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A simple multistage closed-(box+reservoir) model of chemical evolution

Simple closed-box (CB) models of chemical evolution are extended on two respects: (i) simple closed-(box+reservoir) (CBR) models allowing gas outflow from the box into the reservoir or gas inflow into the box from the reservoir with same composition as the preexisting gas and rate proportional to the star formation rate, and (ii) simple multistage closed-(box+reservoir) (MCBR) models allowing different stages of evolution characterized by different inflow or outflow rates. The stellar initial mass function is assumed to be universal, and mass conservation holds for the whole system (box+reservoir) while it is violated for each subsystem (box and reservoir). The theoretical differential oxygen abundance distribution (TDOD) predicted by the model, under the assumption of instantaneous recycling, is a continuous broken line, where different slopes are related to different inflow rates. For an application of the model (a) a fictitious sample is built up from two distinct samples and taken as representative of the inner Galactic halo, and (b) different [O/H]-[Fe/H] empirical relations are deduced from five different samples related to different methods, and two of them are selected for determining the empirical differential oxygen abundance distribution (EDOD) with regard to the fictitious sample. In both cases the EDOD is represented, to an acceptable extent, as a continuous broken line. If the inner halo and the metal-poor bulge (after the inflow stage) are represented by the box and the reservoir, respectively, then the inner halo fractional mass (normalized to the halo) is comparable with, or exceeds by a factor up to 4, the metal-poor bulge fractional mass (normalized to the bulge), for current estimates of the halo-to-bulge mass ratio of about 0.05-0.10.


Introduction
The empirical metallicity distributions of long-lived stars of different populations, constrain models for the formation and the evolution of the Galaxy. Simple closed-box (CB) models make a useful tool in the description of galactic chemical evolution. The original formulation (Searle and Sargent 1972;Pagel and Patchett 1975) relies on instantaneous recycling and homogeneous mixing approximations i.e. gas from dying stars is instantaneously returned to and homogeneously mixed with the interstellar medium. In addition, (1) a null star mass fraction and a null metal abundance are taken as initial values, and (2) mass conservation is assumed to hold, which implies no gas outflow from the box or inflow into the box.
Simple CB models may be extended in many respects, such as nonzero initial metal abundance (prompt initial enrichment) and/or star mass fraction (Truran and Cameron 1971), inhomogeneous mixing (Caimmi 2000(Caimmi + erratum 2001a, hereafter quoted together as C00; Caimmi 2001b, hereafter quoted as C01), and gas outflow (Hartwick 1976, hereafter quoted as H76) or inflow (Caimmi 2007, hereafter quoted as C07) with same composition as the preexisting gas and rate proportional to the star formation rate. Formulations implying more extreme changes, such as reject of instantaneous recycling and unconditioned gas outflow or inflow, can no longer be considered as "simple" models.
The existence of a G-dwarf problem i.e. detection of too few metal deficient G dwarf (or, more generally, of a selected spectral type) with respect to that which could be expected from simple CB models of chemical evolution (e.g., Searle and Sargent 1972;Pagel and Patchett 1975;Haywood 2001) was first established in the solar neighbourhood (van den Bergh 1962; Schmidt 1963). Although in a less extreme form, a G-dwarf problem appears to exist both in the halo (e.g., H76; Prantzos 2003) and in the bulge (e.g., Ferreras et al. 2003). In addition, a G-dwarf problem has been recognized in both bulgedominated and disk-dominated galaxies (Henry and Worthey 1999), which is consistent with the idea that the G-dwarf problem is universal (Worthey et al. 1996).
According to current ΛCDM cosmological scenarios, galaxies were largely built out of disrupted smaller subunits (dSph galaxies whose surviving cores could be massive globular clusters). A similar process is presently occurring on a larger scale: a central bulge (a cD galaxy) is accreting in galaxy clusters, at the expense of infalling smaller galaxies. In this view, a universal Gdwarf problem appears to be closely related to the initial assembling stage of galactic evolution.
With regard to the Galaxy, the empirical differential oxygen abundance distribution (EDOD) shows that (i) at least two different stages of (chemical) evolution exist, and (ii) each stage can be described using an extended simple CB model, for both the inner halo (C07), the bulge (C07), the thick disk (Caimmi and Milanese 2009, hereafter quoted as CM09), and the thin disk (CM09). In particular, the earlier stage is marked by an initially increasing EDOD.
As shown in recent attempts (Carollo et al. 2007(Carollo et al. , 2010, the Galactic halo may be divided into two structural components, namely (1) an inner spheroid with axes, a ≈ 15 kpc, c ≈ 10 kpc, which exhibits a modest prograde rotation and a metallicity peak at [Fe/H] ≈ −1.6, and (2) an outer sphere with radius, R ≈ 20 kpc, which exhibits a clear retrograde net rotation and a metallicity peak at [Fe/H] ≈ −2.2. In addition, the inner halo population dominates within its own volume, while the outer halo population includes a larger fraction of low-metallicity ([Fe/H] < −2.0) stars than does the inner halo population.
For a component of orbital motion measured with respect to the Local Standard of Rest, V < 200 km s −1 , the distribution of [Fe/H] appears similar to what in the past was considered "the halo", with a single metallicity peak at [Fe/H] ≈ −1.6 (Carollo et al. 2007, Fig. 2), which implies the inner halo population is dominant in the related sample. A similar trend is exhibited by the metallicity distribution inferred from an earlier attempt (Ryan and Norris 1991, hereafter quoted as RN91, Fig. 5c), and the related sample (hereafter quoted as the RN91 sample) can also be considered as representative of the inner halo. In both cases, data are biased towards low metallicities, [Fe/H] < −3.0.
The inner halo low-metallicity tail has been determined using data from the Hamburg/ESO survey (Schörck et al. 2009, hereafter quoted as H V, Fig. 11), but the related sample (hereafter quoted as the H V sample) is biased towards higher metallicities, [Fe/H] > −3.0, and extremely low metallicities, [Fe/H] < −4.2. In the comparison of the empirical with the theoretical metallicity distribution, selection effects must be taken into account. Accordingly, each theoretical metallicity distribution under consideration has to be converted into its counterpart as it would be observed by applying the metal-poor star selection criteria used in the survey (H V).
If, on the other hand, the H V sample is dominated by the inner halo population, a correction for the above mentioned selection effects is expected to yield a metallicity distribution with a single peak at [Fe/H] ≈ −1.6, similar to its counterpart related to earlier samples (RN91; Carollo et al. 2007, Fig. 2, V < 200 km s −1 ). In fact, the metallicity distributions related to the RN91 and H V sample can be adjusted to match in the range, −3.4 < [Fe/H] < −2.6, where selection effects are negligible (H V,Figs. 11,12).
The Milky Way inner halo offers a unique opportunity to construct the star formation and mass assembly history of a galactic inner halo, hence providing a unique benchmark for theories of galaxy formation and evolution. To this aim, the sample used has to be complete at least for a wide range of metallicity. In the case under discussion, −3.6 < [Fe/H] < −2.8 for the H V sample and −3.4 < [Fe/H] < −1.0 for the RN91 sample. The remaining ranges are affected by various biases, mainly due to the occurrence of selection effects for the low-metallicity tail above a threshold (H V) and disk contamination for the high-metallicity tail (RN91, H V).
The metallicity distribution of the inner halo can be deduced from a Simple CB models of chemical evolution were first extended allowing for gas outflow (H76) and later for moderate gas inflow, with same metal abundance as in the preexisting gas, yielding negative TDOD slopes in the (Oφψ) plane. On the other hand, the low-metallicity tail of the EDOD is fitted to a straight line with positive slope for both the halo (C07), the bulge (C07), the thick disk (CM09), and the thin disk (CM09), which is achieved by extending simple CB models to strong gas inflow with same metal abundance as the preexisiting gas.
In this view, different stages of evolution are related to different domains in the normalized oxygen abundance, (φ U ) i ≤ φ ≤ (φ U ) f , where the EDOD is fitted to a regression line, ψ = a U φ + b U , with slope, a U , and intercept, b U (U = I, II, ..., is the stage considered, beginning and ending at i and f configurations, respectively). For reasons of continuity, the final values of parameters related to a selected stage must necessarily coincide with the corresponding initial values related to the next stage, with the exception of the outflow or inflow rate, which can be deduced from the EDOD. Accordingly, the knowledge of present values allows the calculation of past values, in the light of the model, getting insight on the formation and evolution of the system under consideration.
In summary, the current paper is aimed to the following main points: (i) metallicity distribution related to a fictitious sample (fs10), supposed to be representative of the inner halo (subsection 2.1); (ii) choice of two different 3); (iv) formulation of extended simple CB models, namely (a) simple CBR models, allowing for gas outflow or inflow with same metal abundance as the preexisting gas, and (b) simple MCBR models, allowing for different stages of evolution (subsections 3.1, 3.2 and 3.4); (v) determination of the best fitting TDOD to the EDOD inferred from the fs10 sample (subsection 3.3); (vi) application to a special stellar system resembling the inner Galactic halo (subsection 3.5). In addition, the discussion and the conclusion make the subject of subsection 3.6 and section 4, respectively.

Data and inferred quantities
The following ingredients are necessary for the determination of the EDOD (in particular, related to the inner Galactic halo), from which input parameters of a selected model are deduced: (i) a representative sample; (ii) a related [

Building up a fictitious sample (fs10)
In building up a fictitious sample (the fs10 sample) of inner halo stars, two different samples shall be considered, namely the RN91 sample (N = 372) of kinematically selected halo subdwarfs and the H V sample (N = 3439) of metal-poor stars selected from the Hamburg/ESO objective prism survey. More specifically, the H V sample has been determined from a subsample (N = 1638) with available spectroscopic follow-up observations, by means of scaling to the full parent sample (N = 3439). For further details refer to the parent paper (H V Contamination from outer halo stars also exists in both samples, but the effect is expected to be negligible. In fact, the inner halo dominates within a spheroid centered on the Galactic centre, with equatorial plane coinciding with the Galactic plane, and semiaxes (a, c) = (15, 10) kpc (Carollo et al. 2007). On the other hand, both the RN91 and H V sample are made mainly of stars placed within the above mentioned spheroid. Then the contamination from outer halo stars is expected to be appreciable only at low ([Fe/H] < −2.0) metal abundances (Carollo et al. 2007 6 (H V); (iv) a bimodal metallicity distribution cannot be fitted to its theoretical counterpart deduced from simple models of chemical evolution, even if the latter is "observed" by applying the metal-poor selection criteria used in the Hamburg/ESO survey (H V).
The H V sample is mainly made of giant stars. A subsequent study on the stellar content of the Hamburg/ESO survey has been focused on a sample (N = 617) of main sequence turnoff stars (Li et al. 2010). Both samples exhibit a quite analogous metal abundance distribution where the halo population dominates ([Fe/H] < −2.0), while the contrary holds for higher values ([Fe/H] > −2.0). In the latter case, the H V sample is also affected from the survey-volume effect. For further details refer to the parent paper (Li et al. 2010).
Basing on the above considerations, the following working hypotheses are made.  (3) The RN91 and H V sample are equally representative (even if belonging 2 The problem was raised via e-mail (December 2009) to two coauthors of the parent paper (H V) and to two coauthors of the paper where inner and outer halo populations were first detected (Carollo et al. 2007), but no reply was received up to-day. It cannot be excluded that related messages have been lost into the spam bin. Accordingly, the fs10 sample can be built up from the RN91 and H V samples, by normalizing to the same number of stars within the metallicity range where the above mentioned samples are supposed to be equally representative of the inner halo. More specifically, the normalization shall be performed with respect to the H V sample, but the Poissonian error related to each bin shall remain unchanged with respect to the parent sample, with the minimum among the two retained for the bin where the parent samples are equally representative of the inner halo. Then the number of stars belonging to the fs10 sample, with regard to a selected metallicity bin, remains

The [O/H]-[Fe/H] empirical relation
In dealing with simple models of chemical evolution, involving the assumption of instantaneous recycling, the predicted metal abundance has to be compared to the observed oxygen (or any other primary element) abundance (e.g., Pagel 1989; C00; C01). Unfortunately, oxygen is more difficult than iron to detect, and an empirical formula may be needed to express the former as a function of the latter. The population of available samples where oxygen abundances are directly determined, does not exceed a few hundreds at most (e.g., Ramirez Fig. 3, are interpolated using the regression models listed above. For heteroscedastic data and homoscedastic models, the typical uncertainties are assigned to all the data points. The results are listed in Table 2. An inspection of Fig. 3 and Table 2 shows that systematic errors related to different methods and/or different spectral lines in oxygen abundance determination, are dominant on both the dispersion due to measurement errors and intrinsic scatter. At this stage, a precise fit to the data extracted from a selected sample would be meaningless, and only acceptable fits related to different situations shall be considered. Accordingly   The oxygen mass abundance normalized to the solar value, φ, to a good extent may be expressed as (e.g., C07): and a selected bin centre and bin semiamplitude, related to the [O/H] distribution, read (e.g., C07):  (1) and (2), by use of Eqs. (6) and (7) in the latter case, are represented in Table 3, where Following recent attempts (Rocha-Pinto and Maciel 1996; C00; C01; C07), the EDOD in a selected class of objects is defined as: where ∆N is the number of objects within a normalized oxygen abundance bin, ∆φ, centered in φ, and N is the number of sample objects. The increment    (Pagel 1989;Malinie et al. 1993) is replaced by its normalized counterpart, ∆N/(N∆φ), to allow comparison between different samples. The uncertainty on ∆N is evaluated from Poisson errors (e.g., RN91), as σ ∆N = (∆N) 1/2 , and the related uncertainty in the EDOD is: where ψ − → −∞ in the limit ∆N → 1. For further details refer to the parent papers (C01; C07). The EDOD related to RN91, H V, and fs10 samples are listed in Table  4 (1) and (2), left and right side, respectively. The EDOD related to the fs10 sample, taken to be representative of the inner halo, is plotted in Fig. 4  A main feature of the EDODs plotted in Fig. 4, is the presence of five regions characterized by a nearly linear trend, which shall be named O, A, F, C, E, respectively, and defined in the following metallicity ranges, each containing n X bins, X=O, A, F, C, E.  [ The regression line related to the EDOD within each populated region, has been determined using the B model listed in Table 2, under the assumption that the intrinsic scatter is dominant (Isobe et al. 1990). A single high-metallicity star from the RN91 sample, considered as due to disk contamination or, in any case, an outlier, has not been included in the fitting procedure. The regression line slope and intercept estimators and related dispersion estimators are listed in Table 5 for each region of the EDODs plotted in Fig. 4. The results are consistent (within ∓σ) with their counterparts determined using the other models listed in Table 2. The selection of a special method is of little relevance, due to the paucity of data within each region.
The regression lines are represented in Fig. 5 for each region (from the left to the right): A (dotted, positive slope), F(dotted, negative slope), C (full), E (dashed), with regard to the EDOD plotted in Fig. 4. In absence of LTE approximation, case SH1 (right panels), a more inclined dashed line fits to the FC region, in alternative to F and C regions separately. To ensure continuity, the above mentioned regions must be redefined by the intersections of regression lines, which make the transition from a selected region to the adjacent one. The results are listed in Table 6 for the EDOD and related regression lines plotted in Fig. 5. A vertical line instead of a regression line has been considered for the O region, where no data exist.
In conclusion, the EDOD related to the inner halo may be approximated, to a satisfactory extent, as the sum of four (A, F, C, E) or three (A, FC, E) regions, within which the trend is linear. An interpretations in terms of simple models of chemical evolution is highly attractive, as the corresponding TDOD shows, in fact, a linear trend (Pagel 1989; Rocha-Pinto and Maciel 1996; C00; C01; C07).

The model
In their original formulation, simple models of chemical evolution are closedbox (CB) i.e. mass conservation (gas + stars) holds (e.g., Searle and Sargent 1972;Pagel and Patchett 1975). In later formulation, a "reservoir" is added to the "box", where mass conservation no longer occurs within the box, but still holds within the system box + reservoir. Accordingly, related models can be conceived as closed-(box+reservoir) (CBR) models. The gas within the box is "active" in the sense that allows star formation according to a specified birth-rate stellar function. The gas within the reservoir is "inhibited" in the sense that no star formation takes place. Gas may outflow from the box into the reservoir (H76) or inflow into the box from the reservoir (C07). The related TDOD is a straight line where the slope (to the knowledge of the author) is negative in all cases studied in literature (e.g., Pagel 1989; Rocha-Pinto and Maciel 1996; C00; C01; C07). For this reason, the formulation of CBR models shall be extended to a TDOD with positive slope, under the standard assumptions of CB models: (i) instantaneous recycling within the box, where stars are divided into two categories, namely (a) short-lived, which instantaneously evolve, and (b) long-lived, all of which are still evolving; (ii) instantaneous mixing within the box, where the gas returned from short-lived stars is instantaneously mixed with the interstellar medium yielding uniform composition; and the standard assumptions of CBR models: (iii) mass conservation within the system (box+reservoir); (iv) gas outflow from the box into the reservoir or inflow into the box from the reservoir, at a rate proportional to the star formation rate; (v) inhibition of star formation within the reservoir; (vi) gas outflow or inflow with same composition with respect to preexisting gas.
In this picture, the oxygen (or any other primary element) mass fraction can be determined, extending the procedure followed for CB models (e.g., Pagel and Patchett 1975;Wang and Silk 1993; C00) to CBR models.

Basic theory
The change in oxygen (or any other primary element) gas mass, M gO , is owing to four contributions, as: related to subtraction via star formation (sf), subtraction via outflow from the box into the reservoir or addition via inflow into the box from the reservoir (gr), addition via unsynthesised gas from short-lived stars (sdu), and addition via newly synthesised gas from short-lived stars (sds), respectively. For simple CBR models, the following relations hold: where M g is the mass in active gas, M S is the global mass in gas which has been turned into stars, M gO is the oxygen mass in active gas, α is the fraction in long-lived stars and stellar remnants within a star generation (lock parameter), κ is the ratio of flowing (outflow from the box into the reservoir or inflow into the box from the reservoir) gas rate to locking (in the form of long-lived stars and stellar remnants) gas rate (flow parameter), Z O and Z are the oxygen and metal mass abundance, respectively, within the active gas, andp is the ratio of the oxygen mass newly synthesised and returned to the interstellar medium, for a metal-free initial composition 5 , to the mass locked up in long-lived stars and stellar remnants (yield parameter). The lock parameter, α, the flow parameter, κ, and the yield parameter,p, for sake of brevity, in the following shall be quoted as the lock, the flow, and the yield, respectively. The definition of oxygen mass abundance, Z O , fractional active gas mass, µ, and fractional star mass, s, read: where M s = αM S is the mass in long-lived stars and stellar remnants, and M 0 is the total mass within the box at the starting configuration. The substitution of Eqs. (11)-(13) into (10) yields: and the fractional oxygen gas mass, via Eq. (14) reads: where, in addition: on the other hand, in terms of fractional masses, Eq. (15) may be cast under the equivalent form: and the combination of Eqs. (17) and (18) yields: wherep ′′ is the effective yield parameter, hereafter quoted as the effective yield (H76).
In the special case of a linear Z O -Z relation: the combination of Eqs. (19) and (21) produces: which can be integrated. After some algebra, the result is: where the index, i, denotes the starting configuration at the cosmic time, t i . Reversing the role of the variables, Eq. (23) reads: which is monotonic in Z O within the domain, 0 ≤ Z O ≤ 1.
Using the MacLaurin series development: under the further assumption that the terms of higher order with respect to the first can be neglected, Eq. (23) reduces to: where the index, f , denotes the ending configuration at the cosmic time, t f , and µ i < µ < µ f or µ i > µ > µ f owing to Eq. (24). The initial oxygen and metal abundance may safely be neglected with respect to unity, (Z O ) i ≤ Z i ≪ 1, which implies A → 1 via Eq. (21). Accordingly, Eq. (26) reduces to: which is the classical formulation (H76).
It is worth emphasyzing that the inequality, expressed by Eq. (27), does not affect the instantaneous recycling approximation, but only the general formulation expressed by Eq. (23) provided the Z O -Z relation is linear, according to Eq. (21). In general, it is assumed the instantaneou recycling approximation holds for sufficiently high fractional active gas mass fraction, µ > ∼ 0.1 (e.g., Prantzos 2007, Fig. 12 therein), but the threshold could be lowered using Eq. In terms of fractional masses, Eq. (11) via Eq. (14) may be cast under the equivalent form: which can be read in the following way: any mass change in active gas is counterbalanced by a change in long-lived stars and stellar remnants plus a change in gas outflow from the box into the reservoir (κ > 0) or a change in gas inflow into the box from the reservoir (κ < 0). More specifically, a number of different flow regimes may be distinguished as listed below.
• Stagnation regime (κ = 0), where star formation is neither inhibited nor enhanced. Accordingly, CBR models reduce to CB models. For an exhaustive description refer to earlier attempts (Searle and Sargent 1972; Pagel and Patchett 1975).
• Moderate inflow regime (−1 < κ < 0), where star formation is enhanced and active gas mass fraction monotonically decreases in time.
For an exhaustive description refer to an earlier attempt (C07).
• Steady inflow regime (κ = −1), where star formation is enhanced and active gas mass fraction remains unchanged.
• Strong inflow regime (κ < −1), where star formation is enhanced and active gas mass fraction monotonically increases in time.
The effective yield,p ′′ , defined by Eq. (20), cannot exceed the real yield, p, both in outflow and in strong inflow regime, while the contrary holds in moderate inflow regime and, a fortiori, in steady inflow regime, where a divergency occurs. For this reason, the effective yield cannot be considered alone, but together with the factor, ln[µ/(µ) i ]. More specifically, the righthand side of Eq. (28), −p ′′ ln[µ/(µ) i ] = −p ln[µ/(µ) i ]/(1 + κ), is positive in all regimes due to the trend exhibited by the active gas mass fraction. In steady inflow regime, the following relation necessarily holds: in terms of the oxygen abundance related to a constant active gas mass fraction, or in terms of the cosmic time.
In any case, mass conservation follows from integration of Eq. (29), as: conformly to Eq. (14), where D is the gas mass fraction which outflowed from the box into the reservoir (κ > 0) or inflowed into the box from the reservoir (κ < 0). An equivalent form of Eq. (31) reads: where variables on the left and the right-hand side member relate to the box and the reservoir, respectively. The birth-rate stellar function (number of stars born per unit volume, mass, and time), needs to be specified for determining the temporal behaviour of gas and star fractional masses. Following an earlier attempt (C00), the selected choice reads: where B is a normalization constant, V the volume of the box, Φ( m) the stellar initial mass function and f (µ) an assigned function of the active gas mass fraction, which is the time-dependent term. The number of stars born per unit volume within an infinitesimal dimensionless mass range, m∓ d m/2, and infinitesimal dimensionless time range, The number of long-lived stars generated (within the box) up to an assigned dimensionless cosmic time, t, is: where m mf is the lower stellar mass limit, and m ℓℓ the upper mass limit of long-lived stars. The mass fraction in stars globally generated (within the box) up to a dimensionless cosmic time, t, is: where m Mf is the upper stellar mass limit. The mass fraction in long-lived stars and in gas outflowed from the box into the reservoir or inflowed into the box from the reservoir (in both cases with equal composition with respect to the preexisting gas), at an assigned dimensionless cosmic time, t, are: according to the definition of lock and flow, respectively. The star formation rate (within the box), the lock rate, and the flow rate at an assigned dimensionless cosmic time, t, are: dD d t = ακCf (µ) ; (41) and the combination of Eqs. (29) and (39) yields: which is equivalent to: in the special case, f (µ) = µ, Eq. (43) can be integrated as: and the combination of Eqs. (20), (28), and (44) yields a linear trend for the oxygen abundance: regardless of the flow, κ.
In general, simple CBR models are described by Eqs. (28) and (43), the latter only for including the temporal behaviour.

Theoretical differential oxygen abundance distribution (TDOD)
For a selected spectral class, the number of long-lived stars generated (within the box) up to an assigned dimensionless cosmic time, is: where m 1 and m 2 are the lower and upper mass limit of the spectral class.
For simple CBR models, the oxygen abundance is monotonically increasing in time owing to Eqs. (23) and (46), which implies the number of stars born up to the cosmic time, t, coincides with the number of stars with oxygen abundance up to the related value, Z O (t), or N(t) = N(Z O ). The same holds for the long-lived star mass fraction (including stellar remnants), s(t) = s(Z O ). By use of Eq. (14), the long-lived star mass fraction and the number of long-lived stars, N ℓℓ , can be related as: where m is the mean mass of long-lived stars (including stellar remnants), m = M s /N ℓℓ . Under the assumption of a universal stellar initial mass function, the fraction of long-lived stars belonging to a selected spectral class with respect to the total, maintains unchanged, N ℓℓ /N = const, and Eq. (50) translates into (Pagel and Patchett 1975): and the combination of Eqs. (31) and (32) yields: (1 + κ)s = 1 − µ ; (52) accordingly, Eq. (51) may be cast under the equivalent form: where the ratio, µ/µ i , is expressed by Eq. (28) with respect to the oxygen abundance, Z O , and by Eqs. (44) and (46) with respect to the dimensionless cosmic time, t.
In terms of the normalized oxygen abundance, and the derivation of Eqs. (51) and (53) with respect to φ yields: which, using Eq. (54), takes the explicit form: where the decimal logarithm of the left-hand side is defined as the TDOD (Pagel 1989; C00; C01): which is represented by a straight line on the (Oφψ) plane. The TDOD slope, a, defined by Eq. (58), depends on the flow regime discussed above. More specifically, a < 0 both in outflow regime and in moderate inflow regime; a = 0 in steady inflow regime; a > 0 in strong inflow regime. The TDOD intercept, b, defined by Eq. (59), must necessarily fulfill the condition, µ f > 0, which implies the inequality: b > b(µ f = 0) = log(− ln 10 a) − aφ i ; a < 0 ; (60) on the other hand, µ f > 0 directly follows from a ≥ 0.

Fitting to empirical differential oxygen abundance distribution (EDOD)
Both the EDOD and the TDOD can be represented on the (Oφψ) plane, but related normalizations are different. More specifically, the former is normalized to the sample population, N, according to Eq. (8), while the latter is normalized to the computed long-lived star population, N f − N i , according to Eq. (57). Then the EDOD and the TDOD differ by a normalization constant, log C N , which must be taken into consideration in performing the fitting procedure. In other words, the TDOD, ψ T , has to be vertically shifted on the (Oφψ) plane by a value, log C N , for matching to the EDOD, ψ E . The TDOD intercept, defined by Eq. (59), translates into: where a and b can be determined as the EDOD regression line slope and intercept, respectively. Using Eq. (58), the empirical counterpart of Eq. (27), which implies the validity of Eq. (54), reads: keeping in mind that the active gas mass fraction, µ, monotonically changes as a function of the normalized oxygen abundance, φ, for simple CBR models. The particularization of Eq. (54) to the final configuration, by use of Eq. (58), reads: and the combination of Eqs. (61) and (63) yields: which may be determined from the knowledge of the EDOD and related regression line.

Different stages of evolution
The mere existence of a G-dwarf problem in different regions of the Galaxy where (s U ) i and (s U ) f are the fractional star mass at the beginning and at the end, respectively, of the U-th stage, and log(C U ) N is the related normalization constant with respect to the EDOD, under the assumption of a universal stellar initial mass function, according to Eq. (51). The combination of Eqs. (51), (56), and (65) yields: which is valid provided the following inequality holds: in agreement with Eq. (62). The particularization of Eq. (63) to the U-th stage reads: and the combination of Eqs. (67), (68), and (70) yields: which may be determined from the knowledge of the EDOD belonging to the U-th stage and related regression line. In general, different samples and/or different stages imply different values of the normalization constant. In the light of the model, region A corresponds to the first stage of evolution, where the system is still assembling, characterized by strong inflow regime (κ < −1, a > 0); region F corresponds to the second stage of evolution, where the system is forming, characterized by outflow regime (κ > 0, a < 0); region C corresponds to the third stage of evolution, where the system is undergoing contraction, characterized by outflow regime; region E corresponds to the fourth step of evolution, where the system has attained dynamical equilibrium, characterized by outflow regime.

Application to a special stellar system
The gas and star mass fraction are left unchanged passing from a selected stage to the next one, which implies the validity of the following relations: where X i , X f , are related to the whole evolution regardless of the stages. Accordingly, mass conservation during the U-th stage may be expressed as: where the initial values are known via Eq. (72). The regression line in an assigned region of the EDOD is defined by the slope, a U , and the intercept, b U , and the intersections between regression lines related to adjacent regions mark initial and final values of normalized oxygen abundance, (φ U ) i and (φ U ) f . The normalization constant, (C U ) N , is determined via Eq. (71). The final value of the active gas mass fraction, (µ U ) f , follows from Eq. (70).
The yield,p, and the lock, α, depend on the birth-rate stellar function. For a power-law stellar initial mass function, the EDOD in different Galactic environments is reproduced to a good extent from simple CBR models with the following output parameters (CM09): In the light of the model, the initial star mass fraction, s i , results from the fractional mass in stars (including stellar remnants) with normalized oxygen abundance, φ < φ i . The above value, though still uncertain at present, may be understimated to a good extent as s i = 0. In addition, D i = 0 without loss of generality, which implies µ i = 1.
The fractional mass of the box (with respect to the initial value) attains the maximum value at the end of A stage, where gas inflows into the box from the reservoir. The related value is (µ A ) f + (s A ) f . The fractional mass in stars at the end of evolution, which coincides with the end of E stage, is (s E ) f . Accordingly, the mass ratio of the box at the end of evolution to the outflowed gas reads: provided the earlier stage A is in inflow regime and the subsequent stages F, C, (or FC), E, are in outflow regime, as it is the case for the EDOD and related TDOD under consideration.
With the above values of input parameters, the flow, the active gas mass fraction, the star mass fraction, and the outflowed or inflowed gas mass fraction, at the end of each stage of evolution, can be computed. The results are listed in Table 7  In any case, the initial mass of the box at the beginning of A stage (protoinner halo) is comparable to the mass of the box at the end of E stage (present inner halo).

Discussion
Though TDODs calculated using simple MCBR models of chemical evolution fit to EDODs with different linear trends in different regions, as shown in Fig. 5, still the application to the inner Galactic halo remains speculative in absence of further improvement. The main reasons are outlined below.
First, the EDOD has been derived from a fictitious sample, fs10, which, in turn, has been deduced from two incomplete samples, RN91 and H V, both biased (to a different extent) towards low metallicities and affected by disk contamination. A homogeneous and unbiased sample would be needed to Table 7: Input parameters (deduced from the regression lines), (φ U ) i , a U , b U , and output parameters,  Third, though the assumption of null star mass fraction holds to a good extent for the starting configuration, still the presence of stars with lower metallicity with respect to sample objects, [Fe/H] < −4.2, has to be considered. This extremely low metallicity has currently been detected in about a dozen of stars down to [Fe/H] ≈ −5.4 (e.g., Beers and Christlieb 2005). Basing on theoretical arguments, a lower limit oxygen abundance in Pop II stars has been determined as [O/H] = −3.05 ∓ 0.20 (Bromm and Loeb 2003), where earlier nucleosynthesis comes from more massive Pop III stars.
If the transition from the latter to the former population was not instantaneous, and coeval Pop III and Pop II stars were generated (Smith et al. 2009), then the stellar initial mass function can no longer be considered as universal in time for Pop II stars. Accordingly, model evolution must be started after the last Pop III star has undergone supernova explosion, which has been assumed to take place when the metal abundance is [Fe/H] = −4.2.
In absence (to the knowledge of the author) of a reliable estimate of the mass fraction in stars and stellar remnants with initial metal abundance, [Fe/H] < −4.2, a null value has been assigned. For a stellar system resembling the inner Galactic halo, a nontrivial question is concerned with the nature of the reservoir where gas outflows after the assembling (A) stage. A natural candidate is the bulge, as suggested by comparison between specific angular momentum distributions related to halo and bulge stars (Wyse and Gilmore 1992). According to recent attempts, (1) the Galactic halo is made of an inner component which exhibits a modest prograde rotation and a metallicity peak at [Fe/H] ≈ −1.6, and an outer component which exhibits a clear retrograde net rotation and a metallicity peak at [Fe/H] ≈ −2.2 (Carollo et al. 2007(Carollo et al. , 2010; and (2) the Galactic bulge is made of an inner component which exhibits bar-like kinematics and metal-rich population, and an outer component which exhibits spheroid kinematics and metal-poor population (Babusiaux et al. 2010). Then a similar specific angular momentum distribution related to inner halo and outer bulge is expected.
In this view, the mass ratio of the box at the end of evolution to the gas outflowed from the box into the reservoir during the evolution, M box /M ofl , expressed by Eq. (75), should reproduce the mass ratio of (inner) halo to (outer) bulge, as: which is equivalent to: where the indices, inn and out, denote the inner and the outer component, respectively, H the halo, and B the bulge. The the inner halo to the outer bulge mass ratio, deduced from the results listed in Table 7, is (M H ) inn /(M B ) out ≈ 0.10-0.20. The halo to bulge mass ratio is currently estimated as M H /M B ≈ 0.05-0.10. Then simple MCBR models considered here yield an inner halo fractional mass (normalized to the halo) which is comparable with, or exceeding by a factor up to 4, the outer bulge fractional mass (normalized to the bulge). On the other hand, quantitative results cannot be expected for the above mentioned reasons. In conclusion, the (inner) halo to (outer) bulge mass ratio appears to be an additional output parameter provided by simple MCBR models of chemical evolution, with regard to the inner Galactic halo.
In the light of the model, the inner Galactic halo can be considered independently of the outer halo and the bulge for the following reasons. The outer halo is both dynamically and chemically decoupled from the inner halo (Carollo et al. 2010), which implies accretion by smaller subunits (mainly dwarf spheroidal galaxies) after the formation of the inner halo. In fact, globular clusters belonging to the inner and outer halo are usually classified as "old halo" and "young halo", respectively (e.g., Mackey and van den Bergh 2005). Similarly, a (inner) halo -(metal-poor) bulge collapse can be inferred by comparison of related specific angular momentum distributions (Wise and Gilmore 1992), which implies bulge formation is subsequent to inner halo formation, and then the (metal-poor) bulge may be considered as a reservoir for the outflowing gas, in the sense specified by the model.

Conclusion
Under the assumption that two samples of halo stars, RN91 and H V, are equally representative of the inner Galactic halo within the metallicity range,  Fig. 3.
A linear trend has been exhibited by related EDODs within three or four regions, as shown in Fig. 4, and the slope and intercept estimators of corresponding regression lines have been determined together with their dispersions, as listed in Table 2 using different interpolation methods. It has been pointed out that the earlier trend, characterized by positive slope, is a signature of a G-dwarf problem.
The main uncertainties on the EDOD have been recognized as related to (1) Fig. 1 for 11 stars in common among the RB09 and the Fal09 sample.
Fitting a theoretical differential oxygen abundance distribution (TDOD) to the above discussed EDOD has needed an extension of simple closed-box (CB) models of chemical evolution on two respects. First, the system has been conceived as made of a box and a reservoir, where the following processes have been allowed: gas outflow from the box into the reservoir (H76), moderate gas inflow (C07), steady and strong gas inflow (current paper) into the box from the reservoir. Simple closed-(box+reservoir) (CBR) models have exhibited mass conservation within the system (box+reservoir), while it has been violated within a single subsystem (box or reservoir). Second, the history of the system has been conceived as a succession of different stages characterized by different outflow or inflow rate. Simple multistage closed-(box+reservoir) (MCBR) models have been found to yield TDODs in the form of continuous broken lines, which can fit to related EDODs.
An application of MCBR models to a special stellar system resembling the inner Galactic halo has been made with fiducial values of input parameters which cannot be deduced from the EDOD. The metal abundance at the beginning and at the end of each stage, have been inferred from the intersection of regression lines fitting to adjacent regions of the EDOD, as shown in Fig. 5. The mass ratio of the box at the end of evolution to the gas outflowed from the box into the reservoir through the evolution, has been determined as M box /M ofl ≈ 0. 11  For current estimates of the halo-to-bulge mass ratio, M halo /M bulge ≈ 0.05-0.10, the inner halo fractional mass (normalized to the halo) has been found to be comparable with, or exceeding by a factor up to 4, the metalpoor bulge fractional mass (normalized to the bulge). On the other hand, it has been considered that quantitative predictions cannot be made for the Galaxy unless complete and unbiased samples of the inner Galactic halo are available, and discrepancies among [O/H]-[Fe/H] empirical relations related to different samples and different methods are removed.                Fig. 4, with regard to the regions (from the left to the right): A (dotted, positive slope, n A = 6), F(dotted, negative slope, n F = 6), C (full, n C = 5), E (dashed, n E = 4). The more inclined dashed line on right panels corresponds to the FC (n FC = 10) region, in alternative to F and C regions separately. Other captions as in Fig. 4. For further details refer to the text.