The Origin of the Relation between Metallicity and Size in Star-forming Galaxies

Based on some 44,000 star-forming galaxies from the Sloan Digital Sky Survey (SDSS), Ellison et al. (2008) found a relation connecting stellar mass (M_⋆), galaxy size (as parameterized by the half-light radius), and gas-phase metallicity. They discovered that at fixed M_⋆, physically smaller galaxies are also metal richer. The metallicity changes by 0.1 dex when the galaxy size changes by a factor of 2. The authors discard observational biases due to the finite size of the central region used to estimate metallicities and to the Hubble type dependence of the radius. A similar relationship between size and gas-phase metallicity was also found by Brisbin & Harwit (2012) and Harwit & Brisbin (2015), and it remains in place at redshift ≃1.4 as measured by Yabe et al. (2012, 2014). Tremonti et al. (2004) and Wu et al. (2015) observed that, given M_⋆, galaxies with higher stellar surface density are also metal richer, which implies a relation between metallicity and size in qualitative agreement with all these other works.

The physical cause of the observed relation remains unclear. Ellison et al. (2008) considered and discarded both metal-poor gas inflows and metal-rich gas outflows. Their argument was based on comparing with the simple chemical evolution models by Finlator & Davé (2008), as well as the fact that they do not satisfactorily explain the observed correlation. Ellison et al. favor differences in star formation (SF) efficiencies. Small galaxies are denser and exhaust their gas faster, and thus they become metal enriched sooner. On the other hand, Sánchez Almeida et al. (2014) pointed out that the relation is a natural outcome of the gas-accretion-driven SF process. In the stationary state the gas-phase metallicity is set by the efficiency of the outflows, which changes systematically with halo mass, that is to say, with the depth of the gravitational potential the baryons have to escape from. The gravitational binding energy depends on the distance to the center of the gravitational well; therefore, at a fixed mass, winds escape easier from larger galaxies. Finally, Yabe et al. (2012) invoke metal-poor gas accretion driven by mergers with no further elaboration.

Whatever the explanation may be, it is telling us about the basic physics underlying the SF process, since the observed anticorrelation between size and metallicity is likely a fundamental property of galaxies. Although less studied than the others, the relation found by Ellison et al. (2008) belongs to the realm of the well-known empirical relations linking global properties of star-forming galaxies, including the main sequence (scaling between star formation rate [SFR] and M_⋆; e.g., Daddi et al. 2007; Noeske et al. 2007), the mass–metallicity relation (e.g., Pagel & Edmunds 1981; Tremonti et al. 2004), the fundamental metallicity relation (connecting M_⋆, gas-phase metallicity, and SFR; Lara-López et al. 2010; Mannucci et al. 2010), or even the lopsidedness–metallicity relation (Reichard et al. 2009; Morales-Luis et al. 2011). All of these together provide the main observational constraints to understand the subtleties of the mechanism by which galaxies form and grow. It is generally accepted that galaxies grow in a self-regulated process controlled by gas accretion and feedback from SF and black holes (e.g., Bouché et al. 2010; Davé et al. 2011, 2012; Silk & Mamon 2012; Lilly et al. 2013; Sánchez Almeida 2017). However, the way in which the global properties of galaxies emerge from the underlying physical processes is not properly understood yet.

Here we revisit the problem of explaining the anticorrelation between galaxy size and gas metallicity. We use the EAGLE cosmological numerical simulations (Crain et al. 2015; Schaye et al. 2015; McAlpine et al. 2016). The model includes a pressure-based law for SF (Schaye & Dalla Vecchia 2008), line cooling in photoionization equilibrium (Wiersma et al. 2009a), stellar evolution (Wiersma et al. 2009b), thermal supernova (SN) feedback (Dalla Vecchia & Schaye 2012), and black hole growth and feedback (Rosas-Guevara et al. 2015). An extensive description of the model, its calibration, and the hydrodynamics solver are given in Schaye et al. (2015), Crain et al. (2015), and Schaller et al. (2015), respectively. They represent state-of-the-art cosmological numerical simulations that self-consistently include baryon physics. Other simulations of this kind are described in, e.g., Hopkins et al. (2014, 2018), Ceverino et al. (2014), Vogelsberger et al. (2014), or Springel et al. (2018). As we will show, the EAGLE galaxies reproduce the trend observed by Ellison et al. (2008), and thus, assuming that they are grasping the essentials of the physical process giving rise to the anticorrelation, we study them to identify what can be causing the observed trend. Obviously, the appropriateness of the explanation depends on whether this working hypothesis is correct, which is a caveat affecting the whole paper.

The EAGLE model galaxies have already proven their potential to reproduce some of the well-known scale relations, which encouraged us to use them in the present context. Explicitly, Schaye et al. (2015) demonstrate that the simulations reproduce a correlation between stellar mass and gas-phase metallicity in agreement with observed data at redshift zero. Lagos et al. (2016) show the existence of a relation between gas fraction, stellar mass, and current SFR, with the galaxies distributed on a plane in the 3D space defined by these three parameters. De Rossi et al. (2017) analyze the fundamental metallicity relation, finding a very good matching with observations up to redshift 5. They also find that the physical parameter that best correlates with metallicity is gas fraction. EAGLE galaxies also provide a relation between mean size and stellar mass in agreement with the relation observed in the local universe (Crain et al. 2015; Schaye et al. 2015; Desmond et al. 2017). This result is reassuring for our analysis, which relies on galaxy sizes.

The paper is organized as follows. Section 2 shows how the EAGLE numerical simulations produce galaxies slightly more metallic when their stellar sizes are smaller. Such difference quantitatively agrees with the difference observed by Ellison et al. (2008). Thus, the physical mechanism responsible for the difference of gas metallicity between small and large galaxies in the simulations may also be responsible for the difference in observed galaxies. The question arises as to what is this physical mechanism. Section 3 explores the three obvious possibilities, namely, metal-poor gas inflows feeding the SF process, metal-rich gas outflows particularly efficient in shallow gravitational potentials, and enhanced efficiency of the SF in compact galaxies. This first exploration is based on simple analytical bathtub models (Section 3 and Appendix A). Then we study these three possibilities in the EAGLE galaxies. It is clear that outflows (Section 4) and SF efficiency controlled by density (Section 5) can be discarded as the underlying cause. We are left with metal-poor gas inflows, which cause the correlation in the model galaxies. Galaxies grow in size with time, so those that receive gas later are both metal poorer and larger, giving rise to the observed correlation (Section 6). We also explore the variation with redshift of the relation between size and gas-phase metallicity, which is shown to be maintained and even strengthened at higher redshifts (up to redshift 8; Section 7). The increase in galaxy size with time is a central ingredient of our explanation. The physical cause of this growth is extensively discussed in the literature, and we examine the various possibilities in Section 8. The general results and conclusions are also summarized in Section 8.

The suite of EAGLE simulations is described in detail by Schaye et al. (2015) and Crain et al. (2015). We use the run covering the largest volume, namely, L100N1504, which corresponds to a 100 comoving-Mpc cube and goes all the way from redshift 127 to redshift 0. It has initial gas mass particles of ∼2 × 10⁶ M_⊙ and initial dark matter (DM) particles of ∼10⁷ M_⊙. All the relevant physical parameters employed in the present study were retrieved by querying the EAGLE SQL³ web interface⁴ (McAlpine et al. 2016).

The selection of all central galaxies at redshift zero renders 16,671 objects. This set was further filtered out to remove objects clearly out of the main sequence, i.e., the well-defined relation between M_⋆ and SFR followed by star-forming galaxies (e.g., Daddi et al. 2007; Noeske et al. 2007). The left panel of Figure 1 shows the full set and the divide we use, so that only the 13,410 galaxies above the line are retained for further analysis. The resulting trimmed main sequence is shown in the right panel of Figure 1. Our analysis is based on the star-forming gas metallicity (the mass fraction in metals, Z_g, which traces dense gas) and the half-mass stellar radius (R_⋆) computed for the mass within a spherical 100 kpc aperture. We checked that the conclusions in the paper do not qualitatively change when using a spherical 30 kpc aperture to measure R_⋆. The EAGLE database does not provide half-light radii, and we use R_⋆ as a proxy for them. Here and throughout the paper, Z_g has been referred to the solar metallicity, which we take as Z_⊙ = 0.02 to comply with the nucleosynthetic yields used in EAGLE (e.g., Portinari et al. 1998; Marigo 2001).

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Figure 2 shows the gas metallicity versus M_⋆ relation for the galaxies in EAGLE. The points are color-coded according to the mean half-mass radius of all the galaxies having the same Z_g and M_⋆. The simulation clearly reveals an anticorrelation between size and metallicity: with fixed M_⋆, the smaller the galaxy, the larger its metallicity.

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The correlation in EAGLE closely resembles the correlation observed by Ellison et al. (2008). They split ∼44,000 SDSS-DR4 star-forming galaxies in bins of equal mass. The objects in each bin were divided into terciles according to their half-light radius—the largest galaxies, the intermediate-size galaxies, and the smallest galaxies. The mean metallicity was computed for each tercile, finding the metallicity–stellar mass relation to differ for the three sets, in the sense that the largest galaxies tend to have the smallest metallicities. The curves found by Ellison et al. for the first and third terciles are reproduced in Figure 3(a) (black lines). We repeat the same exercise with the EAGLE data represented in Figure 2. The curves predicted by the simulations are also included in Figure 3(a) (red lines). Leaving aside a global factor, the fact that the EAGLE simulation shows a metallicity–stellar mass relation too flat compared with the observed relation (see Schaye et al. 2015, Figure 13), and the use of half-mass radii rather than half-light radii, the difference of metallicity between small and large model galaxies, of the order of 0.1 dex, is in good agreement with observations (Figure 3(a)).

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The uncertain scaling factor stems from biases in both observations and simulations. On the one hand, the measurements of O/H were carried out using a strong line ratio method, and this procedure introduces non-negligible systematic errors (e.g., Kewley & Ellison 2008). On the other hand, the metallicity in the simulations depends on the adopted nucleosynthetic yields, which also have significant uncertainties. In addition, Ellison et al. (2008) measure the metallicity in terms of the oxygen abundance, whereas we employ the mass fraction in metals. In order to show them in Figure 3(a), the EAGLE abundances have been transformed to O/H assuming a constant value for the ratio (O/H)/Z_g. We opted for the solar composition given by Asplund et al. (2009); however, this scaling is also arbitrary. In order to discard biases arising from the use of Z_g rather than O/H, Figure 3(a) was repeated with the actual O/H from the EAGLE simulation. The result is almost identical to the curves on display. In fact, O/H is expected to be a good tracer of Z_g since O is the major contributor to the mass in metals.

The curves 12 + log (O/H) versus M_⋆ in EAGLE are flatter than observed (Figure 3(a)). Such a difference seems to be caused by the limited resolution of the numerical simulation. As shown by Schaye et al. (2015), other EAGLE runs at higher spatial resolution produce significantly lower metallicity and a steeper relation, in closer agreement with observations. The merging of the curves for small and large galaxies at log(M_⋆/M_⊙) ∼ 8.5 also reinforces this view—the effect is highest for the lowest-mass bins that are represented by fewer particles in the simulation.

The distribution of metallicities among the galaxies of the first and third terciles is represented in Figure 3(b). Figure 3(c) shows the derivative of the (log) metallicity with respect to the (log) stellar effective radius. It has been estimated from the differences in metallicity and radius of the galaxies in the upper and lower terciles. d log(Z_g)/d log(R_⋆) ≥ −0.6, with the minimum at log(M_⋆/M_⊙) ≃ −9.3. The increase of the slope toward low masses is produced by the aforementioned limited resolution.

The mean metallicity of a galaxy is primarily regulated by (1) the efficiency of the SF process that produces the metals, (2) the presence of outflows carrying away these metals, and (3) the existence of inflows of metal-poor gas fueling the SF (e.g., Larson 1972; Edmunds 1990; Dalcanton 2007; see also Appendix A). We want to determine whether one (or several) of these processes is responsible for the existence of an anticorrelation between gas metallicity and stellar size in the EAGLE model galaxies and, in doing so, to identify a plausible physical scenario that explains the relation observed by Ellison et al. Therefore, we need to examine how these three key processes depend on the galaxy size.

• 1.  
  The efficiency of the SF is related to the galaxy size through the gas density, since denser galaxies are more efficient at transforming gas into stars as reflected by the Kennicutt–Schmidt relation (e.g., Kennicutt 1998). Given a stellar mass, those galaxies that are more efficient at forming stars consume gas sooner, and what is leftover becomes more metallic by mixing with SN ejecta. This effect gives rise to an anticorrelation between gas metallicity and stellar size in qualitative agreement with the observations of Ellison et al.
• 2.  
  Since outflows take gas and metals out of the galaxies, they modulate the gas-phase metallicity. The effectiveness in carrying away gas is related to the power of the winds and the depth of the gravitational potential to be overcome. When the winds are powered by stars, the supply of energy and momentum is set only by the SFR, independently of the galaxy size. However, galaxy size enters into the equation through the depth of the gravitational potential. For a given mass, the depth increases with decreasing size, and therefore smaller galaxies are expected to have less effective winds and so to retain more metals. Thus, for a fixed mass, smaller galaxies become more metallic.
• 3.  
  Galaxies grow in size with time (e.g., Margalef-Bentabol et al. 2016; Nelson et al. 2016; Furlong et al. 2017). Thus, galaxies with late SF are systematically bigger than those formed earlier. If the SF is driven by metal-poor gas accretion, differences in the recent gas accretion rate produce the type of observed relation. Given a stellar mass, younger galaxies are bigger, and since they still preserve recently accreted metal-poor gas, they are metal poorer as well.

In order to understand the physical bases for the correlation in the simulation (Figures 2 and 3), we will rely on a simple self-regulated galaxy model,⁵ where galaxies are characterized by a stellar mass, a gas mass, a metallicity, and so on (e.g., Finlator & Davé 2008; Bouché et al. 2010; Davé et al. 2012; Lilly et al. 2013; Peng & Maiolino 2014; Sánchez Almeida et al. 2014). This kind of toy model is broadly used in the literature because, despite its apparent simplicity, it includes all the key physical ingredients and their interrelations and often reveals the underlaying physical processes in a way that is hard to disclose in the full numerical solutions. We use it to work out the expected variation of metallicity with size if the anticorrelation is created by the depth of the gravitational potential (Section 3.1), the density of the galaxy (Section 3.2), and the recent accretion of gas on an already grown-up galaxy (Section 3.3). The predictions derived from these sections will be used later on to analyze, favor, or discard each of the plausible mechanisms.

3.1. Depth of the Gravitational Well

Under the hypothesis of stationary gas infall, the toy model predicts that the gas metallicity reaches a constant value Z_g0 set only by the stellar yield y (the mass of new metals eventually ejected per unit mass locked into stars), the mass return fraction R (the fraction of mass in stars that returns to the interstellar medium), and the so-called mass-loading factor w,

$$\begin{eqnarray}&&{\rm{\Delta }}{Z}_{g0}={Z}_{g0}-{Z}_{\mathrm{in}}=\displaystyle \frac{y(1-R)}{1-R+w},\end{eqnarray} \tag{ 1 }$$

with Z_in the metallicity of the accreted gas (≪Z_g0 so that ΔZ_g0 ≃ Z_g0). w is defined as the constant of proportionality between the gas outflow rate produced by the starburst, ${\dot{M}}_{\mathrm{out}}$, and its SFR,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{out}}=w\,\mathrm{SFR}.\end{eqnarray} \tag{ 2 }$$

Equation (1) is derived in Appendix A and corresponds to Equation (29). According to Equation (1), differences in w lead to differences in Z_g0. w depends on the depth of the gravitational potential, which we parameterize in terms of the escape velocity, v_esc, defined as the velocity whose kinetic energy balances the (negative) gravitational energy (e.g., Binney & Tremaine 2008, Equation (2.31)). Galaxies with the same mass but smaller radius reside in a deeper gravitational potential. Then the winds produced by stellar feedback will be less effective to escape, lowering w. This is the explanation proposed in Sánchez Almeida et al. (2014). Using Equation (1) with Z_in ≪ Z_g0, one can write down the expected variation of Z_g0 with v_esc as

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{log}{Z}_{g0}}{d\mathrm{log}{v}_{\mathrm{esc}}}=\displaystyle \frac{\beta \,w}{1-R+w},\end{eqnarray} \tag{ 3 }$$

with β parameterizing the relation between w and v_esc,

$$\begin{eqnarray}&&w\propto {v}_{\mathrm{esc}}^{-\beta }.\end{eqnarray} \tag{ 4 }$$

The actual value of β is unknown, but it is expected to go from 2 for energy-driven winds to 1 for momentum-driven winds (e.g., Murray et al. 2005). Explaining the mass–metallicity relation in terms of varying w with v_esc favors low values of β: 0.5–0.9 (e.g., Andrews & Martini 2013; Dayal et al. 2013, with ${v}_{\mathrm{esc}}^{2}\propto {M}_{\star }$). Therefore, even in the most favorable case for w to be important (w ≫ 1), Equation (3) leads to

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{log}{Z}_{g0}}{d\mathrm{log}{v}_{\mathrm{esc}}}\leqslant 1.\end{eqnarray} \tag{ 5 }$$

There is an additional constraint to be satisfied if changes in w are responsible for the observed correlation between gas-phase metallicity and size. In this case, the ratio between SFR and metallicity has to be independent of the mass-loading factor and, thus, independent of the depth of the gravitational potential. According to the simple model in Appendix A, this ratio is set only by the current gas accretion rate, ${\dot{M}}_{\mathrm{in}0},y$, and R (Equation (30)), namely,

$$\begin{eqnarray}&&\displaystyle \frac{{\mathrm{SFR}}_{0}}{{\rm{\Delta }}{Z}_{g0}}=\displaystyle \frac{{\dot{M}}_{\mathrm{in}0}}{y(1-R)}.\end{eqnarray} \tag{ 6 }$$

3.2. Density of the Galaxy

The gas consumption timescale, τ_g, is defined as the ratio between the gas mass, M_g, and the SFR,

$$\begin{eqnarray}&&{\tau }_{g}={M}_{g}/\mathrm{SFR}.\end{eqnarray} \tag{ 7 }$$

This timescale depends on the surface gas density, so that denser systems have shorter timescales (e.g., Kennicutt 1998). Using the Kennicutt–Schmidt relation as parameterized by Kennicutt & Evans (2012), τ_g increases from 0.5 to 5 Gyr when the gas surface density decreases from 5 × 10² M_⊙ pc⁻² to 2 M_⊙ pc⁻² (see Figure 4). For a given mass, smaller systems are denser, and so they should consume the gas faster. Thus, considering an ensemble of galaxies with similar stellar mass and accreting gas, those smaller are expected to be denser, to consume the gas faster, and, consequently, to become metal richer sooner. This mechanism requires the galaxies to be outside equilibrium because, as we pointed out in the previous section, the equilibrium gas-phase metallicity is independent of the gas consumption timescale and is set only by stellar physics and the mass-loading factor (Equation (1)).

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In Appendix A, we derive the variation of the metallicity with time, ΔZ_g(t), when the model galaxy receives an amount of gas. This metallicity depends on τ_g through an effective gas consumption timescale τ_in (Equation (23)),

$$\begin{eqnarray}&&{\tau }_{\mathrm{in}}=\displaystyle \frac{{\tau }_{g}}{1-R+w}.\end{eqnarray} \tag{ 8 }$$

The dependence is given in Equation (26) and is shown in Figure 18 in Appendix A. The probability density function (PDF) of the metallicity that the galaxy presents during its time evolution, P(ΔZ_g), is proportional to the time span spent by the galaxy at each metallicity, i.e.,

$$\begin{eqnarray}&&P({\rm{\Delta }}{Z}_{g})\propto \left|\displaystyle \frac{{dt}}{d{\rm{\Delta }}{Z}_{g}}\right|.\end{eqnarray} \tag{ 9 }$$

Since t(ΔZ_g) is bivalued (it is the inverse to the function shown in Figure 18), P(ΔZ_g) has an involved analytical expression. We evaluate it numerically using a Monte Carlo simulation as follows. We consider a population of galaxies going through gas accretion events, which are detected at random times from the time of accretion. We assume that all of them received the same amount of pristine gas and have the same gas depletion timescale. Using Equation (26), we compute the gas-phase metallicity of each object at the time of observation, and the corresponding PDF is inferred from them. The results for three populations that differ in their gas depletion timescale are given in Figure 5. Note the extended tails of the distributions with large gas consumption timescales. The solid line corresponds to a timescale 30 times longer than the case of the dashed line when the galaxies have not had time to reach the equilibrium metallicity. The equilibrium metallicity in this Monte Carlo simulation is assumed to differ for the different galaxies, following a random Gaussian distribution with its standard deviation 0.2 times its mean value.

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The expected increase in metallicity associated with an increase in stellar mass density, ρ_⋆, can be estimated splitting the derivative as

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{log}{Z}_{g}}{d\mathrm{log}{\rho }_{\star }}=\displaystyle \frac{d\mathrm{log}{Z}_{g}}{d\mathrm{log}{\tau }_{g}}\,\displaystyle \frac{d\mathrm{log}{\tau }_{g}}{d\mathrm{log}{{\rm{\Sigma }}}_{g}}\,\displaystyle \frac{d\mathrm{log}{{\rm{\Sigma }}}_{g}}{d\mathrm{log}{\rho }_{\star }}.\end{eqnarray} \tag{ 10 }$$

The first term on the right-hand side of the equation can be evaluated using the simulations shown in Figure 5, and it amounts to some −0.2 when comparing the dashed and the solid lines. The second term follows from the Kennicutt–Schmidt relation; using the one in Kennicutt & Evans (2012), it results in −0.4. The third term turns out to be 2/3 assuming the surface density of the gas Σ_g to be proportional to the surface density of stars Σ_⋆, and then working out the scaling between surface density and volume density.⁶ All in all, the logarithmic derivative becomes

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{log}{Z}_{g}}{d\mathrm{log}{\rho }_{\star }}\simeq 5\times {10}^{-2}.\end{eqnarray} \tag{ 11 }$$

The terms involved in the evaluation of this equation are uncertain. When the surface density is low, larger exponents in the Kennicutt–Schmidt relation are favored (e.g., Bigiel et al. 2008; Genzel et al. 2010, and references therein), and the estimated derivative could easily be twice as large as the value in Equation (11).

Since the metallicity depends on the gas consumption timescale as expressed in Equation (8), all that is said above also applies to the changes in τ_in induced by variations in the mass-loading factor w. The reader should keep in mind, however, that the trend is opposite to the one described in Section 3.1. According to Equation (8), an increase in w decreases τ_in and thus increases the metallicity. The chain rule yields the change in metallicity when v_esc varies through τ_in, i.e.,

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{log}{Z}_{g}}{d\mathrm{log}{v}_{\mathrm{esc}}}=\displaystyle \frac{d\mathrm{log}{Z}_{g}}{d\mathrm{log}{\tau }_{\mathrm{in}}}\,\displaystyle \frac{d\mathrm{log}{\tau }_{\mathrm{in}}}{d\mathrm{log}w}\,\displaystyle \frac{d\mathrm{log}w}{d\mathrm{log}{v}_{\mathrm{esc}}}=-0.2.\end{eqnarray} \tag{ 12 }$$

The numerical value has been worked out using Equations (4) and (8) and plugging in the parameters used to evaluate the derivative in Equation (5). This dependence of Z_g on v_esc through τ_in may qualitatively explain the correlation shown in the bottom panels of Figure 8, which will be discussed later on.

3.3. Differences in the Recent Gas Accretion Rate

Galaxies with late SF tend to be larger. At the same time, galaxies outside equilibrium have their gas-phase metallicity in proportion to the gas recently accreted, i.e., in proportion to their recent ${\dot{M}}_{\mathrm{in}}$. The metal-poor gas acquired through accretion is still in place, so the gas mass is metal poorer in objects with larger current ${\dot{M}}_{\mathrm{in}}$. This can be shown using the simple model described in the previous section. By increasing the gas mass per clump, one increases ${\dot{M}}_{\mathrm{in}}$. Thus, more massive clumps produce metallicity distributions biased and skewed toward lower metallicities, as illustrated by Figure 6.

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The expected change of gas-phase metallicity due to this process can also be estimated using the toy model worked out in Appendix A. On the one hand, the drop of metallicity after accreting a gas mass ΔM_a scales with the accreted gas mass as

$$\begin{eqnarray}&&{Z}_{g}/{Z}_{g0}\simeq \displaystyle \frac{1}{1+C\,{\rm{\Delta }}{M}_{a}/{M}_{g0}},\end{eqnarray} \tag{ 13 }$$

where M_g0 stands for the mass of gas already present in the object and C is a time-dependent parameter that is of the order of 1 after the accretion event (see Equation (26), with Z_in ≪ Z_g < Z_g0). On the other hand, ${\dot{M}}_{\mathrm{in}}\sim {\rm{\Delta }}$M_a/τ_in, and therefore

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{log}{Z}_{g}}{d\mathrm{log}{\dot{M}}_{\mathrm{in}}}\simeq \displaystyle \frac{d\mathrm{log}{Z}_{g}}{d\mathrm{log}{\rm{\Delta }}{M}_{a}}\simeq \displaystyle \frac{-C\,{\rm{\Delta }}{M}_{a}/{M}_{g0}}{1+C\,{\rm{\Delta }}{M}_{a}/{M}_{g0}}\sim -0.5.\end{eqnarray} \tag{ 14 }$$

The logarithmic derivative has been evaluated assuming ΔM_a ∼ M_g0 and C ∼ 1. Its actual value can go all the way from −1 (when ΔM_a ≫ M_g0) to 0 (when ΔM_a ≪ M_g0).

The short answer to the above question is no. The long answer is elaborated in this section showing that the escape velocity and the metallicity are not correlated in the EAGLE galaxies.

In order to parameterize the depth of the gravitational potential, we use the escape velocity at the half-mass radius,

$$\begin{eqnarray}&&{v}_{\mathrm{esc}}({R}_{\star })=\sqrt{2\,| {\rm{\Phi }}({R}_{\star })| },\end{eqnarray} \tag{ 15 }$$

with Φ(R_⋆) the gravitational potential at R_⋆. We model Φ as a combination of the potential due to DM, Φ_DM, and the potential due to stars, Φ_⋆,

$$\begin{eqnarray}&&{\rm{\Phi }}({R}_{\star })={{\rm{\Phi }}}_{\mathrm{DM}}({R}_{\star })+{{\rm{\Phi }}}_{\star }({R}_{\star }).\end{eqnarray} \tag{ 16 }$$

The EAGLE database does not directly provide the depth of the gravitational potential. We infer the DM component from the DM mass and the half-mass radius of the DM assuming that the density drops with radius following a NFW profile (Navarro et al. 1996). The procedure is sketched in Appendix B. The stellar contribution is also inferred from the stellar mass and the half-mass stellar radius, this time assuming that the stellar density drops exponentially with radius. We follow the work by Smith et al. (2015). We assume the galaxy disks to be thin (scale height of 0.2 kpc), with the escape velocity computed in the plane of the disk. The results are rather insensitive to these assumptions since the potential is dominated by the DM component. This is also the reason why the gas mass is not included in the computation of v_esc, because it represents only a minute fraction of the total mass.

Figure 7 is similar to Figure 2, but this time the symbols are color-coded with the escape velocity at the half-mass stellar radius. The spread in gas-phase metallicity is uncorrelated with the escape velocity, which is mostly set by the stellar mass. For a given stellar mass the galaxies have the same escape velocity irrespectively of the metallicity of their gas (cf. Figures 2 and 7). This fact is even clearer in Figure 8. It shows Z_g versus v_esc for the full set of EAGLE galaxies (top left), as well as for narrow bins in galaxy mass (Δ log M_⋆ ∼ 0.3). Even though the full set shows a global trend for the gas-phase metallicity to increase with the escape velocity, this is just a construct (or a mirage) resulting from the mass–metallicity relation and the superposition of galaxies of all masses in the plot. The panels corresponding to single mass bins show the scatter in metallicity to be independent of v_esc. If the relation between size and gas-phase metallicity is due to changes in the escape velocity, the toy model in Section 3.1 predicts the red solid line in Figure 8 (plotted in the central panel). The fact that the numerical simulations do not follow such a line indicates that the metallicity is not set by v_esc. If anything, there seems to be a weak anticorrelation between Z_g and v_esc in the bins of higher mass (the three bottom panels in Figure 8). Such anticorrelation is contrary to the positive correlation needed to explain the observed anticorrelation between gas-phase metallicity and size. It may arise from the dependence of the gas depletion timescale on the depth of the gravitational potential, as discussed in Section 3.2. However, the predicted slope is much too shallow (see Equation (12)).

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There is one more argument against the depth of the gravitational potential setting the relation between metallicity and size. Should the variation with size be due to the depth of the gravitational potential, then one would expect the ratio SFR/Z_g to be constant at a fixed stellar mass, since in the stationary state this ratio solely depends on the gas accretion rate (Equation (6)). Figure 9 shows SFR/Z_g versus v_esc color-coded with Z_g. It evidences a large variation of SFR/Z_g with v_esc, which indicates that the depth of the gravitational potential by its own cannot be responsible for the variation of Z_g with galaxy size.

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As with the question posed in Section 4, the short answer is no.

Figure 10 shows Z_g versus M_⋆ color-coded with the stellar volume density⁷ of the galaxies. There is a variation of ρ_⋆ with Z_g at a fixed M_⋆ which is significantly larger than the variation with escape velocity (cf. Figures 7 and 10). Given a stellar mass, denser galaxies tend to be metal richer. However, such a trend seems to be a mirage resulting from the superposition of galaxies with very different densities but the same Z_g and M_⋆. Figure 11 shows Z_g versus ρ_⋆ for all the galaxies (top left) and for galaxies within narrow mass bins that cover the whole range of masses from 10⁸ to 10¹¹ M_⊙. The solid line in the central panel represents the expected correlation according to Equation (11). It roughly agrees with the trend followed by the EAGLE galaxies in this mass bin. It indicates that the variation is behaving as expected theoretically. However, it is very mild compared with the range of gas metallicities exhibited by the galaxies. Obviously, the density scales with the size at a given mass, and since there is a correlation between metallicity and size (Figure 2), there should be a correlation between metallicity and density at a fixed mass. However, galaxies with the whole range of metallicities exist for every ρ_⋆, and so density does not seem to be the primary driver of any correlation with metallicity.

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Disk galaxies grow in size with time by accreting gas in their outskirts. Those whose last major gas accretion episode happened earlier are now smaller and more metallic. We think that this physical process is responsible for the anticorrelation between size and gas-phase metallicity in the EAGLE galaxies.

Ideally, one would like to study the dependence of the gas metallicity on the present gas accretion rate ${\dot{M}}_{\mathrm{in}}$. The EAGLE database does not provide the gas accretion rate of the galaxies directly. We have estimated it from the gas mass and the SFR. The database provides the gas mass and thus its variation in time ${\dot{M}}_{g}$. We compute the difference of gas mass in the two last snapshots (redshifts 0 and 0.1) and then divide it by the difference in look-back time (1.35 Gyr). Mass conservation coupled with Equation (2) guarantees

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{in}}={\dot{M}}_{g}+(1-R+w)\mathrm{SFR},\end{eqnarray} \tag{ 17 }$$

so that ${\dot{M}}_{\mathrm{in}}$ can be inferred from ${\dot{M}}_{g}$, SFR, and w. However, w is unknown. Fortunately, this fact is not critical since SFR is usually larger than ${\dot{M}}_{g}$, and its contribution completely dominates ${\dot{M}}_{\mathrm{in}}$. This is shown in Figure 12. The left panel in the figure represents ${\dot{M}}_{g}$ versus SFR, and the galaxies tend to be below the one-to-one line (dashed line). One reaches a similar conclusion by computing ${\dot{M}}_{\mathrm{in}}$ for various w and then noting that, independently of its actual value, ${\dot{M}}_{\mathrm{in}}\propto \mathrm{SFR}$. The middle panel in Figure 12 includes the scatter plots ${\dot{M}}_{\mathrm{in}}$ versus SFR for w = 1. It shows a clear scaling between ${\dot{M}}_{\mathrm{in}}$ and SFR despite the fact that this case is particularly unfavorable. The smaller the value of w, the larger the possible differences between ${\dot{M}}_{\mathrm{in}}$ and SFR (Equation (17) yields ${\dot{M}}_{\mathrm{in}}\propto \mathrm{SFR}$ when w ≫ 1), and w is typically larger than 1 (e.g., Davé et al. 2011; Sánchez Almeida et al. 2014). If a more realistic w = w(M_⋆) is included, the relation tightens even further since most of the scatter at small SFRs in the middle panel of Figure 12 comes from low-mass galaxies, where w ≫ 1. The improvement is shown in the right panel of Figure 12. It has been computed with the semi-empirical w = w(M_⋆) worked out by Dayal et al. (2013) to reproduce the observed mass–metallicity relation. It was chosen because this parameterization of w = w(M_⋆) yields the whole range of expected values, from low values at the high-mass end of the EAGLE mass distribution (w ≃ 0.7 at M_⋆ = 5 × 10¹¹ M_⊙) to large values at the low-mass end (w ≃ 10 at M_⋆ = 1.5 × 10⁸ M_⊙).

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The gas mass used in Figure 12 corresponds to the star-forming gas, i.e., gas dense enough to contain a molecular phase cradling stars.⁸ However, one reaches the same conclusion even if the mass of all the gas is used in this calculation.

In view of the uncertainties in w, and due to the good scaling between the two quantities, from now on we use SFR as a proxy for ${\dot{M}}_{\mathrm{in}}$. Figure 13 shows the gas-phase metallicity versus stellar mass color-coded with the mean SFR. Unlike what happens when the color code reflects v_esc or ρ_⋆ (Figures 7 and 10, respectively), this time there is a clear dependence of Z_g on the SFR for a fixed mass. This trend is even clearer in Figure 14. It contains the scatter plot of metallicity (Z_g) versus SFR color-coded with the half-mass stellar radius of the galaxies. The top left panel represents the full data set. The rest of the panels show the same scatter plot selecting narrow mass bins (as labeled on top of each figure). There is a clear relation between ordinates and abscissae. The solid line in the panel corresponding to masses in the range of (3–6) × 10⁹ M_⊙ shows the anticorrelation expected according to the toy model worked out in the main text (Equation (14) with ${\dot{M}}_{\mathrm{in}}\propto \mathrm{SFR}$). Given a stellar mass, galaxies of higher SFR are also larger (Figure 1, middle panel, but see also the color coding of the panels in Figure 14). This relation between size, SFR, and Z_g is the one that, once averaged over the full population of galaxies, gives rise to the correlation between size and gas metallicity we are trying to explain.

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Note that the correlation between metallicity and SFR, so clear in the panels of the individual mass bins, washes out when considering all the galaxies together (Figure 14, top left). This is due to the fact that the negative correlation between Z_g and SFR at fixed M_⋆ turns into a positive correlation between Z_g and SFR when the variation with mass is considered. Both the mean Z_g and the mean SFR increase with increasing M_⋆ (see Figure 1, left panel). The two tendencies tend to cancel out when averaging over galaxies of all masses, resulting in a lack of correlation.

There is a tight correlation between M_g and SFR in the EAGLE galaxies. Therefore, the above discussion could have been made in terms of M_g rather than SFR. However, we would have reached exactly the same conclusion because M_g is also a proxy for ${\dot{M}}_{\mathrm{in}}$. Both are proportional in the stationary-state solution (see Equation (27)), and, even in general, M_g represents a time average of ${\dot{M}}_{\mathrm{in}}$ over a time lapse τ_in; see Equation (23). Figure 15 is similar to Figure 14 but replacing SFR with M_g, and the behavior and the trends coincide.

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6.1. Sanity Check: Age of the Stellar Populations

If the relation between size and gas metallicity is due to the growth of galaxies with time, then there should be a tight correlation between galaxy size and age of the stellar population. This is indeed the case.

The EAGLE database provides the mean age of the stars in each galaxy, weighted by birth mass. This age estimate is not biased toward recent SF, and it shows a tight correlation with Z_g (see Figure 16, left panel). The right panel of Figure 16 shows galaxy radius versus stellar mass color-coded with mean stellar age. For galaxies with M_⋆ > 5 × 10⁸ M_⊙, bigger galaxies of the same M_⋆ have also younger stellar populations. Again this is consistent with the idea that smaller galaxies formed earlier, at least when the stellar mass is M_⋆ > 5 × 10⁸ M_⊙, a mass limit that coincides with the onset of the relation size versus metallicity in the EAGLE galaxies (see Figures 3(a) and (c)).

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The EAGLE database also provides colors for the model galaxies, which show that galaxies with metal-richer gas have redder stellar populations. They are redder because they have evolved longer, and therefore the colors are also consistent with larger galaxies having younger stellar populations.

The anticorrelation between size and metallicity remains at high redshift. Figure 17 shows Z_g versus M_⋆ for the EAGLE galaxies at redshifts from 0 to 8. (Higher redshifts are not shown because the number of objects decreases drastically, but the trends shown at redshift 8 still remain.) Galaxies of the same M_⋆ have different Z_g according to their radii. The relation changes qualitatively at different redshifts, as does the relation between gas metallicity and mass: the metallicities drop with increasing redshift, and the dependence of metallicity on M_⋆ strengthens. However, all these changes conspire to accentuate the global trend between size and metallicity existing at redshift 0. High-redshift galaxies tend to have their sizes and masses poorly correlated (galaxies of the same size may have a large range of masses, and vice versa). This enhances the dependence of metallicity on mass (see, e.g., the redshift = 3 plot in Figure 17).

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Another notable change of the relation is the emergence of a population of very compact and massive objects at redshift of 1 and larger (the red points at high mass in all the panels, which are absent at redshift 0). These galaxies have to be identified with the so-called blue nuggets, which are thought to be an extreme starburst phase leading to the compaction and eventual quenching of massive galaxies at high redshift (e.g., Zolotov et al. 2015; Tacchella et al. 2016). The end products are compact quenched red nuggets, which seem to be precursors of local massive ellipticals (Bezanson et al. 2009; Hopkins et al. 2009) and/or bulges of massive spirals and S0s (Graham 2013; de la Rosa et al. 2016).

Ellison et al. (2008) found that, for the same stellar mass, physically smaller star-forming galaxies are also more metal-rich. This work explores the possible physical cause of such a relation. The approach is indirect. We first show that the central star-forming galaxies in the EAGLE cosmological numerical simulation reproduce the observed relation qualitatively and quantitatively (see Section 2, where we also discuss existing differences). It is a nontrivial relation, in the sense that it does not follow from other well-known relations such as the SFR–stellar mass relation (i.e., the SF main sequence), the mass–metallicity relation, or the fundamental metallicity relation that connects the three variables mass, metallicity, and SFR. The EAGLE simulation was not tuned in any way to comply with Ellison et al. observations; therefore, the fact that simulated galaxies follow the observed trend is taken as a solid argument supporting that the correlation, in both simulations and observations, results from a common underlying physical cause. Thus, we study the simulation to pinpoint the origin of the correlation, taking as an ansatz that the models already include all the relevant physics. However, one has to keep in mind that the conclusions of our work rely on the validity of this hypothesis.

We consider the three obvious possibilities that may change the metallicity of the star-forming gas in a galaxy in relation to its size (Section 3): (1) SF-driven outflows carry away gas and metals, and the effectiveness of this process is related to the depth of the gravitational well the SN-driven winds escape from. This fact potentially links metallicity with galaxy size, which, given the mass, sets the depth of the potential well. (2) The timescale to deplete the gas depends on the gas density. Denser systems are more efficient at transforming gas into stars, and therefore their gas becomes metal richer sooner. For a given mass, the smaller the galaxy, the denser it is, which provides yet another potential connection between size and gas metallicity. (3) Finally, galaxies systematically grow in size with time, so galaxies with late SF are systematically bigger than those formed earlier. Delayed SF means still having metal-poor gas, which provides a connection between size and gas-phase metallicity. These predictions are analyzed using a simple toy model in Sections 3.1–3.3, which provides physical insight and allows us to estimate the magnitude of the expected effects resulting from each one of these mechanisms.

Aided with this interpretative framework, we analyze the actual EAGLE galaxies in Sections 4–6. Outflows are discarded as the cause of the correlation because, even though there is a trend for the more metal-rich galaxies to have deeper gravitational potential (Figure 8, top left panel), this trend washes out when galaxies of the same mass are considered (the rest of the panels in Figure 8). In a sense, the global trend is a mirage resulting from the global increase of both the depth of the gravitational well and the metallicity with increasing galaxy mass. Varying SF efficiency with mean density is also discarded as the underlying mechanism causing the anticorrelation between metallicity and size (Section 5). We use as proxy for gas density the stellar density. Even though the global trend is similar to that expected from the toy model (Figure 8, central panel), its amplitude is insufficient to explain the range of variability in gas-phase metallicity of the EAGLE galaxies (Figure 8). Finally, the growth of galaxy size with time, coupled with the recent accretion of metal-poor gas, seems to be the cause (Section 6). If so, galaxies of the same mass should present an anticorrelation between the recent gas infall rate and the gas metallicity. We use as proxies for the gas infall rate the SFR and the gas mass, and both indicators show a tight anticorrelation with metallicity (see Figures 14 and 15). Moreover, as expected if the correlation between size and gas metallicity is produced by recent gas accretion, the age of the stellar population of a galaxy is tightly connected with the stellar size and the gas metallicity, so that older stellar populations are characteristic of smaller metal-richer galaxies (Figure 16).

The EAGLE galaxies need to grow in size with time to explain the observation by Ellison et al. This growth in size is extensively discussed in the literature. It already appears in the classical theoretical paper by Mo et al. (1998), although based on hypotheses that may not be realistic (e.g., Sales et al. 2012; Garrison-Kimmel et al. 2017). In the case of the EAGLE simulation, the mechanism of growth depends on the galaxy mass. It is due to in situ SF fueled by gas accretion and minor gas-rich mergers when log(M_⋆/M_⊙) < 10.5, whereas dry mergers play a significant role only at the high-mass end, with log(M_⋆/M_⊙) > 11 (see Qu et al. 2017). Stellar migration is important in the size evolution of the EAGLE massive red compact galaxies, common at redshift 2 and depleted below redshift 1 (Furlong et al. 2017), but these objects and their descendants contribute little to the population of star-forming galaxies at redshift 0. In principle, stellar migration and other secular processes (e.g., El-Badry et al. 2016) redistribute old stars within the galaxy, thus distorting an initial correlation between galaxy size and age of the stellar population. However, these mechanisms do not seem to be effective enough to blur the underlying trend for the star-forming galaxies to increase in size with time. Both observations and simulations show that high-redshift galaxies are smaller. This includes passively evolving galaxies (e.g., Daddi et al. 2005; Trujillo et al. 2006; Buitrago et al. 2008) as well as star-forming galaxies (Ribeiro et al. 2016; Allen et al. 2017). Numerical simulations reproduce the observed trends satisfactorily (e.g., Wellons et al. 2015; Furlong et al. 2017; Genel et al. 2018). Observations also show a clear relation between size and age, so that smaller galaxies generally have older stellar populations. This result holds for passively evolving galaxies (redshift 0.2–0.8 and M_⋆ < 10¹¹ M_⊙, Fagioli et al. 2016; redshift ∼1.2, Williams et al. 2017; redshift 0 and M_⋆ < 3 × 10¹⁰ M_⊙; Shankar et al. 2010), as well as for late-type galaxies (e.g., Bernardi et al. 2010). The same kind of relation between size and age probably explains the difference in size between star-forming and passively evolving galaxies. Given M_⋆, the latter are systematically smaller than the former (e.g., van der Wel et al. 2014, Figures 5 and 6), and the stellar populations in early-type galaxies are systematically older than in late-type galaxies. SF histories derived from spatially resolved spectra clearly show the inside-out growth of the galaxies, with younger stellar populations in the outskirts (e.g., Pérez et al. 2013; García-Benito et al. 2017; Goddard et al. 2017).

It is important to realize that the above explanation is possible only if the galaxies have not reached equilibrium with the average mass accretion rate. Otherwise, the gas-phase metallicity is set only by stellar physics and wind strength, and it is independent of the mass infall rate, the gas mass, or the SFR (Equation (1)). Most galaxies have to be in a transient phase where the gas obtained during the last major gas accretion episode is still in use. A natural way for the galaxies to be systematically out of equilibrium is if the accretion turns out to be very bursty, with discrete accretion events followed by long gas-starved periods in between.

We explore the variation with redshift of the relation between metallicity and size in the EAGLE simulation. It is maintained up to at least redshift 8 (Figure 17). This fact remains to be tested observationally, but it is encouraging that the relation remains in place at redshift ∼1.4 according to Yabe et al. (2012, 2014).

Thanks are due to the EAGLE team for making the simulations publicly available. This work was carried out while J.S.A. was visiting the University of California at Santa Cruz (UCSC). He thanks David Koo for his hospitality during this period and Guillermo Barro, Francesco Belfiore, Nicolas Bouché, and Daniel Ceverino for comments on particular aspects of the analysis. Thanks are also due to an anonymous referee for helping us refine some of the arguments put forward in the paper. The work has been partly funded by the Spanish Ministry of Economy and Competitiveness (MINECO), projects Estallidos AYA2013-47742-C04-02-P and AYA2016-79724-C4-2-P, as well as by the Severo Ochoa Excellence Program granted to the Instituto de Astrofísica de Canarias by MINECO (SEV-2015-0548). C.D.V. acknowledges financial support from MIMECO through grants AYA2014-58308-P and RYC-2015-18078.

Software: TOPCAT, EAGLE interface (McAlpine et al. 2016).

Here, we assume that the galaxies have only two components: gas and stars. Using the equation of mass conservation from chemical evolution models (e.g., Tinsley 1980; Edmunds 1990), the variation with time t of the mass of gas available to form stars, M_g(t), is given by

$$\begin{eqnarray}&&{\dot{M}}_{g}=-(1-R)\mathrm{SFR}+{\dot{M}}_{\mathrm{in}}-{\dot{M}}_{\mathrm{out}},\end{eqnarray} \tag{ 18 }$$

which considers the formation of stars (first term on the right-hand side of the equation), a gas inflow rate ${\dot{M}}_{\mathrm{in}}(t)$, and a gas outflow rate ${\dot{M}}_{\mathrm{out}}(t)$. As usual, dotted quantities represent time derivatives. The SFR is assumed to be proportional to the gas mass,

$$\begin{eqnarray}&&\mathrm{SFR}=\displaystyle \frac{{M}_{g}}{{\tau }_{g}},\end{eqnarray} \tag{ 19 }$$

with the scaling parameterized in terms of a gas consumption timescale τ_g (see Equation (7)). (Note that Equation (19) is quite general since τ_g may depend on the physical properties of the galaxy, including the surface gas density.) We will also assume the outflow rate to scale with the SFR,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{out}}=w\,\mathrm{SFR},\end{eqnarray} \tag{ 20 }$$

with w the so-called mass-loading factor (see Equation (2)). Equation (20) is natural if outflows are driven by stellar winds or SN explosions, but it may also include AGN feedback if the AGN activity is correlated with SF. Under the same approximations leading to Equation (18), the metallicity of the gas that is forming stars, Z_g, follows the differential equation,

$$\begin{eqnarray}&&{\rm{\Delta }}\dot{{Z}_{g}}=(1-R)y/{\tau }_{g}-{\rm{\Delta }}{Z}_{g}\,{\dot{M}}_{\mathrm{in}}/{M}_{g},\end{eqnarray} \tag{ 21 }$$

where ΔZ_g represents the difference between Z_g and the metallicity of the accreted gas Z_in. The symbol y stands for the stellar yield (the mass of new metals eventually ejected per unit mass locked into stars and stellar remnants), and R represents the mass return fraction (the fraction of mass in stars that returns to the interstellar medium).

The gas mass and its metallicity follow from integrating Equations (18) and (21), given Equations (19) and (20), once ${\dot{M}}_{\mathrm{in}}(t)$ is set. We are interested in bursty accretion, where the accretion rate can be approximated as

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{in}}(t)={\dot{M}}_{\mathrm{in}0}+\delta (t-{t}_{a}){\rm{\Delta }}{M}_{a},\end{eqnarray} \tag{ 22 }$$

with ${\dot{M}}_{\mathrm{in}0}$ representing a background accretion rate on top of which the galaxy receives a gas clump of mass ΔM_a at t = t_a. The symbol δ stands for the Dirac delta function. Provided that all scaling factors R, w, and τ_g are constant in time, and t ≫ τ_in, the general solution of Equation (18) is

$$\begin{eqnarray}&&{M}_{g}(t)={\int }_{0}^{t}\,{\dot{M}}_{\mathrm{in}}({t}^{{\prime} }){e}^{-(t-{t}^{{\prime} })/{\tau }_{\mathrm{in}}}\ {{dt}}^{{\prime} },\end{eqnarray} \tag{ 23 }$$

with

$$\begin{eqnarray*}&&{\tau }_{\mathrm{in}}={\tau }_{g}/(1-R+w).\end{eqnarray*}$$

Using the accretion rate in Equation (22), at t ≫ τ_in, the mass of gas turns out to be

$$\begin{eqnarray}&&{M}_{g}(t)={M}_{g0}+{\rm{\Delta }}{M}_{a}\,H(t-{t}_{a})\exp [-(t-{t}_{a})/{\tau }_{\mathrm{in}}],\end{eqnarray} \tag{ 24 }$$

with

$$\begin{eqnarray*}&&{M}_{g0}={\dot{M}}_{\mathrm{in}0}\ {\tau }_{\mathrm{in}},\end{eqnarray*}$$

and with the symbol H(t) standing for the Heaviside step function,

$$\begin{eqnarray}H(t)=\left\{\begin{array}{ll}0 & \mathrm{if}\ {\text{}}t\lt 0,\\ 1 & \mathrm{if}\ {\text{}}t\geqslant 1.\end{array}\right.\end{eqnarray} \tag{ 25 }$$

Equation (21) is a first-order linear differential equation that admits a formal solution similar to Equation (23). It can be integrated using the mass of gas in Equation (24) and the accretion rate in Equation (22), and, after some algebra, the gas metallicity turns out to be

$$\begin{eqnarray} & & {\rm{\Delta }}{Z}_{g}(t)/{\rm{\Delta }}{Z}_{g0}\\ & = & \displaystyle \frac{{M}_{g0}+{\rm{\Delta }}{M}_{a}\ H(t-{t}_{a})\exp [-(t-{t}_{a})/{\tau }_{\mathrm{in}}](t-{t}_{a})/{\tau }_{\mathrm{in}}}{{M}_{g0}+{\rm{\Delta }}{M}_{a}\ H(t-{t}_{a})\exp [-(t-{t}_{a})/{\tau }_{\mathrm{in}}]},\end{eqnarray} \tag{ 26 }$$

with

$$\begin{eqnarray*}&&{\rm{\Delta }}{Z}_{g0}=y(1-R)/(1-R+w).\end{eqnarray*}$$

Equation (26) describes a sudden drop in metallicity at the moment of accretion, which recovers the stationary-state metallicity after τ_in, and then the metallicity keeps increasing, reaches a maximum, and decays again within a timescale significantly larger than τ_in. The behavior is illustrated in Figure 18, which represents a fairly massive burst with a mass contrast ΔM_a/M_g0 = 2.

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The stationary-state solution corresponds to $t\to \infty$, and it renders

$$\begin{eqnarray}&&{M}_{g}(\infty )={M}_{g0}={\dot{M}}_{\mathrm{in}0}\ {\tau }_{\mathrm{in}},\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&\mathrm{SFR}(\infty )={\mathrm{SFR}}_{0}\equiv \,{\dot{M}}_{\mathrm{in}0}/(1-R+w),\end{eqnarray} \tag{ 28 }$$

and

$$\begin{eqnarray}&&{\rm{\Delta }}{Z}_{g}(\infty )={\rm{\Delta }}{Z}_{g0}.\end{eqnarray} \tag{ 29 }$$

We note that in the stationary state, the ratio between SFR and gas-phase metallicity is independent of the mass-loading factor w, explicitly,

$$\begin{eqnarray}&&\displaystyle \frac{{\mathrm{SFR}}_{0}}{{\rm{\Delta }}{Z}_{g0}}=\displaystyle \frac{{\dot{M}}_{\mathrm{in}0}}{y(1-R)}.\end{eqnarray} \tag{ 30 }$$

The escape velocity at distance r from a spherically mass distribution is given by

$$\begin{eqnarray}&&{v}_{\mathrm{esc}}(r)=\sqrt{2\,| {\rm{\Phi }}(r)| },\end{eqnarray} \tag{ 31 }$$

with Φ(r) the gravitational potential (e.g., Binney & Tremaine2008, Equation (2.31)). In the case of a NFW profile,

$$\begin{eqnarray}&&{\rm{\Phi }}(r)=-4\pi \,G\,{\rho }_{0}\,{R}_{s}^{2}\,\displaystyle \frac{\mathrm{ln}(1+r/{R}_{s})}{r/{R}_{s}}\end{eqnarray} \tag{ 32 }$$

(e.g., Binney & Tremaine 2008, Equation (2.67)), where R_s and ρ₀ are the two free parameters of the NFW density profile,

$$\begin{eqnarray}&&\rho (r)=\displaystyle \frac{{\rho }_{0}}{(r/{R}_{s}){(1+r/{R}_{s})}^{2}}.\end{eqnarray} \tag{ 33 }$$

Once R_s and ρ₀ are known, the escape velocity is given by Equations (31) and (32). The EAGLE database does not provide R_s and ρ₀, but instead it provides the total mass of the halo when the mean density is 200 times the critical density ρ_c, M₂₀₀, as well as the corresponding half-mass radius R_1/2. The former parameters can be obtained from the later parameters considering that the mass enclosed within a radius r is (e.g., Binney & Tremaine 2008, Equation (2.66))

$$\begin{eqnarray}&&M(r)=4\pi \,{\rho }_{0}\,{R}_{s}^{3}\left[\mathrm{ln}\left(\displaystyle \frac{r+{R}_{s}}{{R}_{s}}\right)-\displaystyle \frac{r}{r+{R}_{s}}\right].\end{eqnarray} \tag{ 34 }$$

By definition, M(R_1/2) = M₂₀₀/2, and therefore

$$\begin{eqnarray}\mathrm{ln}\left(\displaystyle \frac{{R}_{200}+{R}_{s}}{{R}_{s}}\right) & - & \displaystyle \frac{{R}_{200}}{{R}_{200}+{R}_{s}}\\ & = & \displaystyle \frac{1}{2}\left[\mathrm{ln}\left(\displaystyle \frac{{R}_{1/2}+{R}_{s}}{{R}_{s}}\right)-\displaystyle \frac{{R}_{1/2}}{{R}_{1/2}+{R}_{s}}\right],\end{eqnarray} \tag{ 35 }$$

with

$$\begin{eqnarray}&&{R}_{200}^{3}={M}_{200}/\left(\displaystyle \frac{4\pi }{3}\ 200\ {\rho }_{c}\right).\end{eqnarray} \tag{ 36 }$$

Since ρ_c, R_1/2, and M₂₀₀ are known, Equations (35) and (36) allow us to infer R_s. We solve it iteratively. Then Equation (34) at r = R_1/2 and M = M₂₀₀/2 provides ρ₀.