Breaking the degeneracy between gas inflow and outflows with stellar metallicity: Insights on M101
Xiaoyu Kang, Ruixiang Chang, Rolf-Peter Kudritzki, Xiaobo Gong, Fenghui Zhang
aa r X i v : . [ a s t r o - ph . GA ] J a n MNRAS , 1–7 (2020) Preprint 19 January 2021 Compiled using MNRAS L A TEX style file v3.0
Breaking the degeneracy between gas inflow and outflows withstellar metallicity: Insights on M 101
Xiaoyu Kang , , ⋆ , Ruixiang Chang , Rolf-Peter Kudritzki , , Xiaobo Gong , , , and Fenghui Zhang , , Yunnan Observatories, Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming, 650216, P.R. China Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 396 Yangfangwang,Guandu District, Kunming, 650216, P. R. China Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing,100012, P. R. China Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy ofSciences, 80 Nandan Road, Shanghai, 200030, China LMU M ¨u nchen, Universit ¨a tssternwarte, Scheinerstr. 1, 81679 M ¨u nchen, Germany Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI96822, USA University of Chinese Academy of Sciences, Beijing, 100049, P. R. China
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
An analytical chemical evolution model is constructed to investigate the radial distribu-tion of gas-phase and stellar metallicity for star-forming galaxies. By means of the model,the gas-phase and stellar metallicity can be obtained from the stellar-to-gas mass ratio. Boththe gas inflow and outflow processes play an important role in building the final gas-phasemetallicity, and there exists degeneracy e ff ect between the gas inflow and outflow rates forstar-forming galaxies. On the other hand, stellar metallicity is more sensitive to the gas out-flow rate than to the gas inflow rate, and this helps to break the parameter degeneracy forstar-forming galaxies. We apply this analysis method to the nearby disc galaxy M 101 andadopting the classical χ methodology to explore the influence of model parameters on the re-sulted metallicity. It can be found that the combination of gas-phase and stellar metallicity isindeed more e ff ective for constraining the gas inflow and outflow rates. Our results also showthat the model with relatively strong gas outflows but weak gas inflow describes the evolutionof M 101 reasonably well. Key words: galaxies: evolution – galaxies: abundances – galaxies: stellar content – galaxies:individual (M 101) – galaxies: spiral
Chemical evolution modelling is a powerful tool to explore thegalactic formation and evolution. Analytical chemical evolutionmodels, based on simple parametrisation of key physical processes,such as gas accretion, star formation, nucleosynthesis, stellar massreturn, and gas outflows, have been successfully employed to pre-dict the enrichment of the interstellar medium (ISM) and achieveda series of interesting results. In the Milky Way, the closed-boxedchemical evolution model predicts a higher fraction of metal-poorG-dwarf stars in the solar neighborhood than observed (the clas-sical "G-dwarf problem"), suggesting that the Milky Way disc isnot a closed-boxed system and gas inflow is important for galacticevolution (Chang et al. 1999, and references therein). ⋆ E-mail: [email protected]
The average chemical composition of the stars and the ISMcan both provide constraints on the chemical enrichment historyof galaxies. The gas-phase metallicity provides a snapshot of themetal content at a given time, while the mean stellar metallicity re-flects the time-averaged value of the ISM metal content over the starformation history (SFH) of the galaxy. Most analytical chemicalevolution studies in the literature focused on modelling gas-phasemetallicity, since it is easier to measure than stellar metallicity insystems where individual stars cannot be resolved. Early analyt-ical models derived a relation between gas-phase metallicity andthe ratio of stellar to gas mass (Zahid et al. 2014; Yabe et al. 2015;Kudritzki et al. 2015). However, degeneracy among the model pa-rameters (star formation e ffi ciency (SFE, with the definition of theproportion of gas turns into stellar mass in unit time), the gas inflowand outflow rates (their definitions are in Section 2)), leads to se-vere limitations when using the gas-phase metallicity alone to con- © X. Y. Kang et al.
Table 1.
Basic properties of M 101.Property Valuename M 101, NGC 5457RA 14 h m s . + ◦ ′ ′′ Morphology ScdDistance(adopted) 7 . ◦ R . − strain the model (Kudritzki et al. 2015; Belfiore et al. 2016). Betterconstraints can be obtained if both the stellar and the gas-phasemetallicity are used at the same time. Fortunately, as demonstrated,for instance, by Zahid et al. (2017), the stellar metallicity of star-forming galaxies can be obtained from a population synthesis anal-ysis of spectra of integrated stellar populations. Large samples ofgalaxies observed with spatially resolved spectroscopy are becom-ing available through recent integral field unit (IFU) surveys likeCALIFA (Sánchez et al. 2012) and MaNGA (Bundy et al. 2015).Exploiting early data from such integral field spectroscopy sur-veys, Lian et al. (2018a,b) simultaneously explored the gas-phaseand stellar metallicity, and pointed out that the stellar metallicitymay contribute to breaking the degeneracy between parameters inanalytical chemical evolution models.To further investigate the importance of stellar metallicity inchemical evolution studies we take NGC 5457 (M 101) as an exam-ple and focus on its radial metallicity gradient. M 101 is a nearbyface-on Scd galaxy (Freedman et al. 2001), which is known to becurrently experiencing an inflow of high-velocity gas (Sancisi et al.2008), and has likely been recently subjected to interaction events(Waller et al. 1997; Mihos et al. 2012). Its basic observationalproperties are summarized in Table 1, and their corresponding val-ues are taken from Walter et al. (2008). Observations for gas-phasemetallicity of H ii regions along the disc of M 101 have been carriedout since the 1970s (Searle 1971; Smith 1975; McCall et al. 1985;Kennicutt & Garnett 1996; Kennicutt et al. 2003; Bresolin 2007;Li et al. 2013; Croxall et al. 2016; Hu et al. 2018; Esteban et al.2020). Croxall et al. (2016) carried out the most extensive study todate of oxygen abundance in M101, deriving a metallicity gradientwith slope − . ± . − . Lin et al. (2013) derived thestellar metallicity gradient of M101, using spectral energy distri-bution (SED) fitting of ultraviolet, optical and infrared photometry.They found the stellar metallicity gradient to be flatter than that ofH ii regions. However, model explanation of these observed proper-ties is still lacking.The aim of this work is to investigate whether the analyticalmodel introduced in Section 2 can explain the radial distributionsof both gas-phase and stellar metallicity, and whether the stellarmetallicity can help to relieve the degeneracy between gas inflowand outflows. The structure of this paper is as follows. The mainingredients of the model are described in Section 2. The observa-tions are presented in Section 3. Our main results and discussionare shown in Section 4. The last section summarizes our main con-clusions. Similar to our previous work (Chang et al. 1999; Kang et al. 2016,2017), we assume that a star-forming galaxy is gradually built updue to continuous gas inflow. At the same time, outflows of metalenriched gas are also taken into account. We adopt the instanta-neous recycling approximation (IRA) assuming that the gas returnfrom stars to the ISM happens on a short timescale compared withgalactic evolution, and we assume that the gas is well mixed withstellar ejecta. IRA represents a good approximation for oxygen pro-duced by massive stars with short lifetimes. The chemical evolutionof a galaxy is expressed by the classical set of integro-di ff erentialequations from Tinsley (1980):d M g d t = Φ − Ψ − (1 − R )SFR , (1)d M ∗ d t = (1 − R )SFR , (2)d[ Z g · M g ]d t = y (1 − R )SFR − Z g (1 − R )SFR + Z i Φ − Z o Ψ , (3)where M g and M ∗ are the gas mass and the stellar mass of thegalaxy at evolution time t , respectively; Z g is the gas-phase metal-licity of the system. Φ and Ψ are the gas inflow rate and the gasoutflow rate, respectively. SFR is the star formation rate (SFR). R is the return mass fraction and y is the nucleosynthesis yield. Both R and y depend on the adopted stellar initial mass function (IMF) butare only weakly dependent on metallicity and time (Vincenzo et al.2016). The IMF of Kroupa et al. (1993) is adopted in this work,since this IMF is favored in describing the chemical evolution ofthe disc of spirals similar to the Milky Way (Vincenzo et al. 2016).Neither metallicity nor time dependence of R and y will be fur-ther taken into account in our model. The values of R and y aretaken from Table 2 of Vincenzo et al. (2016) corresponding to thestellar yields of Romano et al. (2010). We obtain R = .
289 and y = .
019 from averaging the values of R and y Z over metallicity,respectively. Z i is the inflowing gas-phase metallicity. We adoptedmetal-free inflow, i.e., Z i = Z o is the outflowing gas-phase metal-licity and assumed to have the same metallicity as the ISM, i.e., Z o = Z g (Chang et al. 2010; Kang et al. 2012, 2016, 2017).Our model requires an assumption about matter inflow andoutflows. Many chemical evolution models adopt an inflow rateof gas which follows an exponential law (Matteucci & Francois1989; Hou et al. 2000; Spitoni et al. 2017, and references therein).An alternative, which we adopt for our model, is to assume thatthe gas inflow rate is proportional to the SFR, since the in-flow of gas provides a continuous reservoir for star formation.Physical arguments in support of this assumption can be foundin Matteucci & Chiosi (1983), Recchi et al. (2008), Bouché et al.(2010), Lilly et al. (2013) and Yabe et al. (2015). For the outflowrate we also assume that it is also proportional to the SFR be-cause the larger is the SFR, the larger is the chance of having alarger-scale outflow (Silk 2003). With these assumptions analyticalsolutions of chemical evolution are straightforward (Recchi et al.2008; Spitoni et al. 2010; Kudritzki et al. 2015). We use Φ = ω (1 − R )SFR, Ψ = λ (1 − R )SFR, where ω ( ω ≥
0) and λ ( λ ≥
0) are the gasmass accretion factor and the outflow loading factor, respectively. ω and λ are two free parameters in our model. It should be pointed outthat, under these assumptions, equation (1) combined with a linearstar formation law from Schmidt (1959) (i.e., SFR = ε M gas , ε isthe so-called SFE in units of Gyr − ) will lead to an exponentially MNRAS , 1–7 (2020) hemical evolution of M 101 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 log(M ∗ /M g ) −1.5−1.0−0.50.00.51.0 l og ( Z g / Z ⊙ ) λ=0.0, ω=0.0λ=0.0, ω=0.5λ=0.0, ω=1.0λ=0.5, ω=0.0λ=0.5, ω=0.5λ=0.5, ω=1.0λ=1.0, ω=0.0λ=1.0, ω=0.5λ=1.0, ω=1.0 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 log(M ∗ /M g ) −1.5−1.0−0.50.00.51.0 l og ( Z ∗ / Z ⊙ ) Figure 1.
Metallicity as a function of the stellar-to-gas mass ratio with di ff erent combinations of gas outflow loading factor ( λ ) and gas mass accretion factor( ω ). The lines with di ff erent color represent di ff erent values of λ , while the di ff erent line-types are corresponding to di ff erent values of ω . Gas-phase metallicityand stellar metallicity are displayed in the left panel and in the right panel, respectively. declining SFR, that is, the inflow rate obeys the decaying exponen-tial law, in agreement with the approach in Matteucci & Francois(1989).With the definition α = λ − ω , equation (1) and (3) can be re-written as follows: d M g = − (1 + α ) · d M ∗ , (4)d[ Z g · M g ] = ( y − Z g − pZ g ) · d M ∗ , (5)In this paper, we define the mean stellar metallicity as themass-weighted average stellar metallicity, h Z ∗ i = (1 − R ) R t Z g ( t ′ )SFR( t ′ )d t ′ (1 − R ) R t SFR( t ′ )d t ′ . (6)Integrating equations (4) and (5), and combining with equation (6),we can obtain the analytical solutions of the chemical evolutionusing appropriate initial conditions. In other words, both the gas-phase metallicity and the mean stellar metallicity solutions can beobtained. There are four di ff erent solutions corresponding to fourspecial cases.The first case includes both gas inflow and outflows, i.e., ω , λ ,
0, and α , −
1. The initial conditions are M ∗ , = M g , = M g + (1 + α ) M ∗ , and the solutions are Z g = y ω { − [1 + (1 + α ) M ∗ M g ] − β } , h Z ∗ i = − y ω (1 + α ) { [1 − + α + λ [1 + (1 + α ) M ∗ M g ] − β ] M g M ∗ − ω + λ [ M g M ∗ + (1 + α )] } . (7)where β = ω + α . We should emphasize that the values of λ and ω areconditional. Since the initial gas mass should not be less than zero,i.e., M g , ≥
0, the value of mass accretion factor is constrained tothe range ω ≤ λ + − µ , where µ is the gas fraction and defined as µ = M g M g + M ∗ .The second case is α = −
1. The initial conditions are M ∗ , = M g , = M g = const, and the corresponding solutions are Z g = y ω (1 − e − ω M ∗ M g ) , h Z ∗ i = y ω [1 + M g ω M ∗ ( e − ω M ∗ M g − . (8) The third case is no gas inflow but with gas outflow, i.e., ω = λ ,
0. The initial conditions are M ∗ , = M g , = M g + (1 + λ ) M ∗ , and the solutions are Z g = y + λ ln[1 + (1 + λ ) M ∗ M g ] , h Z ∗ i = y (1 + λ ) { (1 + λ ) − M g M ∗ ln[1 + (1 + λ ) M ∗ M g ] } . (9)The last one is the closed-box model with neither gas inflowsnor outflows, i.e., ω = λ =
0, the initial conditions are M ∗ , = M g , = M g + M ∗ , and the solutions become Z g = y ln[1 + M ∗ M g ] , h Z ∗ i = y [1 − M g M ∗ ln(1 + M ∗ M g )] . (10)The above equations show that, given the two free parameters λ and ω , the gas-phase and stellar metallicity at time t can be cal-culated by the stellar-to-gas mass ratio at that moment. In order toillustrate this point clearly, Figure 1 displays the gas-phase metal-licity (left) and the stellar metallicity (right) as a function of thestellar-to-gas mass ratio with di ff erent combinations of the param-eters λ and ω . The lines with di ff erent colors correspond to di ff er-ent outflow loading factors (i.e., red λ =
0, blue λ = . λ = . ff erent line-types are corresponding to di ff er-ent mass accretion factors (i.e., solid ω =
0, dashed ω = . ω = . λ and ω exists in the stellar-to-gas mass range of log( M ∗ / M g ) = M ∗ / M g ) = .
4, which makes it di ffi cult to disentangle the MNRAS , 1–7 (2020)
X. Y. Kang et al. role of outflow and inflow from the observations of the gas-phasemetallicity only.On the other hand, the stellar component in the right panel ofFigure 1 shows a di ff erent behavior for − . ≤ log( M ∗ / M g ) ≤ . λ , the mean stellar metallic-ity increases with the mass accretion factor ω , because large in-flow factor means large fraction of star forms at late time andthen having high metallicity. Based on the gas-to-stellar relationof Peeples et al. (2014, Eq.(9)) from ∼
260 star-forming galax-ies, the typical range of the stellar-to-gas ratio is about − . ≤ log( M ∗ / M g ) ≤ .
13 for star-forming galaxies with stellar mass9 . ≤ log( M ∗ / M ⊙ ) ≤ .
5. Therefore, the observed stellar metal-licity may help us to overcome the parameter degeneracy and serveas an important observable to constrain the SFHs of star-forminggalaxies. In the following Sections, we choose M 101 as an exampleto demonstrate the importance of stellar metallicity in the chemicalenrichment studies.
As described in Section 2, with our model, spatially resolved gas-phase and stellar metallicities can be calculated from the spatiallyresolved stellar-to-gas mass ratio. In this Section, we summarize theobserved radial distribution of these properties for M 101, includingthe gas-phase metallicity, the stellar metallicity, the mass surfacedensities of ISM neutral and molecular hydrogen and the stellarmass surface density.
The stellar mass surface density ( Σ ∗ ) is derived from infrared (IR)surface photometry obtained with the 2MASS survey (Jarrett et al.2003) using the K -band at 2 . µ m. The K -band surface bright-ness profile from Muñoz-Mateos et al. (2007) and the fixed K -bandmass-to-light ratio, Υ K ∗ = . ⊙ / L ⊙ , K (see Leroy et al. 2008), isadopted to calculate Σ ∗ .The neutral hydrogen gas mass surface density ( Σ HI ) of M 101is obtained from Very Large Array (VLA) maps of the 21-cm hy-drogen line as part of The H i Nearby Galaxy Survey (THINGS;Walter et al. 2008). The molecular hydrogen gas mass surface den-sity ( Σ H ) is derived by CO ( J = −
1) maps carried out by usingthe IRAM 30m as part of the HERA CO-Line Extragalactic Sur-vey (HERACLES; Leroy et al. 2009). A factor of 1.36 has beenincluded to account for the contribution of helium and heavier ele-ments, and the reader is referred to Schruba et al. (2011) for moredetails about the conversion of emission (21-cm and CO) line into Σ HI and Σ H , respectively. The total gas mass surface density ( Σ gas )is defined as Σ gas = Σ HI + Σ H . Since oxygen is the element most commonly measured and takenas a tracer for the total metal content, and because it is an elementfor which the IRA approximation is appropriate, we will use theoxygen abundance to represent the metallicity of M 101 and adoptthe solar value as 12 + log(O / H) ⊙ = .
69 (Asplund et al. 2009)throughout this work.The radial distribution of gas-phase metallicity of H ii re-gions in M 101 have been obtained in several works, no-tably Kennicutt & Garnett (1996), Kennicutt et al. (2003), Bresolin(2007), Li et al. (2013) and Croxall et al. (2016) and Hu et al. (2018). The gas-phase metallicity from Kennicutt & Garnett (1996)and Hu et al. (2018) are calculated by using the theoretical calibra-tion published by Kobulnicky & Kewley (2004, hereafter KK04),while those from Kennicutt et al. (2003), Bresolin (2007), Li et al.(2013) and Croxall et al. (2016) are derived by using the direct T emethods. Since the absolute gas-phase metallicity depends on thecalibrations used, it is crucial to use the same metallicity calibrationwhen using the observations to constrain the model. T e methodis considered as the most reliable approach to determine the gas-phase matallicity (Izotov et al. 2006), and the gas-phase metallicityobtained from the direct T e method is systemically ∼ . T e cal-ibration to constrain the model. We also add the data based on theKK04 but subtract 0.4 dex to account for the systematic e ff ect ofthis calibration.The radial distribution of stellar metallicity for M 101 is de-rived by Lin et al. (2013), who fitted evolutionary population syn-thesis model (Bruzual & Charlot 2003) to a set of multi-band pho-tometry images from ultraviolet, optical and infrared together withthe 15 intermediate-band images observed in the Beijing-Arizona-Taiwan-Connecticut (BATC) filter system. The BATC photometricsystem covers the wavelength range of 3300 − − ff ective wavelengths of the filters and somestatistics of the stacked images. It should be pointed out thatLin et al. (2013) used a fixed metallicity when performing SEDfitting for each pixel, that is, simple stellar populations of di ff er-ent ages have the same metallicity. Thus, the mean stellar metal-licity they obtained inclines to a luminosity-weighted metallic-ity. The luminosity-weighted metallicity cannot be corrected tothe mass-weighted metallicity by using the mass-to-light ratio ina specific band. Moreover, since the simple model we are usingdoes not contain specific information of SFH, we are not able toconvert a mass-weighted metallicity predicted by our model intoa luminosity-weighted metallicity. We have to tolerate that thismay add some uncertainties to our results. Fortunately, Figure 11in Lin et al. (2013) shows that the disc of M 101 is dominated byintermediate-age stellar populations ( ∼ ff erent fromthose used in Kennicutt & Garnett (1996), Kennicutt et al. (2003),Bresolin (2007) and Li et al. (2013). Consequently, we have scaledall the metallicity distributions of M 101 to the distance used bySchruba et al. (2011), and the value of the distance to M 101 istaken from Karachentsev et al. (2004). The left panel of Figure 2plots the observed radial distribution of gas-phase metallicity ofH ii regions in M 101 from di ff erent authors as di ff erent symbols,and the observed radial distribution of stellar metallicity data fromLin et al. (2013) is displayed in the right panel of Figure 2. It shouldbe pointed out that, since the gas mass surface density taken from MNRAS , 1–7 (2020) hemical evolution of M 101 r (kpc) -1.0-0.8-0.6-0.4-0.2-0.00.20.4 l og ( Z g / Z ⊙ ) both, no err Kennicuttetal.(1996)Kennicuttetal.(2003)Bresolinetal.(2007)Lietal.(2013)Croxalletal.(2016)Huetal.(2018)meanvalueineachbin 0 4 8 12 16 r (kpc) -1.0-0.8-0.6-0.4-0.2-0.00.20.4 l og ( (cid:1) Z ∗ (cid:0) / Z ⊙ ) Linetal.(2013)
Figure 2.
Comparison of model predicted metallicity with observations in M 101 (left: gas-phase; right: stars).
Left panel : di ff erent symbols denote the observedgas-phase metallicity from di ff erent authors. The cyan solid and grey open cycles represent the gas-phase metallicity calculated by using the calibration ofKK04, while the blue open diamonds, magenta solid diamonds, the red solid triangles and green open triangles are corresponding to the gas-phase metallicityderived by using the direct Te calibrations. Note that the KK04 calibration data are corrected to remove the discrepancies between Te and KK04 calibrations(see text). The black solid cycles display the mean values of observed gas-phase metallicity in each bin, and the error bars are the standard deviation of data ineach bin. Right panel : the observed stellar metallicity data are shown as black solid cycles. The solid lines in both panels plot the best-fitting model predictionsadopting ( λ = . ,ω = . λ = . ,ω = . λ = . ,ω = . σ confidence level. Schruba et al. (2011) goes only out to r ∼ r ≤ σ confidence level, which will be described in detailin the following Section. As has been described in Sections 2 and 3, with the observed radialdistributions of the stellar mass, gas mass and metallicity in the discof M 101 together with the metallicity yield y , we can determine theoutflow loading factor λ and the mass accretion factor ω of the ana-lytical chemical evolution model presented in Section 2. For a givengalactic-centric radius r , we use the observed stellar-to-gas ratio atthe present-day to calculate gas-phase and stellar metallicity basedon the model solutions in Section 2. In other words, for each param-eter combination of λ and ω , we generate a model library of radialmetallicity profile for both gas-phase and stellar components. Weuse the classical χ methodology to compare the model predictionswith the corresponding observed data. Refering to the definition byPress et al. (1992, Eq.(15.1.5)), we adopt χ = N P i = C model , i − C obs , i ) σ i ,where σ i is the observed error and N is the number of observeddata. The model that minimizes this reduced χ is considered asthe best-fitting model.In order to ensure gas-phase and stellar metallicity have nearlythe same weight, we divide the radial observed gas-phase metallic-ity data introduced above along the disc ( r ≤ σ i = σ , and that the model does fitwell, that is, we adopt the reduced χ , χ ν =
1, to fit the data. Ac-cording to χ ν = χ N − M , where N − M is the number of degrees offreedom for fitting N data points with M parameters, we can get χ = N − M .We separately calculate the values of χ for gas-phase metal-licity ( χ ), stellar metallicity ( χ ∗ ) and both gas-phase and stellarmetallicity ( χ + χ ∗ ). The boundary conditions are adopted to be0 ≤ λ ≤ . ≤ ω ≤ .
5, respectively. χ contour maps aredisplayed in Figure 3. The left, the middle and the right panels sep-arately plot χ , χ ∗ and χ + χ ∗ contours. The minimum values of χ are shown as red filled asterisks in these three panels of Figure 3.The solid lines display the isocontours of ∆ χ = χ − χ = σ (68.3%),2 σ (95.4%) and 3 σ (99.73%) confidence levels. It should be em-phasized that the oblique line traversing across the red asterisk inthe middle panel arises from the constraint condition ω ≤ λ + − µ in equation 7 as described in Section 2.The left panel of Figure 3 indicates the degeneracy betweenparameters λ and ω , that is, the higher inflow rate and the loweroutflow rate show similar χ values as the lower inflow rate and thehigher outflow rate. The physical reason is that gas-phase metal-licity is diluted by the pristine gas inflow at a fixed radius, andthe enriched outflow process takes a fraction of metals away fromthe disc at a fixed radius. Indeed, the large area of χ contours inthe left panel indicates that it is di ffi cult to determine the modelparameters only using the gas-phase metallicity. The correspond-ing values of the best parameter combinations are λ = .
768 and ω = . λ than to ω . The bestparameter combinations are λ = .
456 and ω = . MNRAS , 1–7 (2020)
X. Y. Kang et al. ω λ only gas ω only star ω gas + star Figure 3. χ contour maps for the outflow loading factor λ and the mass accretion ω determinations. Only gas-phase metallicity (left), only stellar metallicity(middle) and both gas-phase and stellar metallicity (right) are used to constrain the model. The red filled asterisk in each panel denotes the minimum value of χ . The ∆ χ values adopted for these plots are 2.3 (blue), 6.17 (cyan) and 11.8 (brown), which are corresponding to 68 . .
4% and 99 .
73% confidentiallevels, respectively. direction of the degeneracies is di ff erent in two cases. In conse-quence, the combined fitting is most e ff ective for the constraint ofthe parameters.After using both gas-phase and stellar metallicity as con-straints, the right panel shows that the reasonable range of modelparameters is significantly reduced. The minimum value of χ canbe found at λ = .
994 and ω = . ω and λ in the left panel is lifted, which implies that stel-lar metallicity may help us to determine the best combination of ω and λ . In other words, the observed stellar metallicity provides anadditional constraint on the chemical enrichment history of M 101.We name the model with λ = . + . − . and ω = . + . − . as the best-fitting model of M 101. The best-fitting model pre-dicted radial profiles of gas-phase and stellar metallicity are respec-tively shown as solid lines in the left and right panels of Figure 2.The grey shaded regions in both panels display the model resultswithin 1 σ confidence level, that is, ( λ = . ,ω = . λ = . ,ω = . Ψ = . + . − . × SFR) and gas inflow rate (
Φ = . + . − . × SFR)may reasonably describe the fundamental physical processes regu-lating the formation and evolution of M 101.Another point we should emphasize is that, the observedmetallicity gradient of the gas-phase component is much steeperthan that of stellar component. In other words, the di ff erence be-tween Z g and h Z ∗ i (hereafter ∆ Z ) decreases with the increase ofradius. Previous studies have shown that there exists a strong cor-relation between ∆ Z and the mean age of stellar populations h t i inthe sense that larger ∆ Z corresponds to older mean stellar age. Forfurther discussion, we refer the reader to Peng et al. (2015) and toFigure 5 of Ma et al. (2016). Our results indicate that the inner discof M 101 has an older stellar population than the outer disc, whichis consistent with the inside-out formation scenario of stellar discs.Furthermore, we should discuss the influence of the stellaryield y on the determination of model parameters. It can be foundfrom equation sets 7, 8, 9 and 10 in Section 2 that the stellar yield y is proportional to the resulted metallicity, thus the adopted y isexpected to largely influence the resulting model parameters. Fig- y/Z ⊙ m o d e l p a r a m e t e r s ( λ o r ω ) λω Figure 4.
The model parameters λ and ω as a function of the yield y / Z ⊙ . Thesolid and dashed lines are corresponding to λ and ω , respectively. The verti-cal red dotted line denotes the value of the yield adopted in this work, whilethe vertical cyan dash-dotted line marks the yield adopted in Kudritzki et al.(2015). ure 4 plots the best-fitting model parameters λ and ω as a functionof the yield y . The striking feature of Figure 4 is that the value ofoutflow parameter λ is very sensitive to the adopted yield y , sincethe main e ff ect of gas outflow process is to take away part of newlysynthesized metals and reduce the metallicity of the ISM. In fact,this is a common di ffi culty in studies of the chemical evolution ofgalaxies. Therefore, the absolute value of λ derived in this paper isnot robust, and we can only estimate the relative probability of λ for given stellar yield.Finally, we note that the best combination of λ and ω forM 101 derived by Kudritzki et al. (2015) is ω = . λ = . Φ =
Ψ = .
98 in units of SFR, who use the gas-phasemetallicity to constrain the model. The smaller values of λ and ω for M 101 in Kudritzki et al. (2015) than ours (see both left andright panels of Figure 3) mainly due to the fact that they adopted asmaller stellar yield than ours. MNRAS , 1–7 (2020) hemical evolution of M 101 In this work, the radial distribution of gas-phase and stellar metal-licity of star-forming galaxies is investigated by means of an analyt-ical chemical evolution model. We find that the gas-phase and stel-lar metallicity can be derived by the ratio of stellar-to-gas mass sur-face densities. Through comparing the gas-phase metallicity withthe stellar metallicity as a function of the stellar-to-gas mass ratioswith models of di ff erent combinations gas inflow and and outflowrates, it is shown that both gas inflow and outflows can reduce thegas-phase metallicity, but there exists degeneracy e ff ect between ω and λ . On the other hand, stellar metallicity is more sensitive to λ than to ω , and this helps to reduce the degeneracy e ff ect. Theanalytical chemical evolution model is applied to the nearby discgalaxy M 101. By means of the classical χ methodology, ω and λ are better determined by simultaneously using gas-phase and stellarmetallicity as the observed constraints, which further indicates thatstellar metallicity is an important additional observable to constrainthe SFH of star-forming galaxies. Our results also show that rela-tively strong gas outflows but weak inflows occurred on the disc ofM 101 during its evolutionary history.Recent IFU surveys provide large samples of data for star-forming galaxies which include spatially resolved information ofobserved stellar mass, gas mass, gas-phase metallicity and stellarmetallicity. This will provide an opportunity for further tests ourmethod. We plan to apply the analytical chemical evolution modelto a large sample of star-forming galaxies to constrain the gas in-flow and outflows during their evolutionary histories in our futurework. ACKNOWLEDGEMENTS
We thank the anonymous referee for thoughtful comments and in-sightful suggestions that greatly improved the quality of this paper.This work is supported by National Key R&D Program of China(No. 2019YFA0405501). Xiaoyu Kang and Fenghui Zhang are sup-ported by the National Natural Science Foundation (NSF) of China(No. 11973081, 11573062, 11403092, 11390374, 11521303), theYIPACAS Foundation (No. 2012048), the Chinese Academy ofSciences (CAS, KJZD-EW-M06-01), the NSF of Yunnan Province(No. 2019FB006) and the Youth Project of Western Light of CAS.Ruixiang Chang is supported by the National NSF of China (No.11373053, 11390373). Rolf Kudritzki acknowledges support bythe Munich Excellence Cluster Origins Funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foundation) un-der the German Excellence Strategy EXC-2094 390783311.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable re-quest to the corresponding author.
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