1998 measured mean gas masses and SFRs within the optical radii of [MATH] normal spirals and within the central regions of [MATH] ... |
[MATH] and [MATH] Looking at the two subsamples individually, he found [MATH] for the normal spirals and [MATH] for the starbursts. |
This global star formation relation can be compared to a local one which refers to gas densities and SFRs at individual points or in azimuthally averaged bins within a galaxy. Several studies of nearby systems have found that, when examined locally, |
[MATH] (Gottesman & Weliachew, 1977 ; Wong & Blitz, 2002 ; Kennicutt et. al., 2007 ; Boissier et al., 2003 ; Misiriotis et al., 2006 The cause of the observed variation in [MATH] is unclear but possibilities include non-axisymmetric profiles or uncertainties in the extinction correction, the CO- [MATH] conversion facto... |
(Corbelli, 2003 The dependence of [MATH] on molecular fraction is not too surprising since star formation is observed to trace molecular gas quite well |
(e.g., Murgia et al., 2002 ; Heyer et al., 2004 ; Matthews et al., 2005 ; Leroy et al., 2005 ; Gardan et al., 2007 Indeed, the studies mentioned above also found that, when considering the molecular gas alone, [MATH] with much less galaxy-to-galaxy variation than when considering the total or atomic gas. It is beyond t... |
With these considerations in mind, an appropriate empirical KS relation to use would be one derived for M33 involving the total gas density. |
Heyer et al. ( 2004 measured [MATH] and [MATH] using the infrared luminosity to estimate M33’s SFR profile inside [MATH] Similarly, Boissier et. al. ( 2007 studied M33’s KS relation by measuring its UV surface brightness and translating that to a SFR profile. Although Boissier et. al. ( 2007 did not report specific val... |
leads to very poor fits of the CMD because the distributions of stellar age and metallicity are skewed too high and low, respectively (see § 4.2 ). Further tests show that our models are more sensitive to reasonable changes in [MATH] than in [MATH] , so we adopt [MATH] for all models and allow [MATH] to be a free param... |
[MATH] throughout much of the system’s evolution. Finally, we do not include a star formation threshold (see the discussion in Paper II |
and Boissier et. al. ( 2007 for reasons why). Our chemical models can be summarized as follows. Gas inflow deposits gas into the system and drives star formation via the KS relation. Because the inflowing gas has primordial composition, the ISM metallicity is always below what it would be without inflow. About [MATH] o... |
[MATH] is locked up into stellar remnants like white dwarfs, neutron stars, and black holes. The star formation enriches the ISM with metals, but also drives an outflow of gas from the system. All else being equal, the presence of a gas outflow quenches the SFR, slows the chemical enrichment, and suppresses the metalli... |
Method We start with a set of model parameters (e.g., [MATH] [MATH] [MATH] and [MATH] ), which we use to solve Equs. [MATH] We integrate these equations using a 4th-order Runge-Kutta method with a time-step of [MATH] yr. The numerical integration was checked against several analytic solutions and the typical fractional... |
The age-metallicity plane is divided into logarithmic bins of width 0.25 dex in age and 0.3 dex in metal abundance. To cover this plane, we use the same set of synthetic CMDs as in Paper III each of which represents the predicted photometric distribution of stars in the corresponding age and metallicity bins. Each of t... |
The quality of the fit is formally measured by the parameter, [MATH] , which gives the difference between [MATH] and its expectation value in units of its standard deviation. [MATH] value of 0.0 indicates a perfect fit, while, for example, a value of 2.0 indicates a [MATH] departure from a perfect fit (Dolphin, 2002 Pa... |
[MATH] in steps of 0.10 and 0.05 mag, respectively. The global best fit (referred to simply as the best fit) is the weighted average of those distance/extinction combinations whose solutions lie within [MATH] of the best individual solution. |
We use the program, StarFISH (Harris & Zaritsky, 2001 , after incorporating the genetic algorithm, PIKAIA (Charbonneau, 1995 , to efficiently search the full volume of parameter space. Briefly, this algorithm randomly generates an initial population of solutions, which is evolved through successive generations under th... |
Because we are no longer solving for the amplitudes of the basis populations, we had to modify the way StarFISH calculates the confidence intervals. For each parameter, we take small steps in the positive and negative directions away from its optimum value. After each step, we allow the downhill simplex to re-converge ... |
limit of the fitting statistic was reached. To estimate the systematic errors in the stellar evolutionary tracks we created two sets of synthetic CMDs using the Girardi et al. ( 2000 and |
Pietrinferni et al. ( 2004 tracks, which we respectively designate as Padova and Teramo (also referred to as BaSTI in the literature). The conversion from the theoretical to the observational plane is accomplished with the Castelli & Kurucz ( 2003 |
library of bolometric corrections. At this point it is important to note that these stellar tracks have scaled-solar abundances, but the elemental abundance distribution in forming stars changes with time. This is particularly important for the RGB, whose temperature is determined mostly by the abundances of elements w... |
(e.g., VandenBerg et al., 2006 ; Pietrinferni et al., 2006 ; Dotter et al., 2007 Salaris et al. ( 1993 found that [MATH] -enhanced tracks and isochrones are well reproduced by scaled-solar ones with the same global metallicity provided the enhancements are similar for all the [MATH] -elements. Subsequent studies demons... |
(e.g. Salaris & Weiss, 1998 ; Salasnich et al., 2000 ; VandenBerg et al., 2000 ; Kim et al., 2002 We have checked its applicability using several of the most recent isochrone databases |
(Pietrinferni et al., 2006 ; Dotter et al., 2007 ; VandenBerg et al., 2006 and we find that over the range of ages, [Fe/H], and [ [MATH] /Fe] most appropriate for our data, a scaled-solar isochrone in the [MATH] plane is within [MATH] mag of an [MATH] -enhanced isochrone with the same global metallicity. Therefore, we ... |
[EQUATION] In Equ. [MATH] for [MATH] O, Ne, Mg, Si, S, Ca, and Ti and [MATH] Note that this relation implicitly assumes no enhancements in elements other than the [MATH] -elements. This is not a significant problem, however, since the [MATH] -elements comprise most of the metals by mass. In any case, our conclusions ar... |
In principle, the bolometric corrections, color transformations, and evolutionary lifetimes also depend on [ [MATH] /Fe], but in practice, our data are not significantly affected by these dependencies. |
Cassisi et al. ( 2004 showed that the bolometric corrections and color transformations depend negligibly on [ [MATH] /Fe] in [MATH] and redder bands. |
Dotter et al. ( 2007 investigated the effect of abundance variations on stellar evolutionary models and found that when [MATH] , the MS lifetime is decreased by [MATH] We have compared the scaled-solar and [MATH] -enhanced Teramo tracks, which have [MATH] |
(Pietrinferni et al., 2006 , and we find that, in general, the evolutionary phase lifetimes differ by [MATH] These effects are likely to be even smaller for our data since we derive |
[MATH] for almost all ages. Results We tested several different inflow/outflow scenarios using both sets of stellar tracks. In general, they give similar results, so we only show those of the Padova tracks. In the text, we describe any significant differences, since they can help us gauge the systematic errors due to t... |
shows the residual CMD of a particular scenario on a scale where white and black correspond, respectively, to an excess and deficit of model stars at the [MATH] level. The bottom row in these figures shows the best-fitting SFH as the upper line and inflow history (IFH) divided by 10 as the lower line. The IFH is integr... |
Paper III , in which age and metallicity were free parameters. Tables and give the the fit quality ( [MATH] ), reduced [MATH] [MATH] ), number of degrees of freedom ( [MATH] ), and mean distance modulus and extinction with their respective |
[MATH] uncertainties. Tables and give the mean star formation and outflow efficiencies and their upper and lower [MATH] uncertainties. The mean values reported in the tables are averages of the acceptable solutions in the distance/extinction grid as explained in §3. |
4.1 Closed Box Models We began by testing the canonical closed box model in which the inflow and outflow rates were identically zero. The system was initially composed entirely of gas. The total mass remained constant throughout its evolution while the gas mass and SFR decreased monotonically with time. The best-fittin... |
There are several significant discrepancies between the model and data CMDs. First, the model predicts too much star formation at ages [MATH] Gyr which causes an overabundance of model main sequence (MS) stars on the blue plume at [MATH] Second, the model has too few stars in a region below the blue plume centred at [M... |
The Teramo model exhibits similar discrepancies, but the RGB appears slightly too wide, possibly indicating an excessively large metallicity spread, and it has too many stars overall. The Teramo model also has too many stars on the blue horizontal branch, which signals that there is too much star formation at the oldes... |
The fit qualities of the closed box solutions are [MATH] worse than the solutions of Paper III , which had [MATH] values of 6.64 (Padova) and 6.02 (Teramo). This is not surprising since our closed box model has only two free parameters, namely, the star formation efficiency and the total mass. However, as we will see b... |
4.2 Inflow and Outflow Models As described in § , there is evidence that galaxies do not evolve as closed boxes. An exponential inflow rate is one of the simplest and most common forms used in the literature, so it is instructive to see how well it can explain M33’s SFH. Accordingly, we solved for the inflow time-scale... |
This new model provides a better fit to the data than the closed box model, but it still exhibits some large discrepancies with the data. In fact, these discrepancies are very similar to those of the closed box model, but the magnitude of the residuals has been lessened. There is still too much star formation at ages [... |
Paper III solutions. Some numerical simulations of structure formation within the [MATH] CDM framework predict that the average mass accretion rates of dark matter haloes are initially small, grow to some maximum, and decline thereafter (e.g., van den Bosch, 2002 ; Wechsler et al., 2002 With this in mind, we also inves... |
[MATH] where [MATH] is the time between when the inflow starts and when it peaks. This function, which we refer to as Sandage inflow , was first used by Sandage ( 1986 to describe the variation in SFH with galaxy morphology and later explicitly presented by MacArthur et al. ( 2004 By shifting the bulk of the inflow tow... |
Paper III solutions. The Teramo models show a qualitatively similar behavior – the exponential and Sandage functions do a progressively better job at reproducing the observed CMD, but still do not get the overall age distribution correct. The main drawback of these functions is that the IFR today cannot easily be varie... |
These results led us to try three less restrictive inflow models. The first was a double exponential model described by four parameters: a growing time-scale, a decaying time-scale, a transition time between the growing and decaying modes, and the IFR at the transition time. The second was a |
truncated model described by four parameters: the initial IFR, the IFR at a model time of 7 Gyr, a truncation time when the inflow ends, and the IFR at the truncation time. In the third model, which we called free inflow we approximated the true IFH with a discrete function by dividing the entire age range into 4 bins ... |
The solutions using these three inflow models are displayed in Fig. The free inflow model provides the highest quality fits. However, the double exponential and truncated inflow models are [MATH] worse and exhibit qualitatively similar results. The discrepancies between model and data exhibited previously with the expo... |
Paper III solutions. This difference could arise from the approximations made in producing the chemical evolution models and the uncertainties inherent in the stellar yields. Second, age and metallicity are no longer completely free parameters so errors in the stellar tracks cannot be as easily hidden by arbitrary comb... |
(Sellwood & Binney, 2002 ; Roškar et al., 2008 In Fig. we summarize the distribution of stellar ages and metallicities in the three best inflow models for the Padova tracks. The panels show the (a) SFH, (b) age cumulative distribution function (age CDF), (c) Z AMR, and (d) Z metallicity distribution function (Z MDF) of... |
In the best three inflow models, [MATH] of the gas accretion takes place between 3 and 7 Gyr ago. Anything less would produce too few intermediate-age stars. Second, at most [MATH] |
of the gas was accreted in the last 3 Gyr. Any amount in excess of that would lead to a recent SFR that is too high and produce too many young stars on the blue plume MS at [MATH] Third, the preferred outflow efficiency is [MATH] which is smaller than the typical values of [MATH] estimated for dwarf galaxies in the LG ... |
The preferred values for [MATH] tend to be of order [MATH] . The resulting gas depletion timescale, [MATH] [MATH] is on the low end of the equivalent timescale in dwarf galaxies, which ranges from a few to several tens of Gyr |
(Taylor & Webster, 2005 ; Karachentsev & Kaisin, 2007 ; Calura et al., 2008 An [MATH] value as low as 0.0035 (Heyer et al., 2004 is strongly disfavored because, as Fig. |
demonstrates, the resulting distributions of stellar age and metallicity are skewed to higher and lower values than the Paper III solution. If, instead, we fix [MATH] to have a higher value, like 0.084, as implied by the local current gas density and SFR in our field, then the age and metallicity distributions are more... |
larger than the Heyer et al. ( 2004 value and only [MATH] lower than the best-fitting value of 0.62 in the free inflow model. The variation in [MATH] among the best three inflow models indicates the exact parametrization of the IFH can affect it by almost an order of magnitude. Based on the tests in § a similar uncerta... |
By incorporating the chemical evolution equations into the CMD fitting, we can extract more information from the solutions and make more predictions that can be tested against observations. Fig. |
summarizes some predictions for the three best inflow models. Each figure shows (a) the gas mass averaged over each of the 9 SFH bins, (b) the stellar mass-weighted |
[MATH] [MATH] /Fe] [MATH] and [MATH] [Fe/H] [MATH] of all stars formed in each SFH bin, (c) the Fe AMR, and (d) the Fe MDF. In panel (b), the horizontal and vertical lines denote [MATH] [MATH] /Fe] [MATH] |
and [MATH] [Fe/H] [MATH] of all stars ever formed, respectively. The line types are the same as in Fig. Because of the presence of gas inflow, the gas mass rises in the early evolutionary stages and reaches a maximum several Gyr later. The gas mass begins to decline as the inflow rate becomes small but star formation a... |
while that of the youngest bin is [MATH] Because the majority of stellar mass formed within the oldest three age bins, [MATH] [MATH] /Fe] [MATH] of all stars ever formed is [MATH] , even though the majority of age bins have [MATH] [MATH] /Fe] [MATH] |
[MATH] Note that the [ [MATH] /Fe] vs. [Fe/H] relation is not always single-valued, since [ [MATH] /Fe] can increase and [Fe/H] can decrease with time depending on the precise interplay between gas flows, the SFR, and the SN Ia explosion rate (see 7.1 ). |
The Teramo models show an overall similar history as the Padova models, with [MATH] of the total inflow taking place in the last 7 Gyr and [MATH] in the last 3 Gyr. However, the inflow hiatus present in the Padova free inflow model between [MATH] and 9 Gyr is not present in the Teramo model. The mean [ [MATH] /Fe] is a... |
[MATH] dex higher at all ages. This metallicity difference is somewhat larger here than in Paper III We believe this is mostly due to the metallicity of the 6.2 Gyr age bin in the Teramo solution now being [MATH] dex larger than it is in Paper III The overall faster enrichment of the Teramo solutions compared to the Pa... |
Varying the Model Parameters There were several parameters in our chemical models which we fixed to match observations or be consistent with previous studies. These parameters included the initial gas mass, initial chemical composition, composition of the inflowing gas, the KS relation exponent, and the stellar yields.... |
[MATH] (B) the same as A but the gas reservoir was pre-enriched to [Fe/H] [MATH] (C) the inflowing gas was pre-enriched by SNe II to [Fe/H] [MATH] (D) a superposition of A and C, (E) a superposition of B and C, (F) [MATH] was increased by a factor of 2, (G) [MATH] was decreased by a factor of 2, (H) [MATH] , and (I) a ... |
and (K) a free inflow model with a constant IFR (i.e., one inflow bin rather than 4). While these tests are by no means exhaustive, they give a rough sense of the potential systematic errors that could be introduced by the assumptions we made in our chemical models. |
Figs. compare the original best-fitting free inflow model (line with diamonds and error bars) to the new models (lines without error bars). The open circles are the results from Paper III . Note that only the original free inflow model errors are shown for clarity. |
In general, the new results are not significantly different from the original results and the fit qualities are unchanged to within [MATH] The SFH, age CDF, MDF, and Z CDF are the least affected by the new parameter values and they remain close to the results of Paper III More variation occurs in the IFH, inflow CDF, g... |
[MATH] [MATH] /Fe] [MATH] of the oldest age bin. All else being equal, increasing [Fe/H] of the inflowing gas shifts [MATH] [MATH] /Fe] [MATH] of all age bins upward [MATH] dex and shifts the initial and final |
[MATH] [Fe/H] [MATH] upward and downward by [MATH] dex, respectively. Models C – F have some of the largest IFRs and gas mass densities. In model F, a larger IFR of primordial gas is needed to balance the increased stellar yields. When the inflowing gas is not primordial, as in models C – E, more of it is required to c... |
One of the largest sources of uncertainty in chemical evolution models is the stellar yields. Changing the instantaneous yields, [MATH] , by a factor of 2, as in models F and G, shifts the [ [MATH] /Fe] vs. [Fe/H] relation vertically by [MATH] dex with only a small change to the relation’s tilt. Increasing or decreasin... |
appear to be the only ways to significantly move the entire [ [MATH] /Fe] vs. [Fe/H] relation down relative to the original model. |
To demonstrate the binning effects on the results, Model I has a different binning scheme from the original free inflow solution. Instead of 4 bins, Model I uses the same 9 logarithmically spaced bins as the SFH. This model is within [MATH] of the original free inflow model and has a similar inflow history and chemical... |
Myr) amounting to [MATH] of the total inflow. During this burst, the IFR increases by a factor [MATH] over the previous two bins. This burst occurs in order to reproduce the apparent SFR burst in the youngest bin seen in the Paper III solution. As the tests and discussion in Paper III bear out, the reality of such a SF... |
In all the preceding results, we have attributed variations in SFR primarily to variations in IFR because the star formation efficiency, [MATH] , was constant with time. A natural question to ask is, can the SFR variations be due, instead, to variations in [MATH] Allowing [MATH] to vary weakens the link between SFH and... |
(e.g. Elmegreen, 1993 ; Sommer-Larsen et al., 2003 ; Schaye, 2004 ; Schaye & Dalla Vecchia, 2008 ; Robertson & Kravtsov, 2008 Models J and K respectively demonstrate that a closed box or constant IFR model provide an acceptable fit to the CMD if we allow [MATH] to vary with time. However, the amount of variation requir... |
(Kennicutt, 1998 ; Wong & Blitz, 2002 ; Komugi et al., 2005 ; Kennicutt et. al., 2007 ; Boissier et. al., 2007 Also, the initial metallicity in model J must be non-zero, otherwise there are too many metal-poor stars. This model does a poor job of reproducing the observed present-day gas surface density of [MATH] |
(see § ). The gas density in model J decreases from about 3 to 2.3 [MATH] for the Padova tracks and from about 2.2 to 1.5 [MATH] |
for the Teramo tracks. Hence, this quantity is above the range plotted in panel (a) of Fig. In model K, the AMR decreases over the last 2 Gyr because gas inflow continues while very few stars form. Model K also has more gas inflow ( [MATH] ) occurring in the last 3 Gyr than the original free inflow model ( [MATH] ). |
Comparison to Observations in M33 Next, we will examine in further detail the predictions of the original free inflow solutions, since they provide the best fits. The Padova solutions have a present-day gas mass of |
[MATH] From the difference between the Padova and Teramo solutions, we estimate a systematic error of a factor of [MATH] due to uncertainties in the stellar tracks. The projected mean HI column density in this field is |
[MATH] , as measured from a mosaic constructed from Very Large Array and Green Bank Telescope observations (D. Thilker, private communication). If we correct for a disc inclination of [MATH] and for a helium abundance one-third that of hydrogen, then this becomes |
[MATH] The true inclination of the HI disc at this location is somewhat uncertain because of the well-known warp at large radii (Corbelli & Schneider, 1997 A change of [MATH] in the inclination would change the surface density by approximately [MATH] For comparison, the azimuthally-averaged HI column density at this ra... |
[MATH] (Corbelli & Schneider, 1997 ; Corbelli, 2003 The present-day SFR provides another useful check on our results. Using GALEX near-UV and far-UV images of M33, Boissier et al. (2007; private communication) computed the UV surface brightness in our field and then converted that to a SFR using the relation in Kennicu... |
[MATH] where the lower limit corresponds to zero extinction and the upper limit corresponds to an extinction of [MATH] The azimuthally-averaged UV SFR at this radius lies at the lower limit. The chief sources of uncertainty in these limits are the UV flux-SFR calibration, which was based on theoretical isochrone and sp... |
[MATH] smaller. Therefore, the range above becomes [MATH] , which is in good agreement with our predictions of [MATH] and [MATH] for the Padova and Teramo free inflow solutions, respectively. |
Another important check comes from oxygen abundances in M33. Magrini et al. ( 2007 compiled an extensive catalog of previously measured abundances in ii regions, type A-B supergiant stars, and planetary nebulae (PNe). This catalog is plotted in Fig. 10 |
with the predictions of the Padova and Teramo free inflow models for the present-day oxygen abundance (star and diamond, respectively), whose abscissa values are offset from each other for clarity. The ii regions and supergiants probe the present-day ISM abundance whereas the PNe probe the ISM abundance at older ages. ... |
Most of the ii region abundances in the Magrini et al. compilation were made from direct [MATH] measurements, generally considered the most reliable kind. Eight objects have at least two independent measurements and three of those objects have three independent measurements. We have also added the recent gradient deriv... |
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