On the origin of the mass-metallicity gradient relation in the local Universe
Piyush Sharda, Mark R. Krumholz, Emily Wisnioski, Ayan Acharyya, Christoph Federrath, John C. Forbes
MMNRAS , 1–12 (2021) Preprint 22 February 2021 Compiled using MNRAS L A TEX style file v3.0
On the origin of the mass-metallicity gradient relation in the local Universe
Piyush Sharda , ★ , Mark R. Krumholz , † , Emily Wisnioski , ‡ , Ayan Acharyya , , ,Christoph Federrath , , and John C. Forbes Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA
Accepted XXX. Received 2021 February 19; in original form 2020 December 3
ABSTRACT
In addition to the well-known gas phase mass-metallicity relation (MZR), recent spatially-resolved observations have shown thatlocal galaxies also obey a mass–metallicity gradient relation (MZGR) whereby metallicity gradients can vary systematically withgalaxy mass. In this work, we use our recently-developed analytic model for metallicity distributions in galactic discs, whichincludes a wide range of physical processes – radial advection, metal diffusion, cosmological accretion, and metal-enrichedoutflows – to simultaneously analyse the MZR and MZGR. We show that the same physical principles govern the shape of both:centrally-peaked metal production favours steeper gradients, and this steepening is diluted by the addition of metal-poor gas,which is supplied by inward advection for low-mass galaxies and by cosmological accretion for massive galaxies. The MZR andthe MZGR both bend at galaxy stellar mass ∼ − . M (cid:12) , and we show that this feature corresponds to the transition ofgalaxies from the advection-dominated to the accretion-dominated regime. We also find that both the MZR and MZGR stronglysuggest that low-mass galaxies preferentially lose metals entrained in their galactic winds. While this metal-enrichment of thegalactic outflows is crucial for reproducing both the MZR and the MZGR at the low-mass end, we show that the flattening ofgradients in massive galaxies is expected regardless of the nature of their winds. Key words: galaxies: evolution – galaxies: ISM – galaxies: abundances – ISM: abundances – (
ISM :) HII regions – galaxies:fundamental parameters
Metals have a profound impact on galaxy formation and evolutioneven though their contribution to the total visible matter is less thantwo per cent. The symbiotic relationship between galaxies and theirmetal content has now been investigated in detail through numerousobservations, simulations and analytic models. One of the key man-ifestations of this relationship is the correlation between the stellarmass of a galaxy ( 𝑀 ★ , used as a proxy for the total galaxy mass) andits global (gas phase or stellar) metallicity, 𝑍 . It is now well estab-lished that low-mass galaxies have lower 𝑍 as compared to massivegalaxies. This is known as the mass metallicity relation (MZR; e.g.,Kewley & Dopita 2002; Tremonti et al. 2004; Mannucci et al. 2010;Peng et al. 2015; Belfiore et al. 2017; Zahid et al. 2017; Curti et al.2017, 2020). The exact cause of the MZR is still debated; for exam-ple, star formation (Brooks et al. 2007), outflows (Finlator & Davé2008; Chisholm et al. 2018), cosmic accretion or infall (Larson 1972;Davé et al. 2012), feedback (Tissera et al. 2019), and the initial massfunction (IMF, Köppen et al. 2007) can all play a role in setting itsshape. The shape of the MZR seen in observations has now been suc-cessfully reproduced by many simulations (e.g., Brooks et al. 2007; ★ [email protected] (PS) † [email protected] (MRK) ‡ [email protected] (EW) Davé et al. 2011, 2017; Torrey et al. 2019; Tissera et al. 2019) andtheoretical models (e.g., Finlator & Davé 2008; Peeples & Shankar2011; Lilly et al. 2013; De Lucia et al. 2020); however, the absolutenormalisation of the MZR ( i.e., the absolute value of 𝑍 ) remainsuncertain due to difficulties in calibrating 𝑍 from observations (seereviews by Kewley et al. 2019, Maiolino & Mannucci 2019 andSánchez 2020).Since the pioneering works by Searle (1971), Mayor (1976) andShaver et al. (1983), it has been known that galaxies also exhibit agradient in the spatial distribution of metallicity, both in stars andin the gas phase, in the radial direction (e.g., Zaritsky et al. 1994;González Delgado et al. 2015; Goddard et al. 2017; Belfiore et al.2017) as well as variations in the azimuthal direction (e.g., Lucket al. 2011; Li et al. 2013; Ho et al. 2017, 2019; Kreckel et al. 2019).The fact that radial gradients are usually negative ( i.e., the centreof the galaxy is more metal-rich than the outskirts) is a key pieceof evidence for the theory of inside-out galaxy formation (Mo et al.1998; Benson 2010; Naab & Ostriker 2017). Hereafter, we only focuson the metallicities and metallicity gradients in the ionised gas.Thanks to the plethora of galaxies observed in the nearby Universewith large integral field spectroscopy (IFS) surveys like CALIFA(Calar Alto Legacy Integral Field Area, Sánchez et al. 2012), MaNGA(Mapping nearby Galaxies at Apache Point Observatory, Bundy et al.2015), and SAMI (Sydney-AAO Multi-object Integral-field spectro- © a r X i v : . [ a s t r o - ph . GA ] F e b P. Sharda et al. graph, Bryant et al. 2015), we can now study the trends of metallicitygradients with different galaxy properties in a statistical sense. Likethe MZR, of particular interest is the stellar mass–metallicity gradi-ent relation (MZGR). The general consensus is that the metallicitygradient, when measured in absolute units of dex kpc − , either re-mains independent of stellar mass up to 𝑀 ★ ∼ − . M (cid:12) , thenflattens toward zero gradient at higher stellar masses (Maiolino &Mannucci 2019), or shows a mild curvature around ∼ − . M (cid:12) ,with flat gradients on either side (e.g., Belfiore et al. 2017). If thegradients are instead normalised by the effective radius of galaxies( 𝑟 e ) and expressed in dex 𝑟 − , some authors find that the MZGR issteepest around 𝑀 ★ ∼ − . M (cid:12) , with flatter gradients on eitherside (e.g., Belfiore et al. 2017; Mingozzi et al. 2020; Poetrodjojoet al. 2021), whereas others report a constant, characteristic dex 𝑟 − gradient for all galaxies with 𝑀 ★ > . M (cid:12) (Sánchez et al. 2012,2014; Sánchez-Menguiano et al. 2016, 2018; Poetrodjojo et al. 2018).However, these trends in the MZGR are relatively weak as comparedto the MZR, suffer observational and calibration uncertainties (Yuanet al. 2013; Acharyya et al. 2020, 2021; Poetrodjojo et al. 2021), andto date, have received limited theoretical investigation.The goal of this work is to provide a physical explanation for theshape of the MZGR. For this purpose, we use our recently-developedfirst principles model of gas phase metallicity gradients (Sharda et al.2021a). This model is based on the equilibrium between the produc-tion, consumption, loss and transport of metals in galactic discs. Itproduces gas phase metallicity gradients in good agreement with awide range of local and high- 𝑧 galaxies, and shows that these gradi-ents are in equilibrium across a diverse range of galaxy properties.We refer the reader to Sharda et al. (2021a) for a full description of themodel, the gradients produced, as well as applications of the modelto study the cosmic evolution of metallicity gradients and their trendswith galaxy kinematics (Sharda et al. 2021b). The rest of this paperis organised as follows: Section 2 presents a review of the model,Section 3 describes the MZR produced by our model, which we useas a proof of concept to explain the MZGR in Section 4. Section 5introduces the MZR–MZGR space in equilibrium as a new way ofcharacterizing gas phase metallicities, and Section 6 summarizes ourkey results. For the purpose of this paper, we use Z (cid:12) = . + log O / H = .
69 (Asplundet al. 2009), Hubble time at 𝑧 = 𝑡 H ( ) = . Λ CDM cosmology: Ω m = . Ω Λ = . ℎ = .
71, and 𝜎 = .
81 (Springel & Hernquist 2003).
In this section, we provide a brief review of the model of gas phasemetallicity gradients we presented in Sharda et al. (2021a); this isintended to highlight only the results of which we will make use here,and we refer readers to the original paper for full details. In that work,we showed that the evolution of gas phase metallicity is described bythe Euler-Cauchy equation T 𝑠 𝑔 𝜕 Z 𝜕𝜏 (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) equilibriumtime − P 𝑥 𝜕 Z 𝜕𝑥 (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) advection − 𝑥 𝜕𝜕𝑥 (cid:18) 𝑥𝑘𝑠 𝑔 𝜕 Z 𝜕𝑥 (cid:19)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) diffusion = S (cid:164) 𝑠 ★ (cid:124)(cid:123)(cid:122)(cid:125) production+outflows − ZA (cid:164) 𝑐 ★ (cid:124) (cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32) (cid:125) accretion , (1)where Z = 𝑍 / Z (cid:12) is the metallicity normalised to Solar, 𝑥 is theradius of the disc normalised to the radius 𝑟 that we take to be theinner edge of the disc ( i.e., 𝑥 = 𝑟 / 𝑟 where 𝑟 is the galactocentricradius), 𝜏 is the time normalised to the orbital time at 𝑟 , 𝑘 is the nor-malised diffusion coefficient, and 𝑠 𝑔 , (cid:164) 𝑠 ★ , and (cid:164) 𝑐 ★ are the normalised gas mass, star formation rate (SFR), and cosmic accretion rate perunit area of the galactic disc, respectively. From left to right, the dif-ferent terms in equation 1 represent the equilibration time for a givenmetal distribution, radial advection of metals due to inflows, diffu-sion of metals due to concentration gradients, production of metalsthrough star formation and loss via galactic outflows, and cosmic ac-cretion of metal-poor gas from the circumgalactic medium (CGM),respectively. From equation 1, we see that Z is governed by fourdimensionless ratios. These are T – the ratio of orbital to diffusiontimescales, P – the Péclet number of the galaxy that describes theratio of advection to diffusion (e.g., Patankar 1980; Rapp 2017), the‘source’ term S – the ratio of metal production to diffusion, andthe ‘accretion’ term A – the ratio of cosmic accretion (or infall) todiffusion.In equilibrium, the first term goes to zero, and one can find a steady-state solution to equation 1 for any specified profiles of 𝑠 𝑔 , (cid:164) 𝑠 ★ , and (cid:164) 𝑐 ★ versus radius. We set 𝑠 𝑔 and (cid:164) 𝑠 ★ from the unified galaxy disc model ofKrumholz et al. (2018), and (cid:164) 𝑐 ★ based on cosmological simulations(e.g., Colavitti et al. 2008). For these choices, the correspondingequilibrium solution for the metallicity as a function of normalisedgalactocentric radius, Z( 𝑥 ) , is given by Z( 𝑥 ) = SA + 𝑐 𝑥 (cid:104) √P + A−P (cid:105) + (cid:18) Z 𝑟 − SA − 𝑐 (cid:19) 𝑥 (cid:104) −√P + A−P (cid:105) , (2)where 𝑐 is a constant of integration that is determined by the metal-licity of the CGM, Z CGM , and Z 𝑟 is the equilibrium metallicity at 𝑟 that we can determine from other galaxy parameters. We can alsoexpress P , S and A in terms of meaningful galaxy parameters usingthe Krumholz et al. (2018) model, which gives T = 𝜙 𝑄 √︁ ( 𝛽 + ) 𝑓 𝑔,𝑄 𝑄 min (cid:18) 𝑣 𝜙 𝜎 𝑔 (cid:19) , (3) P = 𝜂𝜙 𝑄 𝜙 / 𝑓 𝑔,𝑄 𝑄 (cid:18) + 𝛽 − 𝛽 (cid:19) (cid:18) − 𝜎 sf 𝜎 𝑔 (cid:19) , (4) S = 𝜙 𝑄 𝑓 𝑔,𝑄 𝜖 ff 𝑓 sf 𝜋𝑄 min √︁ 𝑓 𝑔,𝑃 𝜙 mp (cid:18) 𝜙 𝑦 𝑦𝑍 (cid:12) (cid:19) ( + 𝛽 ) (cid:18) 𝑣 𝜙 𝜎 𝑔 (cid:19) , (5) A = 𝐺 (cid:164) 𝑀 ℎ 𝑓 B 𝜖 in 𝜙 𝑄 𝜎 𝑔 [ ln 𝑥 max − ln 𝑥 min ] . (6)Here, 𝜙 𝑄 − 𝑄 parameters(Romeo & Wiegert 2011; Romeo & Falstad 2013), 𝛽 is the rotationcurve index of the galaxy, 𝑓 𝑔,𝑄 and 𝑓 𝑔,𝑃 are two slightly differentmeasures of the effective gas fraction (Ostriker et al. 2010; Krumholzet al. 2018), 𝑄 min is the Toomre 𝑄 parameter (Toomre 1964) belowwhich discs are unstable due to gravity (e.g., Krumholz & Burkert2010; Goldbaum et al. 2015), 𝑣 𝜙 is the rotational velocity of thegalaxy, 𝜎 𝑔 is the gas velocity dispersion, 𝜂 is a dimensional factorof order unity describing the rate of turbulent dissipation (Mac Lowet al. 1998; Forbes et al. 2012), 𝜙 nt is the fraction of total velocitydispersion that is in non-thermal rather than thermal motions, 𝜎 sf is the maximum velocity dispersion that can be maintained by starformation feedback, 𝜖 ff is the star formation efficiency per free-falltime (Krumholz & McKee 2005; Federrath & Klessen 2012; Padoanet al. 2012), 𝑓 sf is the fraction of gas that is molecular (Krumholz et al.2009; Krumholz 2013), 𝜙 mp is the ratio of the total to the turbulent MNRAS000
In this section, we provide a brief review of the model of gas phasemetallicity gradients we presented in Sharda et al. (2021a); this isintended to highlight only the results of which we will make use here,and we refer readers to the original paper for full details. In that work,we showed that the evolution of gas phase metallicity is described bythe Euler-Cauchy equation T 𝑠 𝑔 𝜕 Z 𝜕𝜏 (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) equilibriumtime − P 𝑥 𝜕 Z 𝜕𝑥 (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) advection − 𝑥 𝜕𝜕𝑥 (cid:18) 𝑥𝑘𝑠 𝑔 𝜕 Z 𝜕𝑥 (cid:19)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) diffusion = S (cid:164) 𝑠 ★ (cid:124)(cid:123)(cid:122)(cid:125) production+outflows − ZA (cid:164) 𝑐 ★ (cid:124) (cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32) (cid:125) accretion , (1)where Z = 𝑍 / Z (cid:12) is the metallicity normalised to Solar, 𝑥 is theradius of the disc normalised to the radius 𝑟 that we take to be theinner edge of the disc ( i.e., 𝑥 = 𝑟 / 𝑟 where 𝑟 is the galactocentricradius), 𝜏 is the time normalised to the orbital time at 𝑟 , 𝑘 is the nor-malised diffusion coefficient, and 𝑠 𝑔 , (cid:164) 𝑠 ★ , and (cid:164) 𝑐 ★ are the normalised gas mass, star formation rate (SFR), and cosmic accretion rate perunit area of the galactic disc, respectively. From left to right, the dif-ferent terms in equation 1 represent the equilibration time for a givenmetal distribution, radial advection of metals due to inflows, diffu-sion of metals due to concentration gradients, production of metalsthrough star formation and loss via galactic outflows, and cosmic ac-cretion of metal-poor gas from the circumgalactic medium (CGM),respectively. From equation 1, we see that Z is governed by fourdimensionless ratios. These are T – the ratio of orbital to diffusiontimescales, P – the Péclet number of the galaxy that describes theratio of advection to diffusion (e.g., Patankar 1980; Rapp 2017), the‘source’ term S – the ratio of metal production to diffusion, andthe ‘accretion’ term A – the ratio of cosmic accretion (or infall) todiffusion.In equilibrium, the first term goes to zero, and one can find a steady-state solution to equation 1 for any specified profiles of 𝑠 𝑔 , (cid:164) 𝑠 ★ , and (cid:164) 𝑐 ★ versus radius. We set 𝑠 𝑔 and (cid:164) 𝑠 ★ from the unified galaxy disc model ofKrumholz et al. (2018), and (cid:164) 𝑐 ★ based on cosmological simulations(e.g., Colavitti et al. 2008). For these choices, the correspondingequilibrium solution for the metallicity as a function of normalisedgalactocentric radius, Z( 𝑥 ) , is given by Z( 𝑥 ) = SA + 𝑐 𝑥 (cid:104) √P + A−P (cid:105) + (cid:18) Z 𝑟 − SA − 𝑐 (cid:19) 𝑥 (cid:104) −√P + A−P (cid:105) , (2)where 𝑐 is a constant of integration that is determined by the metal-licity of the CGM, Z CGM , and Z 𝑟 is the equilibrium metallicity at 𝑟 that we can determine from other galaxy parameters. We can alsoexpress P , S and A in terms of meaningful galaxy parameters usingthe Krumholz et al. (2018) model, which gives T = 𝜙 𝑄 √︁ ( 𝛽 + ) 𝑓 𝑔,𝑄 𝑄 min (cid:18) 𝑣 𝜙 𝜎 𝑔 (cid:19) , (3) P = 𝜂𝜙 𝑄 𝜙 / 𝑓 𝑔,𝑄 𝑄 (cid:18) + 𝛽 − 𝛽 (cid:19) (cid:18) − 𝜎 sf 𝜎 𝑔 (cid:19) , (4) S = 𝜙 𝑄 𝑓 𝑔,𝑄 𝜖 ff 𝑓 sf 𝜋𝑄 min √︁ 𝑓 𝑔,𝑃 𝜙 mp (cid:18) 𝜙 𝑦 𝑦𝑍 (cid:12) (cid:19) ( + 𝛽 ) (cid:18) 𝑣 𝜙 𝜎 𝑔 (cid:19) , (5) A = 𝐺 (cid:164) 𝑀 ℎ 𝑓 B 𝜖 in 𝜙 𝑄 𝜎 𝑔 [ ln 𝑥 max − ln 𝑥 min ] . (6)Here, 𝜙 𝑄 − 𝑄 parameters(Romeo & Wiegert 2011; Romeo & Falstad 2013), 𝛽 is the rotationcurve index of the galaxy, 𝑓 𝑔,𝑄 and 𝑓 𝑔,𝑃 are two slightly differentmeasures of the effective gas fraction (Ostriker et al. 2010; Krumholzet al. 2018), 𝑄 min is the Toomre 𝑄 parameter (Toomre 1964) belowwhich discs are unstable due to gravity (e.g., Krumholz & Burkert2010; Goldbaum et al. 2015), 𝑣 𝜙 is the rotational velocity of thegalaxy, 𝜎 𝑔 is the gas velocity dispersion, 𝜂 is a dimensional factorof order unity describing the rate of turbulent dissipation (Mac Lowet al. 1998; Forbes et al. 2012), 𝜙 nt is the fraction of total velocitydispersion that is in non-thermal rather than thermal motions, 𝜎 sf is the maximum velocity dispersion that can be maintained by starformation feedback, 𝜖 ff is the star formation efficiency per free-falltime (Krumholz & McKee 2005; Federrath & Klessen 2012; Padoanet al. 2012), 𝑓 sf is the fraction of gas that is molecular (Krumholz et al.2009; Krumholz 2013), 𝜙 mp is the ratio of the total to the turbulent MNRAS000 , 1–12 (2021) as phase metallicity relations pressure at the mid-plane (Ostriker et al. 2010), (cid:164) 𝑀 ℎ is the dark matteraccretion rate onto the halo (Neistein & Dekel 2008; Bouché et al.2010), 𝑓 B is the universal baryonic fraction (White & Fabian 1995;Planck Collaboration et al. 2016), and 𝜖 in is the baryonic accretionefficiency (Faucher-Giguère et al. 2011). We refer the readers toSharda et al. (2021a, Tables 1 and 2) for full descriptions of andtypical values for all these parameters.In addition to these quantities, the production term S dependson one additional parameter: the yield reduction factor 𝜙 𝑦 , whichdescribes the reduction in the metal yield due to preferential ejectionof metals through galactic outflows. 𝜙 𝑦 = 𝜙 𝑦 = The Sharda et al. (2021a) model is distinct from earlier modelsfor galaxy metallicity distributions in a few ways: (1.) we includeall major transport processes, including advection and diffusion ofmetals, both of which are usually neglected, but which can becomeimportant in some regimes, as we show below; (2.) we do not makethe common assumption that the wind and ISM metallicities areequal, since there is observational evidence that they are not (e.g.,Martin et al. 2002; Strickland & Heckman 2009; Chisholm et al.2018); (3.) we derive model parameters such as the star formationrate, radial advection rate, diffusion rate, etc., from a physical modelof galactic discs that is well tested against observations (Johnson et al.2018; Yu et al. 2019; Übler et al. 2019; Varidel et al. 2020; Girardet al. 2021; Sharda et al. 2021b), rather than adopting parameterisedprescriptions of unknown accuracy; (4.) our model allows us to studyboth global and spatially-resolved metallicity properties.However, the model also has some important limitations that weshould note. First, we derive solutions for Z( 𝑥 ) only for galaxieswhose metal distributions are in equilibrium; we show in Shardaet al. (2021a) that almost all galaxies at 𝑧 = 𝑧 ≈
3. However, a major exception to this may be galaxieswith inverted gradients; for this reason we do not study invertedgradients with this model. We also make a number of simplifyingassumptions in order to obtain our analytic solutions: we assume thatthe rotation curve index 𝛽 is a constant. We use the instantaneousrecycling approximation (Tinsley 1980), which means that the modelis best applied to elements that are returned to the ISM quickly viaType II supernovae, rather than over longer time scales by othernucleosynthetic sources. We assume gas accreting onto the galaxycan be described by a single, fixed metallicity, which implicitly meansthat we neglect galactic fountains, long-term wind recycling throughthe CGM, and other environmental effects (e.g., the presence ofsatellites). Nonetheless, as we show in the next three sections thatthe model can successfully explain the MZR (Section 3), the MZGR(Section 4), and the relationship between the two (Section 5). It is important to clarify that 𝜙 𝑦 is not the same as the metal outflow rateor the metal mass loading factor, since 𝜙 𝑦 only describes how metals arepartitioned between winds and the ISM, not the total metal mass carried bythe winds. For example, a galaxy could have very low mass loading but alsolow 𝜙 𝑦 , if the winds consisted primarily of metal-rich supernova ejecta, withvery little additional ISM mass entrained. Almost all the analytic models that reproduce the observed MZR donot have spatial information of the distribution of metallicities in agalaxy – these are typically developed to study global metallicitiesin galaxies. Although the primary focus of our work is to explainmetallicity gradients by making use of the spatial information ofmetallicity, our model also reproduces the MZR as a proof of concept.To produce an MZR from the model, we need an estimate of themean metallicity in galaxies as a function of 𝑀 ★ . For this purpose,we use the SFR-weighted mean metallicity given by Sharda et al.(2021a, equation 46) Z = ∫ 𝑥 max 𝑥 min 𝑥 (cid:164) 𝑠 ★ Z 𝑑𝑥 ∫ 𝑥 max 𝑥 min 𝑥 (cid:164) 𝑠 ★ 𝑑𝑥 , (7)where (cid:164) 𝑠 ★ ( 𝑥 ) = / 𝑥 is the radial distribution of star formation perunit area (Krumholz et al. 2018). We use the SFR-weighted Z ,because it can be directly compared against available MZRs sincethey are inherently sensitive to the SFR as the nebular metallicitiesare measured in H ii regions around young stars (Zahid et al. 2014).Additionally, semi-analytic models and simulations too use SFR-weighted metallicities to construct MZRs (e.g., Tissera et al. 2019;Torrey et al. 2019; Yates et al. 2020).In order to derive results in terms of 𝑀 ★ , we treat the rotational ve-locity, 𝑣 𝜙 , as the primary quantity that we vary. For each 𝑣 𝜙 , we canestimate the corresponding halo mass 𝑀 h and halo accretion rate (cid:164) 𝑀 h at 𝑧 = 𝑀 ★ following the 𝑀 h − 𝑀 ★ relation from Moster et al. (2013)for the local Universe. Following Sharda et al. (2021a), we keep theyield reduction factor, 𝜙 𝑦 , as a free parameter and vary it between0.1 and 1, though we note that, based on both theory and observa-tions, 𝜙 𝑦 is expected to be close to unity in massive galaxies. For allother parameters, in particular the velocity dispersion 𝜎 𝑔 , we use thefiducial values listed in Sharda et al. (2021a, Tables 1 and 2). Specif-ically, we use local dwarf values for galaxies with 𝑀 ★ ≤ M (cid:12) ,and local spiral values for 𝑀 ★ ≥ . M (cid:12) . For intermediate stellarmasses, we linearly interpolate in log 𝑀 ★ between these two limitsfor all parameters. For example, the velocity dispersions we adoptfor spirals and dwarfs are 10 km s − and 7 km s − respectively, so weadopt 𝜎 𝑔 = (cid:0) 𝑀 ★ / M (cid:12) − (cid:1) km s − for intermediate-massgalaxies with 10 M (cid:12) < 𝑀 ★ < . M (cid:12) . We have verified that theresulting MZR and MZGR are not particularly sensitive to the choiceof the 𝑀 ★ boundaries invoked to classify dwarfs and spirals; we alsodiscuss this further in Section 4. We set Z 𝑟 to its equilibrium value(Sharda et al. 2021a), and set the circumgalactic medium metallicityto Z CGM = . , which sets 𝑐 . The MZR (as wellas the MZGR discussed below) is insensitive to Z 𝑟 and only weaklysensitive to Z CGM as compared to 𝜙 𝑦 , so we do not vary Z GCM separately. Finally, we follow van der Wel et al. (2014) to estimate 𝑟 e as a function of 𝑀 ★ , and set 𝑥 min = . 𝑟 e and 𝑥 max = 𝑟 e as therange of radii 𝑥 over which our model solution applies. This range ofradii roughly mimics that over which metallicities are measured.Figure 1 shows the resulting MZR from our model, color-coded This is slightly lower than the median Z CGM = . 𝑧 ∼ . Z CGM ); however, these surveys do notcover the entire range in galaxy masses we are interested in, and we expect Z CGM to be lower in low mass galaxies. In any case, this difference does nothave a significant effect on the MZGR. MNRAS , 1–12 (2021)
P. Sharda et al. log M /M [ + l og ( O / H )] y = 0.1 y = 1.0 MZR y scaling 1 y scaling 2 Direct T e based MZRsPhotoionization based MZRs l og / Figure 1.
Mass–metallicity relation (MZR) in local galaxies predicted by the Sharda et al. (2021a) model, for different yield reduction factors 𝜙 𝑦 , color-codedby the ratio of the Péclet number ( P ) to cosmic accretion over diffusion ( A ). The MZR displays a curvature around 𝑀 ★ ∼ − . M (cid:12) , corresponding tothe transition from the advection-dominated ( P > A ) to the accretion-dominated ( P < A ) regime. Overlaid on the model are parameter spaces correspondingto MZRs derived from observations, using the direct 𝑇 e method (Pettini & Pagel 2004; Andrews & Martini 2013; Curti et al. 2017, 2020), and photoionizationmodels (Kewley & Dopita 2002; Tremonti et al. 2004; Mannucci et al. 2010), adopted from (Maiolino & Mannucci 2019, Figure 15). Finally, the white markersshow model predictions using two possible empirical scalings of 𝜙 𝑦 with 𝑀 ★ . Scaling 1 is derived from observations (Chisholm et al. 2018), whereas scaling 2is independently derived from the best match between the model MZR and the Curti et al. (2020) MZR; details of these scalings are given in Appendix A. Ourfindings predict a scaling of 𝜙 𝑦 with 𝑀 ★ where massive galaxies prefer a higher value of 𝜙 𝑦 , and vice-versa. This implies that low-mass galaxies have moremetal-enriched winds, consistent with observations (Chisholm et al. 2018) and simulations (Emerick et al. 2018; Tanner 2020). by the ratio P/A that describes the relative strength of advection tocosmic accretion. We remind the reader that both P and A (as wellas S ) are normalised by diffusion in the model. The vertical spreadin the model MZR is a result of varying 𝜙 𝑦 . We also overplot theparameter space of observed MZRs from several other works basedon the direct 𝑇 e method (Pettini & Pagel 2004; Andrews & Martini2013; Curti et al. 2017, 2020) and photoionization modeling (Kewley& Dopita 2002; Tremonti et al. 2004; Mannucci et al. 2010), all ofwhich we adopt from Maiolino & Mannucci (2019, Figure 15). Wesee that the model is able to reproduce the MZR of the local Universealbeit with a large spread due to 𝜙 𝑦 . There are several factors behindquantitative differences between the model MZR and MZRs in theliterature. From the perspective of the model, these differences areattributed to the choice of the metal yield 𝑦 , excluding the galaxynucleus while finding the mean metallicities, and the absolute size ofthe galaxy disc. From the perspective of the MZRs we compare themodel with, these differences are due to calibration and observationaluncertainties, as well as limited coverage of the galaxy discs.In order to match with the measured MZRs, the model prefershigher 𝜙 𝑦 for massive galaxies and lower 𝜙 𝑦 for low-mass galax-ies. This implies that metals are well-mixed in the ISM in massivegalaxies before they are ejected through outflows, whereas in dwarfgalaxies, some fraction of metals are ejected directly before they can mix in the ISM; in other words, the best match between the modelMZR and the literature MZRs predicts that dwarf galaxies have moremetal-enriched winds than massive galaxies. This finding is not newand has been theorized in several works (e.g., Larson 1974; Dekel& Silk 1986; Dalcanton 2007; Finlator & Davé 2008; Lilly et al.2013; Forbes et al. 2019), simulations (Creasey et al. 2015; Ma et al.2016; Christensen et al. 2018; Emerick et al. 2018, 2019), and alsohas some observational evidence (Martin et al. 2002; Chisholm et al.2018).To further treat the question of how 𝜙 𝑦 scales with 𝑀 ★ quantita-tively, we also plot two models for this scaling. We obtain the first ofthese from available observations that directly constrain the ratio ofwind metallicity to ISM metallicity (Chisholm et al. 2018), and thesecond simply by forcing the model to reproduce the observed MZRprovided by Curti et al. (2020). Appendix A describes how we ob-tain these scalings (and the associated uncertainties) in detail. Whilethe shape of the first scaling is consistent with observed MZRs, thesecond is almost identical to the direct 𝑇 𝑒 based MZRs by construc-tion; we include the second scaling nonetheless because there is noguarantee that the scaling we have enforced to produce the MZR willalso yield the correct MZGR, a question we explore below.Figure 1 shows that the MZR bends roughly where the ratio P/A passes through unity. We can understand this behaviour as follows:
MNRAS000
MNRAS000 , 1–12 (2021) as phase metallicity relations the total metallicity is set by a competition between metal production(the term S ) and dilution by metal-poor gas, which can be suppliedeither by direct cosmological accretion onto the disc ( A ) or advectionof gas from the weakly-star-forming outskirts to the more rapidly-star-forming centre ( P ). Each of these terms varies differently withrotation curve velocity 𝑣 𝜙 , which in turn correlates with stellar mass;as shown in Sharda et al. (2021a), P is independent of 𝑣 𝜙 , while S ∝ 𝑣 𝜙 and A ∝ 𝑣 . 𝜙 . In the low-mass regime, correspondingto small 𝑣 𝜙 , we have P > A , implying that the metallicities areprimarily set by the balance between source and advection. Since P ∝ 𝑣 𝜙 and S ∝ 𝑣 𝜙 , as we go to smaller 𝑀 ★ and 𝑣 𝜙 , the equilibriummetallicity drops because of lower 𝑣 𝜙 and lower 𝜙 𝑦 as compared tomassive galaxies. On the contrary, in the high-mass regime A > P ,implying that the metallicities are set by the balance between A and S . Since A ∝ 𝑣 . 𝜙 , which is stronger than the dependence of S on 𝑣 𝜙 , the metallicity, which is proportional to S/A , ceases to risewith 𝑀 ★ , and instead reaches a maximum and starts to decrease.However, the decrease is rather mild, because shortly after passingthe value of 𝑣 𝜙 where we move into the A > P regime, galaxiesbecome so massive that they cease to be star-forming altogether.Thus, among star-forming galaxies, the trend of Z with 𝑀 ★ is simplythat Z ceases to increase and reaches a plateau. For less massivegalaxies the dominant source of metal-poor gas is advection ratherthan accretion. However, this only holds as long as advection is non-zero; for low mass galaxies where there is no advection ( i.e., there isno turbulence due to gravity), it falls upon cosmic accretion to balancemetal production. Since cosmic accretion is much weaker in low massgalaxies, it can take a long time for this balance to approach a steady-state, which can push the gradients out of equilibrium (Sharda et al.2021a, Section 5.1). The existence of a local gas phase MZR has been known since earlyanalysis of data from the Sloan Digital Sky Survey (SDSS, Tremontiet al. 2004), although the absolute normalisation of the MZR remainsan unsolved issue due to systematic calibration uncertainties (Kewley& Ellison 2008; Pilyugin & Grebel 2016; Brown et al. 2016; Curtiet al. 2017; Barrera-Ballesteros et al. 2017; Teimoorinia et al. 2021).Despite these uncertainties, however, it is clear both that a relation-ship exists, and that it has a characteristic mass scale of ∼ . M (cid:12) at which the curvature of the relation changes (Blanc et al. 2019).Not surprisingly, there have been numerous attempts to explain theserelations theoretically, and it is interesting to put our model in thecontext of these works. However, we caution that what follows isonly a partial discussion of the (vast) literature on this topic, andrefer readers to the comprehensive review by Maiolino & Mannucci(2019, Section 5.1).The basic result from theoretical models to date is that galaxiestend to approach equilibrium between inflows, accretion, star forma-tion and outflows, which naturally gives rise to the observed MZR(Finlator & Davé 2008; Davé et al. 2012; Lilly et al. 2013; Forbeset al. 2014). Our results are broadly consistent with this picture. How-ever, there are some subtle differences among published models, andbetween existing models and ours. One important point of distinction Recall that each of these terms is expressed as the relative importance of aparticular process compared to metal diffusion; thus,
P ∝ 𝑣 𝜙 does not meanthat advection is equally rapid in all galaxies independent of stellar mass, justthat the ratio of advection to diffusion does not explicitly depend on stellarmass. is the extent to which outflows are metal-enriched relative to the ISM( i.e., 𝜙 𝑦 < i.e., theoutflow metallicity is the same as the ISM metallicity, 𝜙 𝑦 = 𝜙 𝑦 < 𝜙 𝑦 <
1, particularly in low-mass galaxies, is consistent withthe findings of the latter group of investigators. However, many ofthese authors do not study the relative importance of metal-enrichedoutflows for dwarfs versus spirals, which we find to be important.It is also debated whether the MZR really has a curvature at inter-mediate stellar masses, and if it does, whether it simply flattens outor starts to bend. While some simulations do find curvature in theMZR around 10 − . M (cid:12) (e.g., Davé et al. 2017; Torrey et al.2019), others do not (e.g., Torrey et al. 2014; De Rossi et al. 2015;Ma et al. 2016). Our model is consistent with the former, especiallyif we look at the empirical scalings of 𝜙 𝑦 with 𝑀 ★ . Moreover, recentresults also show that the curvature is physical and persists in the dataeven after observational uncertainties are accounted for (Blanc et al.2019). However, the cause behind the curvature is not completelyunderstood, and factors like Active Galactic Nuclei (AGN) feedback(De Rossi et al. 2017), gas recycling (Brook et al. 2014), effective gasfraction (Torrey et al. 2019), chemical saturation in the ISM of mas-sive galaxies (Zahid et al. 2013), and a transition in galaxy regimestogether with metal-enriched outflows as we show in this work canall play a role.In addition to the models above, to which our results are directlycomparable, a number of authors have studied the dependence ofthe MZR on factors not included in our work, like downsizing, time-dependent outflows, variations in star formation efficiencies and IMF,presence of satellites, environmental effects, etc. (e.g., Köppen et al.2007; Cooper et al. 2008; Maiolino et al. 2008; Calura et al. 2009;Spitoni et al. 2010; Bouché et al. 2010; Hughes et al. 2013; Peng &Maiolino 2014; Genel 2016; Wu et al. 2017; Bahé et al. 2017; Lianet al. 2018a,b). However, unlike the current work, most models onlystudy the MZR and not the MZGR, thus it is difficult to reconcilewhether their conclusions hold or are self-consistent with spatially-resolved galaxy properties. We use the same metallicity distributions described in Section 3 tocompute metallicity gradients. To be consistent with the proceduremost commonly used in analysing observations, we obtain the gra-dient by performing a linear fit to log Z from 0 . − . 𝑟 e (e.g.Sánchez et al. 2012, 2014; Sánchez-Menguiano et al. 2016; Poetrod-jojo et al. 2018). Following the discussion on inverted gradients inSharda et al. (2021a, Section 5.2.3) and the uncertainty around thembeing in equilibrium, we restrict the model to produce only flat or To be consistent with observations, we only utilize metallicities till 2 . 𝑟 e to measure the gradients, as opposed to 3 𝑟 e that we use to measure Z .MNRAS , 1–12 (2021) P. Sharda et al. negative gradients for the purposes of studying the MZGR. Figure 2shows the MZGR from our model, again color-coded by the ratio ofadvection to accretion (
P/A ). The top and the bottom panels showthe metallicity gradients in dex kpc − and dex 𝑟 − units, respectively.The spread, as for the MZR, is a result of 𝜙 𝑦 . The transition from theadvection-dominated to the accretion-dominated regime, as in theMZR, is also visible in the MZGR. When the gradients are measuredin dex kpc − , this transition corresponds to the slight curvature in theMZGR that appears around 𝑀 ★ ∼ − . M (cid:12) (top panel inFigure 2). When they are measured in dex 𝑟 − , it corresponds to thesomewhat sharper curvature around the same stellar mass (bottompanel in Figure 2). This finding is strong evidence for the links be-tween the MZR and the MZGR, and also reveals that it is the sameunderlying physical mechanism that controls the shape of both.While the stellar mass of the accretion-advection transition influ-ences the location at which our model curves bend, it is not the onlyfactor that does so. The precise location of the bend is also sensitiveto parameters like Z CGM and 𝜙 𝑦 , and both of the MZGR bend andthe mass where P/A = 𝑀 ★ we se-lect for smoothly interpolating between the dwarf and spiral regimes:for example, if we lower the threshold for spirals from 10 . M (cid:12) to10 M (cid:12) , both shift to lower stellar mass. Similarly, if we increase thethreshold for dwarfs from 10 M (cid:12) to 10 . M (cid:12) , both shift to higherstellar mass. However, irrespective of the interpolation limits in 𝑀 ★ ,both the curvature of the MZGR and the transition from P > A to P < A are always present. The existence of these features is a robustprediction of the model independent of uncertain parameter choices.The physical origin for the behaviour of the MZGR is also thesame as for the MZR: gradients are at their steepest when both of theprocesses for smoothing them – accretion, A , and inward advectionof gas, P , are at their weakest compared to metal production, S .Diffusion also helps smooth gradients, but is always subdominantcompared to either accretion or advection, as evidenced by the factthat we never have P < A < P > A , to accretion-dominated, P < A . We emphasise that, while the exact stellar massat which this transition occurs can be somewhat sensitive to choicesof model parameters (for example, the Toomre 𝑄 of galactic discs),its existence is not; the bends in the coloured bands in Figure 2that describe our model always occur irrespective of our parameterchoices. Additionally, note that the minimum of the model MZGR isnot always coincident with P/A =
1; the position of the minimumis dependent on the model parameters, in particular, 𝜙 𝑦 .In Figure 2 we also plot MZGRs from the MaNGA (Belfiore et al.2017), CALIFA (Sánchez et al. 2014; Sánchez-Menguiano et al.2016) and SAMI (Poetrodjojo et al. 2018, 2021) surveys, homoge-nized and corrected for spatial resolution by Acharyya et al. (2021).We adopt the dex kpc − values from Acharyya et al. (2021), and con-vert to dex 𝑟 − following the 𝑟 e – 𝑀 ★ scaling relations from van derWel et al. (2014) to be consistent with our assumptions elsewhere .We also overplot results from MaNGA based on three different metal-licity calibrations by Mingozzi et al. (2020): Pettini & Pagel (2004,PP04), Maiolino et al. (2008, M08), and Blanc et al. (2015, IZI).The first thing to notice is that the qualitative trend found in the The qualitative trend of the MZGR remains the same for the dex 𝑟 − gra-dients reported by Acharyya et al. (2021) as compared to the ones shown inthe bottom panel of Figure 2 using the scaling relation between 𝑟 e – 𝑀 ★ , witha change in the overall normalisation of the metallicity. We have also verifiedthat the 𝑟 e we find from van der Wel et al. (2014) is in very good agreementwith that measured in, for example, the SAMI sample we use. data is in good agreement with that predicted by our model: gradientsare steepest at 𝑀 ★ ∼ − . M (cid:12) , and flatten at both lower andhigher masses. However, the location of the curvature in the data andthe model differ by as much as 0 . − 𝑀 ★ , ourconstant adopted value of Z CGM , and the scaling of 𝜙 𝑦 with 𝑀 ★ ).Moreover, it is important to recall that the data themselves are notfully secure, due to uncertainties caused by the choice of metallicitydiagnostic; Poetrodjojo et al. (2021, their Figure 11) show that theexact mass at which the MZGR bends depends on which diagnosticis used to determine the metallicity, and that these variations arereduced but still persist even after the diagnostics are homogenised.Thus, it is presently difficult to accurately determine the location ofthe curvature, especially given its mildness. Nonetheless, the pres-ence of a bend seems to be robust in the data, as it is in our model.Second, we see that similar to the MZR, this comparison of themodel to the observed MZGR reveals that low-mass galaxies preferlow 𝜙 𝑦 . However, the spread due to 𝜙 𝑦 in the MZGR at the high-mass end is quite narrow; thus, gradients in massive galaxies arenot particularly sensitive to 𝜙 𝑦 , although the data suggests higher 𝜙 𝑦 for the MZGR in massive galaxies (note the inverted arrowsfor 𝜙 𝑦 on Figure 2 as compared to Figure 1). Our findings on 𝜙 𝑦 being ineffective at setting gradients in massive galaxies is consistentwith earlier works (e.g., Fu et al. 2013). However, our proposedexplanation for the flattening of gradients in massive galaxies basedon the advection-to-accretion transition differs from these studies thatattributed the observed flattening to saturation of ISM metallicities(Phillipps & Edmunds 1991; Mollá et al. 2017), radially-varying starformation efficiency (Belfiore et al. 2019), or past mergers (Rupkeet al. 2010; Perez et al. 2011; Fu et al. 2013).In Figure 2, we also plot model predictions using the two scalingsof 𝜙 𝑦 with 𝑀 ★ that we described in Section 3. These scalings areable to reproduce the high mass end of the MZGR, and yield aqualitative trend similar to that seen in the data, but quantitatively thepredicted gradients from the scalings are steeper than that observedat the low mass end. In retrospect, this is not entirely unexpectedgiven the uncertainties in the two approaches, and the fact that thesescalings are sensitive to the absolute metallicity (see Appendix A).Judging from Figure 2, we slightly prefer scaling 2, since it is closerto the observations at intermediate stellar masses; we revisit thecomparison between the two scalings in Section 5. Nevertheless, thefact that both the MZR and the MZGR suggest a qualitatively similarscaling between 𝜙 𝑦 and 𝑀 ★ is an encouraging sign of consistency.However, it is difficult to derive quantitative similarities given theuncertainties in these empirical scalings. Only a handful of models exist in the literature that focus on gasphase metallicity gradients rather than global metallicities (Mott et al.2013; Jones et al. 2013; Ho et al. 2015; Carton et al. 2015; Kudritzkiet al. 2015; Pezzulli & Fraternali 2016; Schönrich & McMillan 2017;Kang et al. 2021), and even fewer that actually study the local MZGRor its equivalent (Lian et al. 2018b, 2019; Belfiore et al. 2019). Ofthese, the models by Lian et al. (2018b) and Belfiore et al. (2019)are closest in spirit to ours. Quantitative comparison between ourresults and those of Lian et al. is challenging, because they do not Lian et al. (2019) focus only on low-mass satellites, so our results are noteasily comparable.MNRAS000
1; the position of the minimumis dependent on the model parameters, in particular, 𝜙 𝑦 .In Figure 2 we also plot MZGRs from the MaNGA (Belfiore et al.2017), CALIFA (Sánchez et al. 2014; Sánchez-Menguiano et al.2016) and SAMI (Poetrodjojo et al. 2018, 2021) surveys, homoge-nized and corrected for spatial resolution by Acharyya et al. (2021).We adopt the dex kpc − values from Acharyya et al. (2021), and con-vert to dex 𝑟 − following the 𝑟 e – 𝑀 ★ scaling relations from van derWel et al. (2014) to be consistent with our assumptions elsewhere .We also overplot results from MaNGA based on three different metal-licity calibrations by Mingozzi et al. (2020): Pettini & Pagel (2004,PP04), Maiolino et al. (2008, M08), and Blanc et al. (2015, IZI).The first thing to notice is that the qualitative trend found in the The qualitative trend of the MZGR remains the same for the dex 𝑟 − gra-dients reported by Acharyya et al. (2021) as compared to the ones shown inthe bottom panel of Figure 2 using the scaling relation between 𝑟 e – 𝑀 ★ , witha change in the overall normalisation of the metallicity. We have also verifiedthat the 𝑟 e we find from van der Wel et al. (2014) is in very good agreementwith that measured in, for example, the SAMI sample we use. data is in good agreement with that predicted by our model: gradientsare steepest at 𝑀 ★ ∼ − . M (cid:12) , and flatten at both lower andhigher masses. However, the location of the curvature in the data andthe model differ by as much as 0 . − 𝑀 ★ , ourconstant adopted value of Z CGM , and the scaling of 𝜙 𝑦 with 𝑀 ★ ).Moreover, it is important to recall that the data themselves are notfully secure, due to uncertainties caused by the choice of metallicitydiagnostic; Poetrodjojo et al. (2021, their Figure 11) show that theexact mass at which the MZGR bends depends on which diagnosticis used to determine the metallicity, and that these variations arereduced but still persist even after the diagnostics are homogenised.Thus, it is presently difficult to accurately determine the location ofthe curvature, especially given its mildness. Nonetheless, the pres-ence of a bend seems to be robust in the data, as it is in our model.Second, we see that similar to the MZR, this comparison of themodel to the observed MZGR reveals that low-mass galaxies preferlow 𝜙 𝑦 . However, the spread due to 𝜙 𝑦 in the MZGR at the high-mass end is quite narrow; thus, gradients in massive galaxies arenot particularly sensitive to 𝜙 𝑦 , although the data suggests higher 𝜙 𝑦 for the MZGR in massive galaxies (note the inverted arrowsfor 𝜙 𝑦 on Figure 2 as compared to Figure 1). Our findings on 𝜙 𝑦 being ineffective at setting gradients in massive galaxies is consistentwith earlier works (e.g., Fu et al. 2013). However, our proposedexplanation for the flattening of gradients in massive galaxies basedon the advection-to-accretion transition differs from these studies thatattributed the observed flattening to saturation of ISM metallicities(Phillipps & Edmunds 1991; Mollá et al. 2017), radially-varying starformation efficiency (Belfiore et al. 2019), or past mergers (Rupkeet al. 2010; Perez et al. 2011; Fu et al. 2013).In Figure 2, we also plot model predictions using the two scalingsof 𝜙 𝑦 with 𝑀 ★ that we described in Section 3. These scalings areable to reproduce the high mass end of the MZGR, and yield aqualitative trend similar to that seen in the data, but quantitatively thepredicted gradients from the scalings are steeper than that observedat the low mass end. In retrospect, this is not entirely unexpectedgiven the uncertainties in the two approaches, and the fact that thesescalings are sensitive to the absolute metallicity (see Appendix A).Judging from Figure 2, we slightly prefer scaling 2, since it is closerto the observations at intermediate stellar masses; we revisit thecomparison between the two scalings in Section 5. Nevertheless, thefact that both the MZR and the MZGR suggest a qualitatively similarscaling between 𝜙 𝑦 and 𝑀 ★ is an encouraging sign of consistency.However, it is difficult to derive quantitative similarities given theuncertainties in these empirical scalings. Only a handful of models exist in the literature that focus on gasphase metallicity gradients rather than global metallicities (Mott et al.2013; Jones et al. 2013; Ho et al. 2015; Carton et al. 2015; Kudritzkiet al. 2015; Pezzulli & Fraternali 2016; Schönrich & McMillan 2017;Kang et al. 2021), and even fewer that actually study the local MZGRor its equivalent (Lian et al. 2018b, 2019; Belfiore et al. 2019). Ofthese, the models by Lian et al. (2018b) and Belfiore et al. (2019)are closest in spirit to ours. Quantitative comparison between ourresults and those of Lian et al. is challenging, because they do not Lian et al. (2019) focus only on low-mass satellites, so our results are noteasily comparable.MNRAS000 , 1–12 (2021) as phase metallicity relations log ( M /M ) ( l og ) [ d e xkp c ] y = 0.1 y = 1.0 MZGR y scaling 1 y scaling 2MaNGACALIFASAMIMaNGA+CALIFA+SAMI log ( M /M ) ( l og ) [ d e x r e ] y = 0.1 y = 1.0 Mingozzi+2020[PP04]Mingozzi+2020[IZI]Mingozzi+2020[M08] l og / l og / Figure 2.
The mass–metallicity gradient relation (MZGR) for the local Universe. The coloured band shows model predictions for different yield reductionfactors, 𝜙 𝑦 (note the opposite direction of the arrow as compared to Figure 1), color-coded by the ratio of the Péclet number ( P ) to cosmic accretion overdiffusion ( A ) in galaxies. The data to which we compare this model (orange points) are taken from a homogeneous analysis of metallicity gradients from theSAMI (Poetrodjojo et al. 2021), MaNGA (Belfiore et al. 2017) and CALIFA (Sánchez et al. 2014) surveys, corrected for spatial resolution by Acharyya et al.(2021). To give a sense of the systematic uncertainty, grey markers denote gradients measured with different metallicity calibrations (Pettini & Pagel 2004,PP04, Maiolino et al. 2008, M08, and Blanc et al. 2015, IZI) for the MaNGA survey by Mingozzi et al. (2020). Finally, we show model predictions with twopossible empirical scalings of 𝜙 𝑦 with 𝑀 ★ (white markers); these scalings are the same as in Figure 1. The important conclusion from this plot is that metallicitygradients in local galaxies transition from the advection-dominated regime ( P > A ) to the accretion-dominated regime ( P < A ) as the stellar mass increases,and it is this transition that drives the shape of the MZGR. Note that the range in stellar mass covered by this figure is different than that shown in Figure 1, dueto differences in the mass ranges covered by the available observations. quote measurements in dex / kpc or equivalent. Examining their plots,it seems that they also find slightly steeper gradients for intermediatemass galaxies, consistent with our findings. Similarly, Belfiore et al.find that observed gradients in local dwarfs and spirals are bestreproduced by a model where the star formation timescale at eachradius is proportional to the local orbital period. For massive galaxies,this scaling is quite similar to that in the Krumholz et al. (2018)galaxy model that is embedded in our metallicity model, and thus at first glance is also consistent with our findings. However, thereremain substantial differences between our model and those of Lianet al. and Belfiore et al.. Neither of these studies include the effectsof radial inflow or metal diffusion. Neither adopt our approach ofsystematically varying the highly-uncertain yield reduction factor 𝜙 𝑦 : Lian et al. adopt a parameterised, time-dependent functionalform that they tune in order to match stellar and gas metallicitygradient data, while Belfiore et al. assume that the ISM and outflow MNRAS , 1–12 (2021)
P. Sharda et al. metallicities are equal ( 𝜙 𝑦 = 𝜙 𝑦 . Thus, it is unclearto what extent the agreement between the models is simply a matterof their being enough adjustable parameters to make them behavesimilarly.In addition to analytic models, semi-analytic models like L-Galaxies 2020 have also investigated the local MZGR, finding some-what flatter gradients for massive galaxies as compared to low massgalaxies (Yates et al. 2020). The authors attribute their findings toinside-out star formation that increases the gas phase metallicity inthe inner disc in massive galaxies. In the outer disc in these galax-ies, Yates et al. either find metal-rich accretion from the CGM thatenhances the metallicity (their ‘modified’ model), or metal-poor ac-cretion that dilutes the metallicity at every radius (their ‘default’model). The combined effect is to produce flatter metallicity pro-files in massive galaxies in each case. They further conclude thatflattening of the metallicity profiles in massive local galaxies is ex-pected regardless of the mass-loading factors of outflows. Thus, theirexplanations for the trends seen in the local MZGR are consistentwith the findings of our model. It is worth noting that while workingwith an earlier version of L-Galaxies, Fu et al. (2013) found relativemetal enrichment of outflows to be more important than advectionin driving gas phase metallicities. These authors also find a trend inthe MZGR consistent with Yates et al. and ours.It is also helpful to compare our results to simulations that havestudied the local MZGR. For example, both Tissera et al. (2016)and Ma et al. (2017) find slightly flatter gradients for massive localgalaxies in their simulations, consistent with our model and availableobservations. The EAGLE simulations (Schaye et al. 2015) find thatmetallicity gradients in their simulated galaxies are systematicallyshallower at 𝑧 = In this section, we introduce a new way of looking at galaxy metallic-ities, by studying the MZR − MZGR correlation space. The two-foldmotivation behind this is to: (1.) understand how global metallicitiescorrelate with metallicity gradients in galaxies, because this can in-form us about the correlations between global and internal dynamicsof galaxies, and (2.) given that both the MZR and the MZGR requiresimilar scaling of 𝜙 𝑦 with 𝑀 ★ to reproduce the observations, we canstudy the relative importance of 𝜙 𝑦 for both of these relations. Anadditional advantage of studying this parameter space is that it canbe constructed both in observations and simulations. In order to construct the MZR–MZGR correlation space in themodel, we simply plot ∇( log Z) from Figure 2 as a function of Z from Figure 1. We show this in the left panel of Figure 3, wherewe color-code the model points by 𝑀 ★ , with different curves cor-responding to different 𝜙 𝑦 . Note that the range in 𝑀 ★ is slightlydifferent in this plot as compared to that in Figure 1 and Figure 2;thus, there are some differences visible in this plot as compared toprevious figures. It is clear from this plot that 𝜙 𝑦 has two distincteffects. At the high-mass end, it simply shifts the overall metallic-ity − Z ∝ 𝜙 𝑦 − without significantly affecting the gradient. At thelow-mass end, it affects the overall metallicity, but also affects thegradient, by making it steeper for larger 𝜙 𝑦 . It is also clear that therelationship between Z and ∇( log Z) is non-monotonic becauseof the same P/A split we have seen in the MZR and the MZGR, i.e., there are two typical branches where Z and ∇( log Z) changemonotonically with respect to one another, but the curves bend whengalaxies transition from the advection-dominated to the accretion-dominated regime. Irrespective of the value of 𝜙 𝑦 , this bend alwaysoccurs around 10 − . M (cid:12) because it is dictated by the ratio P/A crossing unity. To demonstrate the robustness of this feature, we alsooverplot results for the two empirical scalings of 𝜙 𝑦 with 𝑀 ★ that wediscussed in previous sections. We see that both empirical scalingsalso produce a bend in the Z − ∇( log Z) plane, but with rather dif-ferent amounts of curvature. Thus, a generic prediction of our modelis that galaxies should lie along a bent track in Z − ∇( log Z) space, with one arm closer to vertical and one closer to horizontal,but we cannot predict the exact shape of this track without a betterunderstanding of how 𝜙 𝑦 varies with 𝑀 ★ . The trends in the modelwe identify in the MZR-MZGR space remain qualitatively the samewhen the gradients are plotted in units of dex 𝑟 − , so we do notdiscuss them separately.We create a parameter space similar to that above by plotting themeasured metallicity gradients as a function of the measured gasphase metallicity at 𝑟 e from Acharyya et al. (2021) . We show this inthe right panel of Figure 3, color-coded with 𝑀 ★ . The main takeawayfrom this figure is that the data shows a qualitatively similar bendat 𝑀 ★ ∼ . M (cid:12) as the model. While this is not a one-to-onecomparison between the model and the data given the former usesglobal metallicity whereas the latter uses metallicity at a specificlocation in the disc, we expect the qualitative trend ( i.e., the presenceof the bend) to be robust given the findings in the previous sections.Similar to our observations in Section 4, we find that scaling 2 betterreproduces the trend seen in the data. Further, like the model, the sametrends in the data are also present when the gradients are plotted inunits of dex 𝑟 − . Thus, the model is able to identify and recover thepresence of this bend in the metallicity–metallicity gradient space,and sets clear predictions for future work that will enable us to re-construct this space and facilitate a direct comparison with the model.Hence, in addition to our findings in Section 3 and Section 4, weconclude that metal-enriched outflows play a crucial role in settingboth the MZR and the MZGR for low-mass galaxies, while for high-mass galaxies, outflows play a significant role only for the MZR. The conversion from metallicity at 𝑟 e to mean metallicity is non-trivial andsuffers considerable calibration uncertainties, both in the observations and inthe model (which does not use 𝑟 𝑒 as a parameter or make an independentprediction of its location in the disc), which is why we do not attempt to createan MZR from the same observations for which we have the MZGR to directlystudy the MZR–MZGR space.MNRAS000
P. Sharda et al. metallicities are equal ( 𝜙 𝑦 = 𝜙 𝑦 . Thus, it is unclearto what extent the agreement between the models is simply a matterof their being enough adjustable parameters to make them behavesimilarly.In addition to analytic models, semi-analytic models like L-Galaxies 2020 have also investigated the local MZGR, finding some-what flatter gradients for massive galaxies as compared to low massgalaxies (Yates et al. 2020). The authors attribute their findings toinside-out star formation that increases the gas phase metallicity inthe inner disc in massive galaxies. In the outer disc in these galax-ies, Yates et al. either find metal-rich accretion from the CGM thatenhances the metallicity (their ‘modified’ model), or metal-poor ac-cretion that dilutes the metallicity at every radius (their ‘default’model). The combined effect is to produce flatter metallicity pro-files in massive galaxies in each case. They further conclude thatflattening of the metallicity profiles in massive local galaxies is ex-pected regardless of the mass-loading factors of outflows. Thus, theirexplanations for the trends seen in the local MZGR are consistentwith the findings of our model. It is worth noting that while workingwith an earlier version of L-Galaxies, Fu et al. (2013) found relativemetal enrichment of outflows to be more important than advectionin driving gas phase metallicities. These authors also find a trend inthe MZGR consistent with Yates et al. and ours.It is also helpful to compare our results to simulations that havestudied the local MZGR. For example, both Tissera et al. (2016)and Ma et al. (2017) find slightly flatter gradients for massive localgalaxies in their simulations, consistent with our model and availableobservations. The EAGLE simulations (Schaye et al. 2015) find thatmetallicity gradients in their simulated galaxies are systematicallyshallower at 𝑧 = In this section, we introduce a new way of looking at galaxy metallic-ities, by studying the MZR − MZGR correlation space. The two-foldmotivation behind this is to: (1.) understand how global metallicitiescorrelate with metallicity gradients in galaxies, because this can in-form us about the correlations between global and internal dynamicsof galaxies, and (2.) given that both the MZR and the MZGR requiresimilar scaling of 𝜙 𝑦 with 𝑀 ★ to reproduce the observations, we canstudy the relative importance of 𝜙 𝑦 for both of these relations. Anadditional advantage of studying this parameter space is that it canbe constructed both in observations and simulations. In order to construct the MZR–MZGR correlation space in themodel, we simply plot ∇( log Z) from Figure 2 as a function of Z from Figure 1. We show this in the left panel of Figure 3, wherewe color-code the model points by 𝑀 ★ , with different curves cor-responding to different 𝜙 𝑦 . Note that the range in 𝑀 ★ is slightlydifferent in this plot as compared to that in Figure 1 and Figure 2;thus, there are some differences visible in this plot as compared toprevious figures. It is clear from this plot that 𝜙 𝑦 has two distincteffects. At the high-mass end, it simply shifts the overall metallic-ity − Z ∝ 𝜙 𝑦 − without significantly affecting the gradient. At thelow-mass end, it affects the overall metallicity, but also affects thegradient, by making it steeper for larger 𝜙 𝑦 . It is also clear that therelationship between Z and ∇( log Z) is non-monotonic becauseof the same P/A split we have seen in the MZR and the MZGR, i.e., there are two typical branches where Z and ∇( log Z) changemonotonically with respect to one another, but the curves bend whengalaxies transition from the advection-dominated to the accretion-dominated regime. Irrespective of the value of 𝜙 𝑦 , this bend alwaysoccurs around 10 − . M (cid:12) because it is dictated by the ratio P/A crossing unity. To demonstrate the robustness of this feature, we alsooverplot results for the two empirical scalings of 𝜙 𝑦 with 𝑀 ★ that wediscussed in previous sections. We see that both empirical scalingsalso produce a bend in the Z − ∇( log Z) plane, but with rather dif-ferent amounts of curvature. Thus, a generic prediction of our modelis that galaxies should lie along a bent track in Z − ∇( log Z) space, with one arm closer to vertical and one closer to horizontal,but we cannot predict the exact shape of this track without a betterunderstanding of how 𝜙 𝑦 varies with 𝑀 ★ . The trends in the modelwe identify in the MZR-MZGR space remain qualitatively the samewhen the gradients are plotted in units of dex 𝑟 − , so we do notdiscuss them separately.We create a parameter space similar to that above by plotting themeasured metallicity gradients as a function of the measured gasphase metallicity at 𝑟 e from Acharyya et al. (2021) . We show this inthe right panel of Figure 3, color-coded with 𝑀 ★ . The main takeawayfrom this figure is that the data shows a qualitatively similar bendat 𝑀 ★ ∼ . M (cid:12) as the model. While this is not a one-to-onecomparison between the model and the data given the former usesglobal metallicity whereas the latter uses metallicity at a specificlocation in the disc, we expect the qualitative trend ( i.e., the presenceof the bend) to be robust given the findings in the previous sections.Similar to our observations in Section 4, we find that scaling 2 betterreproduces the trend seen in the data. Further, like the model, the sametrends in the data are also present when the gradients are plotted inunits of dex 𝑟 − . Thus, the model is able to identify and recover thepresence of this bend in the metallicity–metallicity gradient space,and sets clear predictions for future work that will enable us to re-construct this space and facilitate a direct comparison with the model.Hence, in addition to our findings in Section 3 and Section 4, weconclude that metal-enriched outflows play a crucial role in settingboth the MZR and the MZGR for low-mass galaxies, while for high-mass galaxies, outflows play a significant role only for the MZR. The conversion from metallicity at 𝑟 e to mean metallicity is non-trivial andsuffers considerable calibration uncertainties, both in the observations and inthe model (which does not use 𝑟 𝑒 as a parameter or make an independentprediction of its location in the disc), which is why we do not attempt to createan MZR from the same observations for which we have the MZGR to directlystudy the MZR–MZGR space.MNRAS000 , 1–12 (2021) as phase metallicity relations [12 + log (O/H)] ( l og ) [ d e xkp c ] y =0.1 y =0.2 y =0.3 y =0.4 y =0.5 y =0.6 y =0.7 y =0.8 y =0.9 y =1.0 y scaling 1 y scaling 2 y scaling 1 y scaling 2 l og M / M r e [12+log (O/H) at r e ] ( l og ) [ d e xkp c ] CALIFAMaNGASAMI 9.009.259.509.7510.0010.2510.5010.7511.00 l og M / M Figure 3.
Left panel:
MZGR–MZR space from the model for the local Universe, defined by the metallicity gradient (in dex kpc − ) as a function of the global(SFR-weighted) galaxy metallicity (defined as in equation 7). Points are color-coded by stellar mass, and different curves represent the different yield reductionfactor, 𝜙 𝑦 , which describes the metal-enrichment of galactic outflows. Both the MZR and the MZGR predict a scaling of 𝜙 𝑦 with 𝑀 ★ such that low-mass galaxiesprefer low 𝜙 𝑦 , implying that these galaxies lose a higher proportion of the metals they produce to winds, as compared to massive galaxies. Also overlaid are thetwo empirical scalings of 𝜙 𝑦 with 𝑀 ★ that are shown in Figure 1 and Figure 2. The bend seen at intermediate masses corresponds to the advection-to-accretiontransition identified in Figure 1 and Figure 2. The range in 𝑀 ★ covered in this plot is slightly different from that in Figure 1 and Figure 2. Right panel:
Meanmetallicity gradients as a function of metallicity at the effective radius 𝑟 e in the CALIFA, MaNGA and SAMI surveys that we adopt from Acharyya et al. (2021).The observations show a similar bend compared to the predictions of the model in the MZR–MZGR space. Note, however, the differences in the axes rangesbetween this panel and the left panel, reflecting the difficulty of putting metallicity measurements at specific radius ( 𝑟 e ) and “global” metallicities on a commonscale. The trends in the model as well as the data in the MZR-MZGR space remain qualitatively similar when the gradients are plotted in units of dex 𝑟 − insteadof dex kpc − . In this work, we present a physical explanation for the observedrelation between the stellar mass and the gas phase metallicity gra-dient (MZGR) for galaxies in the local Universe, using the recently-developed first-principles model of gas phase metallicity gradients ingalaxies given by Sharda et al. (2021a). We show that the shape of theMZGR is driven by the balance between metal advection and produc-tion for low-mass galaxies, and between cosmic accretion and metalproduction for massive galaxies. The point where the MZGR beginsto curve as the galaxy mass increases corresponds to the transition ofgalaxies from the advection-dominated to the accretion-dominatedregime. Additionally, the best match between the model and thedata naturally recovers the expected dependence of the MZGR onmetal-enrichment of galactic outflows: low-mass galaxies have moremetal-rich winds as compared to massive galaxies, implying thatmetals in low mass galaxies are not well-mixed with the ISM beforeejection. This is in good agreement with observations (Martin et al.2002; Chisholm et al. 2018) and simulations (Emerick et al. 2018;Christensen et al. 2018; Tanner 2020).We also present the first joint explanation for the mass-metallicityrelation (MZR) and the MZGR. We find that in addition to the modelsuccessfully reproducing both the MZR and the MZGR, it has twoprimary commonalities: (1.) the curvature observed in both the MZRand the MZGR around a stellar mass 𝑀 ★ ≈ − . M (cid:12) have thesame underlying cause, which is the shift between radial advection(in low-mass galaxies) and cosmological accretion (in more massivegalaxies) as the dominant agent supplying metal-poor gas to galaxycentres, and (2.) both the MZR and the MZGR produced by themodel predict that supernova-produced metals in low-mass galaxiesare largely ejected before mixing with the ISM, while metals in high-mass galaxies are well-mixed with the ISM. The fact that the MZRand MZGR results are qualitatively consistent with each other isevidence for the links between global and spatially-resolved galaxy properties, though our ability to check this quantitatively is currentlylimited by the large uncertainties in observed metallicites.In studying these relations, we also introduce a new way of char-acterizing gas phase metallicities via the MZR–MZGR correlationspace. We find that the relation between the global metallicity andmetallicity gradient in galaxies is non-monotonic, and bends as aresult of the advection-to-accretion transition identified above. Wealso retrieve this bend in the available data (in metallicity gradient–metallicity at 𝑟 e space), although limitations due to the mismatchbetween model and data techniques prevent us from constructingthe observed MZR–MZGR space exactly as we do for the model.Moreover, the MZR–MZGR space also disentangles the relative im-portance of metal-enriched outflows for the global metallicities andmetallicity gradients: while metal-enrichment of the outflows signifi-cantly influences both the global metallicity and metallicity gradientsin low-mass galaxies, in massive galaxies only the absolute metal-licity is sensitive to the properties of the outflows, and gradients areflat regardless of outflow metallicity. ACKNOWLEDGEMENTS
We thank the anonymous reviewer for their feedback, which helpedto improve the paper. We also thank Lisa Kewley for going through apreprint of this paper and providing comments, and Roland Crockerfor useful discussions. PS is supported by the Australian Govern-ment Research Training Program (RTP) Scholarship. MRK and CFacknowledge funding provided by the Australian Research Coun-cil (ARC) through Discovery Projects DP190101258 (MRK) andDP170100603 (CF) and Future Fellowships FT180100375 (MRK)and FT180100495 (CF). MRK is also the recipient of an Alexandervon Humboldt award. PS, EW and AA acknowledge the support ofthe ARC Centre of Excellence for All Sky Astrophysics in 3 Di-mensions (ASTRO 3D), through project number CE170100013. CF
MNRAS , 1–12 (2021) P. Sharda et al. further acknowledges an Australia-Germany Joint Research Cooper-ation Scheme grant (UA-DAAD). JCF is supported by the FlatironInstitute through the Simons Foundation. Analysis was performed us-ing numpy (Oliphant 2006; Harris et al. 2020) and scipy (Virtanenet al. 2020); plots were created using
Matplotlib (Hunter 2007).This research has made extensive use of NASA’s Astrophysics DataSystem (ADS) Bibliographic Services. The ADS is a digital libraryportal for researchers in astronomy and physics, operated by theSmithsonian Astrophysical Observatory (SAO) under a NASA grant.This research has also made extensive use of
Wolfram|Alpha and
Mathematica for numerical analyses, and the image-to-data tool
WebPlotDigitizer . DATA AVAILABILITY STATEMENT
No data were generated for this work.
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APPENDIX A: SCALING OF THE YIELD REDUCTIONFACTOR WITH STELLAR MASS
Comparing our analytic model with observations of the MZR andthe MZGR discussed in the main text suggests a scaling of the yieldreduction factor 𝜙 𝑦 with stellar mass 𝑀 ★ . In this appendix, we exploreways to directly retrieve this scaling using two different methods. Thetwo scalings introduced below capture the qualitative essence of how 𝜙 𝑦 should scale with 𝑀 ★ , albeit with significant uncertainties.(i) Scaling 1:
We make use of the observations reported inChisholm et al. (2018) to derive a scaling of 𝜙 𝑦 with 𝑀 ★ . The authorsreport on the ratio of the wind to the ISM metallicity, Z 𝑤 /Z , as wellas the metal mass loading factor for galaxies of different masses. Weperform a linear fit to their data to obtain Z 𝑤 /Z as a function of 𝑀 ★ . To find how the metal mass loading factor varies as a functionof 𝑀 ★ , we use the scaling provided by Denicoló et al. (2002) whichprovides the best fit to the data. Then, we use Z 𝑤 /Z and the metalmass loading factor to find the mass loading factor 𝜇 as a functionof 𝑀 ★ . Using Z 𝑤 /Z and 𝜇 , it is straightforward to compute 𝜙 𝑦 (Sharda et al. 2021a, equations 10 and 13) 𝜙 𝑦 = − 𝜇 Z 𝑦 (cid:18) Z 𝑤 Z − (cid:19) , (A1)where 𝑦 is the yield of metals from core collapse supernovae. Beforewe proceed further, it is important to point out the caveats of thisapproach. Firstly, Chisholm et al. only observed 7 galaxies acrossa wide range of 𝑀 ★ ( ∼ − M (cid:12) ), so the coverage in stellarmass is very sparse. Secondly, the ISM metallicities for the galaxiesused in Chisholm et al. are non-homogeneous; for example, someare stellar metallicities and some are gas metallicities. Thirdly, somegalaxies in this dataset are undergoing mergers, and show dilutedmetallicities as compared to isolated galaxies of the same mass.Keeping these caveats in mind, and noting that 𝜙 𝑦 is sensitive to theabsolute value of Z as we see from equation A1, we simply increase MNRAS , 1–12 (2021) P. Sharda et al. log M /M y Scaling 1 [Chisholm et al. 2018]Scaling 2 [Curti et al. 2020]
Figure A1.
Scalings of the yield reduction factor 𝜙 𝑦 with 𝑀 ★ , obtained usingthe two approaches described in Appendix A. Scaling 1 is from observations(Chisholm et al. 2018) whereas scaling 2 is from the best match between themodel MZR and the Curti et al. (2020) MZR. the ISM metallicities quoted in Chisholm et al. by 0.3 dex, which hasthe effect of bringing them into closer alignment with the observedMZR; without this increment, the least and the most massive galaxiesin the sample ( 𝑀 ★ ≈ and 10 . M (cid:12) , respectively) would have ametallicity Z = .
03 and 0 .
5, respectively, placing them well belowthe observed MZR. We do not re-scale the ratio Z 𝑤 /Z because itis not sensitive to the absolute value of Z . With this adjustment, weshow the resulting scaling of 𝜙 𝑦 with 𝑀 ★ in Figure A1. This is ourfirst model scaling.(ii) Scaling 2:
In this approach, we simply find the best matchbetween the model MZRs and the Curti et al. (2020) MZR by eye,where we take the latter to be the representative MZR in the localUniverse. We note that there is no particular reason to prefer the latterMZR over other available MZRs, especially given the uncertaintiesin the absolute normalisation of metallicities. However, for the sakeof developing a scaling of 𝜙 𝑦 with 𝑀 ★ from this approach, we willcontinue with this MZR. We plot the resulting scaling in Figure A1.Interestingly, while the general trend of 𝜙 𝑦 increasing with 𝑀 ★ stillholds, we find an inflection at intermediate masses where 𝜙 𝑦 isthe lowest. However, we do not place great weight on this finding,given the large uncertainties in both the choice of MZR and itsabsolute value. From the standpoint of our model predictions, themain difference between this scaling and our first scaling is that thisscaling gives a shallower trend in 𝜙 𝑦 with 𝑀 ★ , such that 𝜙 𝑦 reachesa minimum value of only ≈ . This paper has been typeset from a TEX/L A TEX file prepared by the authors.MNRAS000