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Chemical and Photometric Evolution Models for Disk, Irregular, and Low Mass Galaxies
We summarize the updated set of multiphase chemical evolution models performed with 44 theoretical radial mass initial distributions and 10 possible values of efficiencies to form molecular clouds and stars. We present the results about the infall rate histories, the formation of the disk, and the evolution of the radial distributions of diffuse and molecular gas surface density, stellar profile, star formation rate surface density, and elemental abundances of C, N, O, and Fe, finding that the radial gradients for these elements begin steeper and flatten with increasing time or decreasing redshift, although the outer disks always show a certain flattening for all times. With the resulting star formation and enrichment histories, we calculate the spectral energy distributions (SEDs) for each radial region by using the ones for single stellar populations resulting from the evolutive synthesis model POPSTAR. With these SEDs we may compute finally the broad band magnitudes and colors radial distributions in the Johnson and in the SLOAN/SDSS systems which are the main result of this work. We present the evolution of these brightness and color profiles with the redshift.
Chemical evolution models to study the evolution of spiral galaxies have been the subject of a high number of works for the last decades. From the works by Lynden-Bell , Tinsley , Clayton [3, 4], and Sommer-Larsen and Yoshii , many other models have been developed to analyze the evolution of a disk galaxy. The first attempts were performed to interpret the G-dwarf metallicity distribution and the radial gradient of abundances [6–16] observed in our Milky Way galaxy (MWG). A radial decrease of abundances was soon observed in most of external spiral galaxies, too (see Henry and Worthey  and references therein), although the shape of the radial distribution changes from galaxy to galaxy, at least when it is measured in dex kpc−1 (recent results seem to indicate that the radial gradient may be universal for all galaxies when it is measured in dex , being the radius enclosing the half of the total luminosity of a disk galaxy, or with any other normalization radius, something already suggested some years ago by Garnett  although the statistics was not large enough to reach accurate conclusions).
It was early evident that it was impossible to reproduce these observations by using the classical closed box model  which relates the metallicity of a region with its fraction of gas over the total mass, (stars, , plus gas, ), . Therefore infall or outflows of gas in MWG and other nearby spirals were soon considered necessary to fit the data. In fact, such as it was established theoretically by Goetz and Koeppen  and Koeppen  a radial gradient of abundances may be created only by 4 possible ways: (1) a radial variation of the initial mass function (IMF), (2) a variation of the stellar yields along the galactocentric radius, (3) a star formation rate (SFR) changing with the radius, and (4) a gas infall rate variable, , with radius. The first possibility is not usually considered as probable, while the second one is already included in modern models, since the stellar yields are in fact dependent on metallicity. Thus, from the seminal works from Lacey and Fall , Güsten and Mezger , and Clayton  most of the models in the literature [25–32], including the multiphase model used in this work, explain the existence of this radial gradient by the combined effects of a SFR and an infall of gas which vary with the galactocentric radius in the galaxy.
Most of the chemical evolution models of the literature, included some of the recently published, are, however, only devoted to the MWG, (totally or only for a region of it, halo or bulge) [33–38] or to any other individual local galaxy as M 31, M 33 or other local dwarf galaxies [35, 39–50]. These works perform the models in a Tailor-Made Models way, done by hand for each galaxy or region. There are not models applicable to any galaxy, except our grid of models shown in Mollá and Diaz (, hereinafter MD05) and these ones from Boissier and Prantzos [32, 52], who presented a wide set of models with two parameters, the total mass or rotation velocity, and the efficiency to form molecular clouds and stars in MD05 and a angular momentum parameter in the last authors grid.
Besides that, these classical numerical chemical evolution models only compute the masses in the different phases of a region (gas, stars, elements, etc.) or the different proportions among them. They do not give the corresponding photometric evolution, preventing the comparison of chemical information with the corresponding stellar one. There exist a few consistent models which calculate both things simultaneously in a consistent way, as those from Vazquez et al. , Boissier and Prantzos  or those from Fritze-Von Alvensleben et al. , Bicker et al. , and Kotulla et al. (, hereafter GALEV). The latter, GALEV evolutionary synthesis models, describe the evolution of stellar populations including a simultaneous treatment of the chemical evolution of the gas and of the spectral evolution of the stellar content. These authors, however, treat each galaxy as a whole for only some typical galaxies along the Hubble sequence and do not perform the study of radial profiles of mass, abundances, and light simultaneously. The series of works by Boissier and Prantzos [32, 52], Prantzos and Boissier , and Boissier et al.  seems one of few that give models allowing an analysis of the chemical and photometric evolution of disk galaxies.
Given the advances in the instrumentation, it is now possible to study high redshift galaxies as the local ones with spatial resolution are good enough to obtain radial distributions of abundances and of colors or magnitudes and thus to perform careful studies of the possible evolution of the different regions of disk galaxies. For instance to check the existence of radial gradients at other evolutionary times different than the present [58–61] and their evolution with time or redshift is now possible. It is also possible to compare these gradients with the radial distributions from the stellar populations to study possible migration effects. It is therefore important to have a grid of consistent chemo-spectro-photometric models which allows the analysis of both types of data simultaneously.
The main objective of this work is to give the spectrophotometric evolution of the theoretical galaxies presented in MD05, for which we have updated the chemical evolution models. In that work we presented a grid of chemical evolution models for 440 theoretical galaxies, with 44 different total mass, as defined by its maximum rotation velocity and radial mass distributions, and 10 possible values of efficiencies to form molecular clouds and stars. Now we have updated these models by including a bulge region and by using a different relation mass-life-meantime for stars now following the Padova stellar tracks. These models do not consider radial flows nor stars migration since no dynamical model is included. The possible outflows by supernova explosions are not included, too. We check that with the continuous star formation histories resulting of these models, the supernova explosions do not appear in sufficient number as to produce the energy injection necessary to have outflows of mass. With these chemical evolution model results, we calculate the spectrophotometric evolution by using each time step of the evolutionary model as a single stellar populations at which we assign a spectrum taken from the POPSTAR evolutionary synthesis models . Our purpose in to give as a catalogue the evolution of each radial region of a disk and this way the radial distributions of elemental abundances, star formation rate, gas, and stars will be available along with the radial profiles of broad band magnitudes for any time of the calculated evolution.
The work is divided as follows: we summarize the updated chemical evolution models in Section 2. Results related to the surface densities and abundances are given in Section 3. We describe our method to calculate the SEDs of these theoretical galaxies and the corresponding broad band magnitudes and colors in Section 4. The corresponding spectrophotometric results are shown in Section 5 where we give a catalog of the evolution of these magnitudes in the rest-frame of the galaxies. Some important predictions arise from these models which are given in the conclusions.
2. The Chemical Evolution Model Description
The models shown here are the ones from MD05 and therefore a more detailed explanation about the computation is given in that work. We started with a mass of primordial gas in a spherical region representing a protogalaxy or halo. The initial mass within the protogalaxy is the dynamical mass calculated by means of the rotation velocity, , through the expression : with in, in kpc, and in km. We used the universal rotation curve from (URC) from Persic et al.  to calculate a set of rotation velocity curves and from these velocity distributions we obtained the mass radial distributions (see MD for details and Figure 2 showing these distributions). It was also possible to use those equations to obtain the scale length of the disk , the optical radius, defined as the one where the surface brightness profile is , which, if disks follow the Freeman’s  law, is and the virial radius, which we take as the galaxy radius . The total mass of the galaxy is taken as the mass enclosed in this radius . The expression for the URC was given by means of the parameter , the ratio between the galaxy luminosity, , and the one of the MWG, , in band I. This parameter defines the maximum rotation velocity, , and the radii described above. Thus, we obtained the values of the maximum rotation velocities and the corresponding parameters and mass radial distributions for a set of values such as it may be seen in Table 1 from MD05.
To the radial distributions of disks calculated by means of (1) described above, we have added a region located at the center to represent a bulge. The total mass of the bulge is assumed as a 10% of the total mass of the disk. The radius of this bulge is taken as . Both quantities are estimated from the correlations found by Balcells et al.  among the disk and the bulges data.
2.1. The Infall Rate: Its Dependence on the Dynamical Mass and on the Galactocentric Radius
We assume that the gas falls from the halo to the equatorial plane forming out the disk in a scenario ELS . The time-scale of this process, or collapse-time scale , characteristic of every galaxy, changes with its total dynamical mass , following the expression from Gallagher III et al. : where is the total mass of the galaxy and is its age. We assume that all galaxies begin to form at the same time and evolve for a time of Gyr. We use the value of 13.8 Gyr, given by the Planck experiment  for the age of the Universe and therefore galaxies start to form at a time .
Normalizing to MWG, we obtain where is the total mass of MWG and Gyr (see details in the next paragraph) is the assumed characteristic collapse-time scale for our galaxy.
The above expression implies that galaxies more massive than MWG form in a shorter time-scale, that is more rapidly, than the least massive galaxies which will need more time to form their disks. This assumption is in agreement with the observations from Jimenez et al. , Heavens et al. , and Pérez et al.  which find that the most massive galaxies have their stellar populations in place at very early times while the less massive ones form most of their stars at . This is also in agreement with self-consistent cosmological simulations which show that a large proportion of massive objects are formed at early times (high redshift), while the formation of less massive ones is more extended in time, thus simulating a modern version of the monolithic collapse scenario ELS.
The calculated collapse-time scale is assumed that corresponds to a radial region located in a characteristic radius, which is /2 ~ 6.5 kpc for the MWG model which uses the distribution with and number 28, with a maximum rotation velocity . The value Gyr was determined by a detailed study of models for MWG. We performed a large number of chemical evolution models changing the inputs free-parameters and comparing the results with many observational data [27, 28] to estimate the best value. Similar characteristic radii (all these radii and values are related to the stellar light and not to the mass, but we clear that we do not use them in our models except to define the characteristic radius for each theoretical mass radial distribution. The free parameters are selected for the region defined by this but taking into account that we normalize the values after a calibration with the solar neighborhood model, a change of this radius would not modify our model results) for our grid of models were given in Table 1 from MD05 too, with the characteristic (from now we will use the expression for for the sake of simplicity) obtained for each galaxy total mass .
By taking into account that the collapse time scale depends on the dynamical mass and that spiral disks show a clear profile of density with higher values inside than in the outside regions, we may assume that the infall rate, and therefore the collapse time scale , has a radial dependence, too. Since the mass density seems to be an exponential in most of cases, we then assumed a similar expression: where is the scale-length of the collapse time-scale, taken as around half of the scale-length of surface density brightness distribution, ; that is .
Obviously the collapse time scale for the bulge region is obtained naturally from the above equation with . We show in Figure 1(a) the collapse time scale , in natural logarithmic scale, as a function of the galactocentric radius, for seven radial distributions of total mass, as defined by their maximum rotation velocity, , and plotted with different colors, as labeled. These seven theoretical galaxies are used as examples and their characteristics are summarized in Table 1, where we have the number of the distribution, , corresponding to column 2 from Table 1 in MD05 in column 1, the maximum rotation velocity, , in , in column 2, the total mass, , in units, in column 3, the theoretical optical radius , following Persic et al.  equations, in kpc, in column 4, the collapse time scale in the characteristic radius, , in Gyr in column 5, the value which defines the efficiencies (see Equation (7) in Section 2.2) in column 6, and the values for these efficiencies in columns 7 and 8.
The red line corresponds to a MWG-like radial distribution. The long-dashed black line shows the time corresponding to 2 times the age of the Universe. Such as we may see, the most massive galaxies would have the most extended disks, since the collapse timescale is smaller than the age of the universe for longer radii, thus allowing the formation of the disk until radii as larger as 20 kpc, while the least massive ones would only have time to form the central region, smaller than 1-2 kpc, as observed. The dashed (gray) lines show the time corresponding to 2 times the age of the Universe, Gyr. The dotted black line defines the collapse time scale for which the maximum radius for the disk of the MWG model would be 13 kpc, the optical radius, and it corresponds to a collapse time scale of 5 times the age of the Universe.
Other authors have also included a radial dependence for the infall rate in their models [23, 26, 27, 52, 80] with different expressions. In fact this dependence, which produces an in-out formation of the disk, is essential to obtain the observed density profiles and the radial gradient of abundances, such as it has been stated before [26, 28, 52]. In Figure 1(b) we show the collapse time scale , in natural logarithmic scale, assumed in different chemical evolution models of MWG, as a function of the galactocentric radius. The red solid line corresponds to our MWG model ( and number 28 of the mass distributions of Table 1) from MD05. The other functions, straight lines, are those used by Chang et al. , Boissier and Prantzos , Chiappini et al. , Renda et al. , Carigi and Peimbert , and Marcon-Uchida et al. , as labeled. Since they use straight lines the collapse time scale for our model results shorter for the inner disk regions (except for the bulge region 3-4 kpc) and longer for the outer ones, than the ones used by the other works. This will have consequences in the radial distributions of stars and elemental abundances as we will see.
2.2. The Star Formation Law in Two-Steps: The Formation of Molecular Gas Phase
The star formation is assumed different in the halo than in the disk. In the halo the star formation follows a Kennicutt-Schmidt law. In the disk, however, we assume a more complicated star formation law, by creating molecular gas from the diffuse gas in a first step, again by a Kennicutt-Schmidt law. And then stars from the cloud-cloud collisions. There is a second way to create stars from the interaction of massive stars with the surrounding molecular clouds.
Therefore the equation system of our model is
These equations predict the time evolution of the different phases of the model: diffuse gas, , molecular gas, , low mass stars, , and intermediate mass and massive stars, , and stellar remnants, , (where letters and correspond to disk and halo, resp.). Stars are divided in 2 ranges, being the low mass stars and the intermediate and massive ones, considering the limit between both ranges stellar a mass . are the mass fractions of the 15 elements considered by the model: H, D, He, He, C, O, N, C, Ne, Mg, Si, S, Ca, and Fe and the rich neutron isotopes created from C, O, N, andC.
Therefore we have different processes defined in the galaxy as follows,(i)Star formation by spontaneous fragmentation of gas in the halo: , where we use .(ii)Clouds formation by diffuse gas: with too.(iii)Star formation due to cloud collision: .(iv)Diffuse gas restitution due to cloud collision: .(v)Induced star formation due to the interaction between clouds and massive stars: .(vi)Diffuse gas restitution due to the induced star formation: .(vii)Galaxy formation by gas accretion from the halo or protogalaxy: ,where , , , and are the proportionality factors of the stars and cloud formation and are free input parameters (since stars are divided in two groups: those with and , the parameters are divided in the two groups too; thus, , , ).
Thus, the star formation law in halo and disk is
Although the number of parameters seems to be large, actually not all of them are free. For example, the infall rate, , is the inverse of the collapse time , as we described in the above section. Proportionality factors , , , and have a radial dependence, as we show in the study of MWG Ferrini et al. , which may be used in all disks galaxies through the volume of each radial region and some proportionality factors called efficiencies. These efficiencies or proportionality factors of these equations have a probability meaning and therefore their values are in the range [0, 1]. The efficiencies are then the probability of star formation in the halo, , the probability of cloud formation, , cloud collision, , and the interaction between massive stars . This last one has a constant value since it corresponds to a local process. The efficiency to form stars in the halo is also assumed constant for all of them. Thus, the number of free parameters is reduced to and .
The efficiency to form stars in the halo, , is obtained through the selection of the best value to reproduce the SFR and abundances of the Galactic halo (see Ferrini et al. , for details). We assumed that it is approximately constant for all halos with a value –6 × 10−3. The value for is also obtained from the best value for MWG and assumed as constant for all galaxies since these interactions massive stars-clouds are local processes. The other efficiencies and may take any value in the range [0-1]. From our previous models calculated for external galaxies of different types , we found that both efficiencies must change simultaneously in order to reproduce the observations, with higher values for the earlier morphological types and smaller for the later ones. In MD05 there is a clear description about the selection of values and the relation . As a summary, we have calculated these efficiencies with the expressions: selecting 10 values between 1 and 10 (we suggest to select a value similar to the Hubble type index to obtain model results fitting the observations). The efficiencies values computed for the grid from MD05 are shown in Table 2 from that work.
2.3. Stellar Yields, Initial Mass Function, and Supernova Ia Rates
The selection of the stellar yields and the IMF needs to be done simultaneously since the integrated stellar yield for any element, which defines the absolute level of abundances for a given model, depends on both ingredients. In this work we used the IMF from Ferrini et al.  with limits and . The stellar yields are from Woosley and Weaver  for massive stars ( and from Gavilán et al. [84, 85] for low and intermediate mass stars (). Stars in the range have no time to die, so they still live today and do not eject any element to the interstellar medium. The mean stellar lifetimes are taken from the isochrones from the Padova group [86–89] instead of using those from the Geneva group Schaller et al. . This change is done for consistency since we use the Padova isochrones on the POPSTAR code that we will use for the spectrophotometric models. The supernova Ia yields are taken from Iwamoto et al. . The combination of these stellar yields with this IMF produces the adequate level of CNO abundances, which is able to reproduce most of observational data in the MWG galaxy [84, 85], in particular the relative abundances of C/O, N/O, and C/Fe, O/Fe, N/Fe. The study of other combinations of IMF and stellar yields will be analyzed in Mollá et al. . The supernova type Ia rates are calculated by using prescriptions from Ruiz-Lapuente et al. .
3. Results: Evolution of Disks with Redshift
The chemical evolution models are given in Tables 2 and 3. We show as an example some lines corresponding to the model and , the whole set of results will be given in electronic format as a catalogue. In Table 2 we give the type of efficiencies and the distribution number in columns 1 and 2, the time in Gyr in column 3, the corresponding redshift in column 4, the radius of each disk region in kpc in column 5, and the star formation rate in the halo and in the disk, in , in columns 6 and 7. In next columns 8 and 9 we have the total mass in the halo and in the disk. In columns 10 to 16 we show the mass in each phase of the halo, diffuse gas, low-mass stars, massive stars, and mass in remnants (columns 10 to 13) and of the disk, diffuse and molecular gas, low-mass stars, massive stars, and mass in remnants (columns 14 to 18).