Galactic Isotopic Decomposition for the Sculptor Dwarf Spheroidal Galaxy
MMNRAS , 1–20 (2021) Preprint 5 February 2021 Compiled using MNRAS L A TEX style file v3.0
The Galactic Isotopic Decomposition for the Sculptor Dwarf SpheroidalGalaxy
Kanishk Pandey, ★ Christopher West , Department of Physics and Astronomy, Carleton College, Northfield, MN, 55057, USA Joint Institute for Nuclear Astrophysics—Center for the Evolution of the Elements, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Stellar evolution models require an initial isotopic abundance set as input, but these abundances have not been thoroughlyestablished outside of our sun. Nucleosynthesis studies require reliable isotopic abundances which are challenging to infer fromelemental observations and are galaxy specific. Despite the challenges, accurate GCE models for dSphs can provide significantinsight on the galactic hierarchy. We present an isotopic history model for the Sculptor dSph galaxy based on astrophysicalprocesses, which is a complementary approach to GCE models. We estimate the isotopic composition of Sculptor’s late stageevolution using OMEGA, and we use BBN as the other boundary condition. Each astrophysical process was assigned a parametricfunction with free parameters according to the underlying physics that dictate their average chemical evolution. The isotopicyields were summed into elemental yields and fit to observational Sculptor abundance data to fix the free parameters. Thisprocedure gives an average isotopic history of Sculptor for massive star, Type 1a SNe, main s -process peak, and r -processcontributions, which can be compared with the chemical history of the MW. We find that Type 1a SNe contribute approximately86 per cent to the late stage evolution Fe abundance, which is greater than typical MW solar values of approximately 70 per cent,and in agreement with other dSph chemical evolution studies. The model also finds that NSMs only contribute approximately30 per cent to the late stage evolution Eu abundance, further suggesting that CCSNe are the more dominant r -process progenitorin dSphs. Key words:
Galaxy: abundances, evolution, dwarf — stars: abundances, massive, supernovae: general
Stellar simulations require an initial isotopic composition, and thisinput is crucial for obtaining the correct nucleosynthesis outputs.When modelling compositions similar to the sun, this is aided by arobust solar isotopic abundance composition. Many studies continueto refine this composition (Lodders 2003; Lodders et al. 2009; Lod-ders 2019; Asplund et al. 2005, 2009; Grevesse et al. 2005, 2007).However, isotopic abundances are more challenging to determine forenvironments outside the solar neighborhood due to the lack of iso-topic data, which cannot be inferred from elemental measurements.Galactic chemical evolution (GCE) models are one way to con-struct an initial isotopic composition input. GCE models typicallyintegrate stellar yields across time-steps to build abundances frombig bang nucleosynthesis (BBN) to the solar composition. In doingso, these models predict average isotopic abundances which can facil-itate our understandings of stellar nucleosynthesis (Prantzos 2008).Traditionally, GCE models have included, for example, an initialmass function (IMF) and star formation rate (SFR) to build isotopicabundances from H to Zn (Timmes et al. 1995). Further models havecalculated the evolution of heavy elemental abundances from C toZn in our solar neighborhood (Kobayashi et al. 2006).Whereas an accurate and complete isotopic decomposition model ★ E-mail: [email protected] is difficult to find for the Milky Way (MW), doing so for dwarfspheroidal galaxies (dSphs) suffers additional challenges. Thesegalaxies have a low luminosity because they are gas-poor systems(Lanfranchi & Matteucci 2003; Grebel et al. 2003) and have anolder stellar population, possibly due to being "fossils" from thepre-reionization era (Ricotti & Gnedin 2005; Theler et al. 2020).Furthermore, it is difficult to understand the star formation history(SFH), gas infall and outflow, and stellar populations of dSphs whichare often vital components in GCE models (Lanfranchi & Matteucci2003; Calura & Menci 2009). Lack of observational stellar data fromdSphs also makes it difficult to assess GCE yields. However, paststudies have compiled partial observational elemental abundancesfor various dSphs (e.g., Shetrone et al. 2001, 2003; Venn et al. 2004;Bonifacio et al. 2004; Sadakane et al. 2004; Monaco et al. 2005;Geisler et al. 2005; Battaglia et al. 2006, 2011; Bosler et al. 2007;Kirby et al. 2008, 2009, 2010, 2018; Aoki et al. 2009; Tafelmeyeret al. 2010; Letarte et al. 2010; Starkenburg et al. 2013; Jablonkaet al. 2015; Simon et al. 2015; Skúladóttir et al. 2017; Mashonkinaet al. 2017; Duggan et al. 2018; Hill et al. 2019).Despite these challenges, attempts have also been made to prop-erly understand the evolution of dSphs over the past two decades.One of the first standardized models for an average dSph was de-vised by Lanfranchi & Matteucci (2003) who developed a chemicalevolution model for 8 dSphs in the Local Group. Other substantialmodels have been proposed (e.g., Recchi et al. 2001; Venn et al. © a r X i v : . [ a s t r o - ph . GA ] F e b Pandey & West
Several astrophysical processes govern the isotopic abundances ofstars and galaxies and are an important tool for establishing an iso-topic abundance decomposition for the Sculptor dSph. Here the iso-topic solar abundance pattern was taken from a new data set byLodders (2019) which gives updated values relative to their 2010publication (Lodders et al. 2009). In the following, we organize theprocesses roughly by the mass range of isotopes to which they con-tribute. More information on the specifics of each astrophysical pro-cess can be found in West & Heger (2013).
In the very early Universe, temperatures of about 10 Kelvin causedall matter, mainly bare protons and neutrons, to remain in thermalequilibrium through weak interactions with neutrinos. However, afterapproximately 3 minutes after the Big Bang, the temperature cooledto about 10 Kelvin, resulting in weak interaction rates that were lessthan the Hubble expansion rate of the Universe (Cyburt et al. 2016).At this “freeze out” temperature, the neutron-to-proton ratio wasabout 1 /
5, and free neutron decays further decreased the value to 1 / → d + 𝛾 , was delayed due to a deuteriumbottleneck (Cyburt et al. 2016). After the deuterium production ratesurpassed the destruction rate, several reactions followed. Almostall the neutrons ended up in He, or Y P , and this resulted in aY P abundance of ∼ Hewere produced at a mass fraction of about 10 − , and Li at about10 − (Cyburt et al. 2016). However, the Li abundance requiresfurther attention due to the discrepancy in its abundance from currentobservations of metal-poor halo stars in the MW and the theoreticalamount that is thought to have existed from BBN, which is knownas the “Lithium Problem.” Before it could produce higher isotopes,BBN was halted due to low temperatures, Coulomb barriers, and themass gap at 𝐴 = = . · − , He = . · − , Li = . · − , and Li = . · − fromCyburt et al. (2016). We also use updated values for Y P = 0.2470from Pitrou et al. (2018). MNRAS , 1–20 (2021) he Galactic Isotopic Decomposition for the Sculptor dSph Most helium was made during BBN, though some is also madethrough hydrostatic hydrogen burning in stars through pp chains andthe CNO cycle (Burbidge et al. 1957). As a result, it is a primaryprocess since it is made directly from hydrogen and deuterium. 𝜈 -process The 𝜈 -process can be split into the "light" 𝜈 -process and "heavy" 𝜈 -process. The "light" 𝜈 -process occurs when neutrinos and lighternuclei interact in core-collapse supernova (CCSNe). The intendednuclei that experience these interactions are CNO isotopes which aremade from hydrogen and helium, such as Li and Be (Heger et al.2005; Prantzos 2010; Kusakabe et al. 2019). As a result, the light 𝜈 -process is considered to be primary. The "heavy" 𝜈 -process differsfrom the "light" 𝜈 -process in that the target nuclei are made fromneutron-capture processes in a prior environment. Since this requirespre-existing metals to proceed, the "heavy" 𝜈 -process is consideredto be secondary, and behaves more like the 𝛾 -process (West & Heger2013). As a result, we have decided to let the 𝜈 -process include onlythe "light" 𝜈 -process, while the "heavy" 𝜈 -process contributes to the 𝛾 -process, similar to West & Heger (2013). Since elements past He cannot be made through proton-protonchains due to the unstable nature of Li and Be (Bertulani & Ka-jino 2016), and since the triple-alpha process produces C and noelements in between He and C (Coc et al. 2012), the LiBeB fam-ily of elements cannot be produced in stellar interiors, but in morecold and dilute environments (Viola 2000). Galactic Cosmic Ray(GCR) Spallation produces much of the LiBeB abundances (Viola2000) and occurs when highly energetic particles, such as protons or 𝛼 -particles, collide with CNO nuclei. It is responsible for the produc-tion of some light elements, such as Li, Be, B, and B (West &Heger 2013). While GCR spallation requires previous CNO nucleito occur, the process can be considered both secondary and primaryas it has been suggested that accelerated winds of rotating massivestars can be sufficient in CNO nuclei, and these nuclei can then inter-act with particles by GCR Spallation in a primary nature (Prantzos2010, 2012). As a result, we consider two types of GCR spallationin this paper: standard GCR spallation that is secondary, as well asprimary GCR that occurs in primary events due to accelerated windsin massive stars.
Classical novae are formed from binary star systems between a whitedwarf and a main sequence star. The white dwarf accretes mate-rial from the companion star, forming a layer on the surface of thewhite dwarf. When the pressure and temperature is large enough tostart nuclear fusion reactions, the hydrogen is converted into heavierelements which leads to a thermonuclear runaway (Starrfield et al.2016). Classical novae are considered primary processes becausethey synthesize hydrogen, helium, and primary CNO nuclei from thecompanion star (West & Heger 2013). CO novae produce Li, C, N, O, and F, whereas ONeMg novae produce heavier isotopesup until K (José & Hernanz 2007). However, the contributionsof classical novae to these heavier elements are likely negligible compared to the contributions from massive stars, so we disregardcontributions from classical novae for isotopes above Li.
Low and intermediate-mass stars are rich in C and N isotopes; how-ever, contributions from massive stars are difficult to separate fromlow and intermediate-mass stars (West & Heger 2013). As a result,we combine low and intermediate-mass stars and massive stars intoone category, as both of them are primary. We define our massivestar isotopic range to be from C to Zn.
Most of the isotopes from helium to the iron-peak are synthesizedby hydrostatic burning in massive stars, CCSNe, and Type 1a SNe.Hydrostatic burning is the process that fuses lighter isotopes intoheavier isotopes within stellar interiors, and is responsible for themajority of the isotopes up until iron (Woosley et al. 2002). CCSNeoccur after core Fe burning. The core collapses and rebounds afterachieving nuclear densities. The released energy is shock-driven andimpacts other in-falling material, ejecting the products of hydrostaticburning and the remaining composition of the star. As a result, thestar releases the elements that were synthesized during its lifetime,which mostly consist of the 𝛼 elements from O to the iron-peakelements (Heger & Woosley 2010; West & Heger 2013). CCSNe aresaid to produce anywhere between one-third and two-thirds of theiron-peak elements (West & Heger 2013, and references therein).These 𝛼 elements can also be formed through explosive processingin 𝛼 -rich freezeout which occurs when the CCSNe shockwave travelsthrough the Si-rich shell of the star (Jordan et al. 2003).Many massive star simulations exist (e.g., Nomoto et al. 2006;Heger & Woosley 2010), but it is essential to use simulations thataccurately represent the environments in dSphs. Kirby et al. (2019)compared inferred yields from dSphs to yields from massive starsimulations by Nomoto et al. (2006) and Heger & Woosley (2010)and found that the inferred yields of the 𝛼 and iron-peak elementsgenerally agreed with both simulations for numerous dSphs (SeeSection 4.1.1). We use yields from Heger & Woosley (2010) for ourmodel with a Salpeter IMF, a low mixing parameter of 0.02512 usedin a running boxcar method Heger & Woosley (2010), an explosionenergy of E = 10 erg, and stellar mass bounds of 10 −
110 M (cid:12) (formore details, see Section 4.1).Type 1a SNe occur when, in a binary system between a whitedwarf and a main sequence star, the white dwarf collapses afterexceeding the Chandrasekhar limit from accreting too much materialfrom the companion star. However, this progenitor model is stilldebated, which can influence the isotopic yields (see Section 4.2.1).Type 1a SNe products mostly consist of the iron-peak elements fromexplosive burning during nuclear statistical equilibrium, but C, O,and Ne can also be produced from CO and CNeO white dwarfs.Kirby et al. (2019) modelled the most dominant explosion mech-anism of Type 1a SNe in dSphs. They compared theoretical Type1a SNe yields from numerous simulations to their inferred yieldsin dSphs and determined that sub-Chandrasekhar detonations werelikely the most dominant Type 1a progenitors in dSphs. For ourmodel, we choose yields from Leung & Nomoto (2020) who per-formed double detonation simulations on C/O white dwarfs (for moredetails, see Section 4.2.1).We assume that both Type 1a SNe and massive stars contribute afraction of each intermediate-mass and iron-peak isotope abundance.
MNRAS , 1–20 (2021)
Pandey & West
We define a fraction, 𝑓 , to be the contribution of Type 1a SNe to theobserved Fe (cid:12) abundance. Therefore, the scaling factor for eachisotope depends on 𝑓 and is given as 𝑓 · X (cid:12) / X Ia56 , where X (cid:12) is the late stage evolution abundance of Fe in Sculptor and X
Ia56 is the Type 1a contribution taken from Leung & Nomoto (2020).This shifts the isotopic abundance pattern for Type 1a SNe to fitthe yield for Fe. The fraction 𝑓 is a free variable that must be fitusing observational data (see Table 1). The massive star yields werescaled in a similar fashion with the factor given by (1 − 𝑓 ) · X (cid:12) / X massive56 , where X (cid:12) is the late stage evolution abundance of Fe inSculptor and X massive56 is the massive contribution taken from Heger& Woosley (2010). The purpose of this scaling is to normalize the Feabundances for both massive stars and Type 1a SNe to the late stageevolution of Sculptor (see Section 4 for how the late stage evolutionabundance pattern was obtained).An additional scaling was required to ensure that all other isotopiccontributions from both massive stars and Type 1a SNe summed tothe late stage evolution abundance for each isotope between 12 ≤ 𝐴 ≤
70, and is given as follows: 𝑋 massive 𝑖, 𝑓 = (cid:32) 𝑋 (cid:12) 𝑖 𝑋 massive 𝑖, + 𝑋 𝑖, (cid:33) 𝑋 massive 𝑖, (1) 𝑋 𝑖, 𝑓 = (cid:32) 𝑋 (cid:12) 𝑖 𝑋 massive 𝑖, + 𝑋 𝑖, (cid:33) 𝑋 𝑖, (2)where X 𝑖, and X 𝑖, 𝑓 are the original and scaled abundances of iso-tope i , respectively, and 𝑋 (cid:12) 𝑖 is the late stage evolution abundanceof isotope i . While the abundance pattern within each model is dis-torted due to the scaling processes, the isotopic ratios are preservedacross each model. For example, even elements still generally have agreater abundance than odd elements, and Type 1a yields still mostlycontribute to the iron-peak elements (See Figure 1 and Figure 2). Elements that are heavier than iron are not formed by charged particlefusion reactions because iron has the maximum value for the bindingenergy per nucleon (e.g., Bertulani & Kajino 2016). Rather, they areformed mainly through neutron-capture processes by adding neutronsto iron-peak elements through (n, 𝛾 ) reactions, thus forming heavierisotopes. If the neutron capture rate is small compared to the betadecay rate, then the nucleosynthesis follows the isotopic stability line,and is called the s -process (Bertulani & Kajino 2016). Conversely,if the neutron capture rate is large compared to the beta decay rate,then the nucleosynthesis is driven far to the neutron-rich side ofstability, and the unstable isotopes beta-decay back to stability oncethe neutron exposure ends (Bertulani & Kajino 2016). This is the r -process. Since the s -process follows the path of isotopic stability, it doesnot contribute to neutron-rich nor neutron deficient stable nuclei(Bertulani & Kajino 2016). Furthermore, Fe often plays the role ofthe seed nucleus due to its large abundance as well as its large neutron-capture cross section relative to neighboring isotopes (as discussedin West & Heger 2013). The s -process only occurs on existing iron-peak isotopes that formed in a previous stellar environment, so theprocess is considered to be a secondary process. The s -process is usually split into the main, weak and strong com-ponents. The weak s -process occurs in massive stars from convectivecore He burning and shell C burning and it is believed to contributeto roughly half of the isotopes beyond iron up to a mass numberof A ≈
100 (West & Heger 2013). The main s -process is initiatedthrough hydrogen burning where an abundance of protons can causea C pocket. The C( 𝛼 , n) O reaction is initiated and contributesto a greater abundance of neutrons which fuels the main s -processin low mass Asymptotic Giant Branch (AGB) stars (e.g., Bisterzoet al. 2015). The neutron exposure in the main s -process is longerthan than that of the weak s -process which contributes mainly to theheavier isotopes with mass numbers A ≥
88 (West & Heger 2013).Originally, the addition of the strong s -process was to account forPb abundances that were under produced at solar metallicities (West& Heger 2013). However, it is now believed to be part of the main s -process at low metallicities.Ideally, all three components of the s -process should be constrainedseparately in our model. However, we do not differentiate these dif-ferent components because of the absence of s -process contributionsfor non s -only isotopes in our decomposition of Sculptor’s late stageevolution (see Section 4) and due to lack of observational data. Assuch, in the present work we explicitly model the main s -processonly. The lighter element primary process (LEPP) was introduced to ac-count for the excess abundance of some lower mass elements, suchas Sr, Y, and Zr, that are under produced through neutron-captureprocesses (Travaglio et al. 2004). Similar to the r -process, LEPP isbelieved to be primary as it originates from ultra-metal poor stars(West & Heger 2013). However, the LEPP process is still uncertain,and we absorb its contributions into the massive star group. Supernovae are the some of the main sites in the Universe that have ahigh neutron flux due to their explosive environments. As a result, the r -process is usually associated with CCSNe. However, the productionsites of the r -process are still uncertain (e.g., Goriely 1999; Beniaminiet al. 2016; Skúladóttir et al. 2019), and recent studies show collapsarsand neutron star mergers could also be such production sites (seeReichert et al. (2020), and references therein).Despite continued debate on the progenitor, data suggests that the r -process contributes to the chemical enrichment of the galaxy atvery low ([Fe/H] < −
3) metallicities. In addition, r -process elementsare correlated with the 𝛼 elements since they both share a plateauat low metallicities and decrease in abundance after the onset ofType 1a SNe (e.g., Reichert et al. 2020, and also found in West& Heger 2013). Since the beta decay rate is small compared to theneutron-capture rate, the r -process drives the composition of neutron-rich isotopes before they decay back to stability. Bottlenecks arise atclosed neutron shells (Käppeler et al. 2011; Coc et al. 2015); however,these bottlenecks exist at lower mass numbers at A = 80, 130, and195 corresponding to atomic numbers of roughly Z = 32 (Ge), Z =63 (Eu), and Z = 78 (Pt) (West & Heger 2013). Since the r -processdoes not depend on the initial metallicity of the production site, it isconsidered to be primary. We consider the r -process to contribute tothe mass range 69 ≤ A ≤ ≤ A ≤
68 and instead combine them with the massivestar category as the two are difficult to separate in simulations (West& Heger 2013).
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MNRAS000 , 1–20 (2021) he Galactic Isotopic Decomposition for the Sculptor dSph Figure 1.
The Type 1a yields from Leung & Nomoto 2020 scaled to a nominal value of Fe / Fe (cid:12) = 0.85. Shown are the yields before (Stars) and after(Circles) applying fits with massive star yields. Figure 2.
The Massive star yields from Heger & Woosley 2010 fitted to observed Sculptor data and scaled to a nominal value of Fe massive / Fe (cid:12) = 0.15.Shown are the yields before (Stars) and after (Circles) applying fits with Type 1a yields. p -Isotopes The 𝛾 -process contributes to proton-rich isotopes beyond a massnumber A ≈
100 and occurs during photo-disintegration events onpre-existing metals in SNe (Woosley & Howard 1978). The 𝜈 p -process occurs in CCSNe when proton-rich ejecta are created byweak reactions (Martínez-Pinedo et al. 2006), and appear to synthe-size proton-rich isotopes up to a mass number of A ≈
100 (West& Heger 2013). While we model the 𝛾 -process in this work, we do not account for the 𝜈 p -process as they are not computed in OMEGA(Côté et al. 2017). dSphs in the local group are said to be the most dark matter dominatedsystems in the Universe and were formed by the infall of enriched gas MNRAS , 1–20 (2021)
Pandey & West in the outer region of the galactic halo of the MW, Andromeda, andother host galaxies (Vincenzo et al. 2014). Past chemical evolutionmodels suggest dSphs have a high wind efficiency (Lanfranchi &Matteucci 2003). This wind can transport enriched gas outside thegalaxy, lowering the total gas content and the SFR. Furthermore, thelower gas content suggests a lower metallicity distribution function(MDF) for dSphs which can lead to a broad metal-poor tail (seeFigure 3 of Lanfranchi & Matteucci 2003 and Figure 9 of Homma& Murayama 2014). These characteristics must result in notablydifferent isotopic histories between dSphs and the MW. The resultsof previous chemical evolution models of dSphs and their comparisonto the MW are outlined in this section. 𝜈 -Process There is insufficient indication that GCRs, classical novae, and the 𝜈 -process behave differently in dSphs than in the MW. While cos-mic rays (CRs) are thought to be sub-dominant in dwarf galaxies,Hopkins et al. (2020) found that an intermediate value of the CR dif-fusion coefficient suppressed star formation in dwarf galaxies sincethey have a smaller virial radius. Furthermore, CRs may be moreimportant for intermediate galaxies such as the MW. However, thedifferences appear minor. Classical novae have also not been studiedrigorously in dSphs. Only a couple classical novae have been discov-ered in dSphs, such as a possible novae in the M32 dSph (Darnleyet al. 2004), and the unusual nova V5852 Sgr, which may be the firstnova to be discovered in a dSph (Aydi et al. 2016). The 𝜈 -process hasnot been addressed in great detail in dSphs. Since these processesare not the main focus of this work and since there is no clear indi-cation that these processes behave differently in dSphs, we assumethat GCR, classical novae, and the 𝜈 -process behave similarly in boththe MW and Sculptor. The consequence of this assumption mainlyimpacts the LiBeB isotopes. The 𝛼 elements are primarily produced by massive stars and productsfrom CCSNe. At low metallicities, the iron production from CCSNeis relatively low for both dSphs and the MW (Tolstoy et al. 2009;Simon et al. 2015). However, iron production begins to dominatedue to the onset of Type 1a SNe at an observed metallicity of [Fe/H] ≈ − 𝛼 /Fe] trends, despitetheir different SFHs. In fact, Kirby et al. (2011) found the abundancedistributions of the 𝛼 elements of dSphs to evolve in a similar fashion.Numerous studies have also estimated the onset to occur around − . ≤ [Fe/H] ≤ − . 𝛼 /Fe] ratios are predicted to be above solar at [Fe/H] ≤ − . ≥ − . ∼ − .
8. Other studies have observed a knee for [ 𝛼 /Fe]ratios at [Fe/H] = − . − . − . 𝛼 /Fe] trend as well. Sextansstars have been observed to have a knee at [Fe/H] = − − . − . 𝛼 /Fe] ratios are lower than MW ratios at − . < [Fe/H] < − . 𝛼 /Fe] forall observed dSphs (Fornax, Leo I, Sculptor, Leo II, Sextans, Draco,Canes Venatici I, Ursa Major) drops from + . − . − . 𝛼 /Fe] knee. They calculated the position of the onset to occurat − . ≤ [Fe/H] ≤ − .
55 for 5 different dSphs (see Table 7 ofReichert et al. 2020 for specific values). For Sculptor, they found 𝛼 -knees of [Fe/H] = − − . − .
59, and − .
55 for Mg, Sc, Ti,and 𝛼 elements.There are several reasons for the earlier onset in dSphs. In a chem-ical evolution model devised by Lanfranchi & Matteucci (2003), thewind parameter was found to be the most contributive parameter toreproduce observational data. Their models incorporated high windefficiencies and low SFRs to accurately model dSphs. The high windefficiency is caused when the thermal energy of the enriched gasis greater than its binding energy (Bradamante et al. 1998). SincedSphs are comprised of a more densely concentrated dark matterhalo than the MW, the binding energy of the gas is less in dSphsthan in the MW which results in more efficient winds in dSphs thatcarry enriched gas out of the galaxy (Vincenzo et al. 2014). Kirbyet al. (2011) suggested that low-mass systems, such as dSphs, aremore sensitive to CCSNe yields than higher mass systems, such asthe MW halo. Indeed, in lower mass systems, SNe products have agreater influence on the chemical enrichment of the galaxy as notmuch mass was present in the system initially.The lower SFR in dSphs reflects fewer massive stars, which leavesthe 𝛼 elements, the primary products of massive stars, to decreasein abundance, notably at lower metallicities (Matteucci 2008). Thelower SFR also positions Type 1a SNe to govern the chemical enrich-ment of dSphs, and this could lead to a greater abundance of iron-peakelements (Tolstoy et al. 2009; Romano & Starkenburg 2013; Kirbyet al. 2011). The ejection velocity of CCSNe also tends to be greaterthan the dSphs escape velocity (Cohen & Huang 2009). As a result,CCSNe products have a reduced impact on the chemical enrichmentcompared to spiral galaxies. Type 1a SNe have also been found to bemore dominant in dSphs at higher metallicities than the MW (Kirbyet al. 2020). Since dSphs are often near a larger, host galaxy, tidalstripping can remove gas from dSphs (Tolstoy et al. 2009; Homma& Murayama 2014).After the Type 1a onset, the decrease of the [ 𝛼 /Fe] ratio has beenshown to be steeper than the corresponding decrease in the MW(Recchi et al. 2001; Matteucci 2008; Vincenzo et al. 2014; Theleret al. 2020). Galactic winds have an impact on the onset behavior,and it has been found in numerous chemical evolution models thatincreasing the wind parameter causes the slope of [ 𝛼 /Fe] to becomesteeper (Matteucci 2008; Vincenzo et al. 2014). This is because thegreater wind efficiency further decreases the SFR after the onset ofType 1a SNe, which affects the products of CCSNe more than Type1a SNe (Vincenzo et al. 2014). The relation between the SFR andthe slope of [ 𝛼 /Fe] vs [Fe/H] after Type 1a onset has also been ex- MNRAS000
55 for Mg, Sc, Ti,and 𝛼 elements.There are several reasons for the earlier onset in dSphs. In a chem-ical evolution model devised by Lanfranchi & Matteucci (2003), thewind parameter was found to be the most contributive parameter toreproduce observational data. Their models incorporated high windefficiencies and low SFRs to accurately model dSphs. The high windefficiency is caused when the thermal energy of the enriched gasis greater than its binding energy (Bradamante et al. 1998). SincedSphs are comprised of a more densely concentrated dark matterhalo than the MW, the binding energy of the gas is less in dSphsthan in the MW which results in more efficient winds in dSphs thatcarry enriched gas out of the galaxy (Vincenzo et al. 2014). Kirbyet al. (2011) suggested that low-mass systems, such as dSphs, aremore sensitive to CCSNe yields than higher mass systems, such asthe MW halo. Indeed, in lower mass systems, SNe products have agreater influence on the chemical enrichment of the galaxy as notmuch mass was present in the system initially.The lower SFR in dSphs reflects fewer massive stars, which leavesthe 𝛼 elements, the primary products of massive stars, to decreasein abundance, notably at lower metallicities (Matteucci 2008). Thelower SFR also positions Type 1a SNe to govern the chemical enrich-ment of dSphs, and this could lead to a greater abundance of iron-peakelements (Tolstoy et al. 2009; Romano & Starkenburg 2013; Kirbyet al. 2011). The ejection velocity of CCSNe also tends to be greaterthan the dSphs escape velocity (Cohen & Huang 2009). As a result,CCSNe products have a reduced impact on the chemical enrichmentcompared to spiral galaxies. Type 1a SNe have also been found to bemore dominant in dSphs at higher metallicities than the MW (Kirbyet al. 2020). Since dSphs are often near a larger, host galaxy, tidalstripping can remove gas from dSphs (Tolstoy et al. 2009; Homma& Murayama 2014).After the Type 1a onset, the decrease of the [ 𝛼 /Fe] ratio has beenshown to be steeper than the corresponding decrease in the MW(Recchi et al. 2001; Matteucci 2008; Vincenzo et al. 2014; Theleret al. 2020). Galactic winds have an impact on the onset behavior,and it has been found in numerous chemical evolution models thatincreasing the wind parameter causes the slope of [ 𝛼 /Fe] to becomesteeper (Matteucci 2008; Vincenzo et al. 2014). This is because thegreater wind efficiency further decreases the SFR after the onset ofType 1a SNe, which affects the products of CCSNe more than Type1a SNe (Vincenzo et al. 2014). The relation between the SFR andthe slope of [ 𝛼 /Fe] vs [Fe/H] after Type 1a onset has also been ex- MNRAS000 , 1–20 (2021) he Galactic Isotopic Decomposition for the Sculptor dSph plored specifically for the Sextans dSph, which experiences a greaterdecrease in SFR compared to other dSphs like Sculptor or Fornax,and therefore the slope of [ 𝛼 /Fe] ratio is steeper in Sextans than inSculptor or Fornax (Theler et al. 2020). Since dSphs have a lowerSFR than the MW, it follows that the slope of [ 𝛼 /Fe] is steeper indSphs than the MW.It has also been discovered that Type 1a SNe have a higher ejectionefficiency than CCSNe, and this results in a greater enrichment of theiron-peak elements rather than the 𝛼 elements, further contributingto the steep [ 𝛼 /Fe] ratio after the Type 1a onset (Recchi et al. 2001).Since the slope of the [ 𝛼 /Fe] ratio corresponds to the ratio betweenthe contributions of CCSNe and Type 1a SNe (Kirby et al. 2019),the steeper slope in dSphs is due to the greater chemical enrichmentof Type 1a SNe rather than CCSNe at higher metallicities. Finally,Kirby et al. (2020) studied dSphs near the M31 galaxy and found acorrelation between the mass and metallicity of dSphs. In particular,a greater mass suggests a higher SFR which delays the Type 1a onset,leading to a steeper [ 𝛼 /Fe] slope.Despite different [ 𝛼 /Fe] knee positions and slope values, it is worthnoting that dSphs (Sculptor, Ursa Minor, Sextans, and Fornax) and theMW exhibit a plateau of [ 𝛼 /Fe] ∼ ≤ − Na abundances have been disputed between numerous past studies.They have been observed to be relatively similar in both dSphs andthe MW below [Fe/H] = −
1, but Na abundances are lower in dSphsthan the MW at metallicities greater than [Fe/H] > − − [Ni/Fe] correlationin dSphs which was first suggested by Nissen & Schuster (1997) whodiscovered that stars in the MW halo that were deficient in 𝛼 elementswere also deficient in Na, and to a lesser extent, Ni. This is attributedto Type 1a SNe, since the neutron excess constrains the products ofmassive stars after 𝛼 -rich freeze out. This correlation was also foundin dSphs, especially in Fornax (Tolstoy et al. 2009).Venn et al. (2012) suggests [Na/Fe] follows the same rise as the s -process. This indicates that dSphs have a greater abundance of AGBstars because Na increases in abundance as AGB stars enrich thegalaxy. Na has subsolar values at low metallicities but then reachessolar at higher metallicities. Mn and Cr abundances have also ob-served to be similar to Na, since they increase in abundance whenAGB stars and Type 1a SNe contribute significantly to the chemicalenrichment of dSphs at around [Fe/H] = − The abundance of iron-peak elements, mainly Cr, Mn, Fe, Co, andNi, have been observed to be similar in Sculptor to the MW. Sculptorhas iron-peak abundances that compare well with the MW (Simonet al. 2015). Similarly, other dSphs show similar trends, including theCarina dSph (Venn et al. 2012) and the Sextans dSph (Theler et al.2020).However, other studies show that the iron-peak elements are solar at low metallicities and subsolar at higher metallicities. For example,the Draco and Ursa Minor dSphs have abundances of [Cr/Fe], [Ni/Fe],[Zn/Fe], and perhaps [Co/Fe] that overlap with MW abundancesat low metallicities. However, they decrease in abundance and aresubsolar at higher metallicities (Cohen & Huang 2009, 2010). [Zn/Fe]has also shown to decrease with metallicity in Sculptor (Skúladóttiret al. 2017). North et al. (2012) also measured [Mn/Fe] in 4 dSphsand found that they were subsolar. Kirby et al. (2019) also notedthe same pattern and found that [Co/Fe], [Ni/Fe], and [Zn/Fe] alldecreased with metallicity, and the ratios were lower than those ofthe MW. They attributed this to the greater ratio of Type 1a SNe toCCSNe in dSphs than the MW. It is worth noting that Kirby et al.(2019) found the [Ni/Fe] ratio to be subsolar in ancient galaxies likeSculptor, but this ratio increases with greater [Fe/H] for more activegalaxies.The heavy iron-peak element Zn was studied by Reichert et al.(2020), and they found the ratios [Zn/Ba] and [Zn/Eu] to be constantat low metallicities up to a metallicity of [Fe/H] ∼ −
It is widely believed that AGB stars are the host for s -process pro-duction in both the MW and dSphs (Fenner et al. 2006; Tolstoy et al.2009; Cohen & Huang 2009; Venn et al. 2012; Skúladóttir et al.2019, 2020; Theler et al. 2020; Reichert et al. 2020). Since dSphsgenerally have a lower metallicity due to a lower gas content, theneutron-to-seed ratio is greater at low metallicities and thus is morelikely to produce heavier s -process elements (Fenner et al. 2006).Large values of [Ba/Y] (Fenner et al. 2006) and [Y/Eu] (Venn et al.2012) confirms this behaviour. In fact, Sculptor stars have a value of[Ba/Fe] = 0.5 at [Fe/H] = −
1, and this is above the MW value by 0.8dex (Fenner et al. 2006).It has also been suggested that AGB stars begin dominating thechemical evolution of dSphs at the same time as Type 1a SNe (Vennet al. 2012; Theler et al. 2020; Skúladóttir et al. 2020). The MW hasa flat [Ba/Fe] ratio at higher metallicities (Venn et al. 2012), whichimplies that the timescale of AGB enrichment and Type 1a SNeenrichment are similar. Similar trends have been shown to occur inSculptor, since the [Ba/Fe] ratio plateaus at the solar value at [Fe/H]= − − − < [Fe/H] < − r -process to s -process transition in theMW and Sculptor. However, it is difficult to quantify exact metallicityranges for this transition as past studies on dSphs have mostly usedqualitative means to determine this value. In the MW, this transitionbegins around [Fe/H] = − − s -process and the stellarmass of galaxies. In particular, they performed a least squares fit on[Ba/H] to [Fe/H] and found that [Eu/Ba] decreased at a metallicityof − − − MNRAS , 1–20 (2021)
Pandey & West (Reichert et al. 2020). As a result, the more massive galaxies havean s -process onset that occurs later than less massive galaxies. Thisrelation is similar to the one they found for the 𝛼 elements, whichsupports the idea that the time delay of Type 1a SNe is on the samescale as the time delay of AGB stars. Based on these results, the MWshould have an s -process onset that is later than that of Sculptor.These results agree with Cohen & Huang (2009) who suggested thatthe s -process begins to dominate at a lower metallicity in the DracodSph than the MW. Other studies have also suggested metallicityranges for this transition. Fenner et al. (2006) modelled the chemicalevolution of Sculptor and found the [Ba/Eu] ratio to rapidly increaseat [Fe/H] = − r -process to s -process transition atthis metallicity. Tolstoy et al. (2009) observed that dSphs transitionfrom the r -process to the s -process at [Fe/H] = − s -process occurs between − < [Fe/H] < − −
2, and the ratio of all neutron-capture elements to Ba ([X/Ba]) became subsolar at [Fe/H] = − r -process in Sculptor has been shown to be similar to the MW.The [Eu/Fe] ratio is super-solar at low metallicities in both Sculptorand the MW because the r -process dominates over contributionsfrom Type 1a SNe. However, as the contribution from Type 1a SNeincreases, [Eu/Fe] decreases, similar to the [ 𝛼 /Fe] ratio. In the MW,[Eu/Fe] begins to decrease at [Fe/H] = −
1, which is congruent withthe Type 1a onset value (Skúladóttir et al. 2020).The contribution of AGB stars has been shown to be stronger inSculptor than in the MW. This is likely due to the slower and ex-tended SFR of Sculptor (de Boer et al. 2012a), meaning that delayedprocesses (like Type 1a SNe and s -process contribution through AGBstars) enrich the gas more at these high metallicities due to the lowermass environment of dSphs (Salvadori et al. 2015). In contrast, theMW has a higher SFR so gas is enriched up to higher metallacities.Therefore, AGB stars are not as dominant at higher metallicities inthe MW (Skúladóttir et al. 2020). This is confirmed by looking at theratios of neutron-capture elements relative to Ba ([X/Ba]). The ratiosare comparable to the MW at [Ba/H] < − r -process (Skúladót-tir et al. 2020). However, at [Ba/H] > − s -processcontributes more to Sculptor than the MW at high metallicities.There is evidence that suggests that CCSNe, and not NSM, arethe dominant r -process progenitors in dSphs (Tolstoy et al. 2009;Duggan et al. 2018; Skúladóttir et al. 2019; Reichert et al. 2020;Skúladóttir et al. 2020). Skúladóttir et al. (2019) observed that the r -process contributes throughout the evolution of Sculptor, since the[Eu/Mg] ratio was relatively constant throughout all metallicitiesfor both the MW and Sculptor, and this suggests that the r -processcontributes to the chemical evolution of these galaxies at the sametime as massive stars. Their results also indicate that neutron starmergers (NSM) may not be the most dominant site for the r -process.Rather, the dominant r -process sites may be closely associated withCCSNe (Skúladóttir et al. 2019). The r -process has also been shownto have little delay compared to CCSNe (Skúladóttir et al. 2020).Finally, Reichert et al. (2020) suggested that r -process material isco-produced with 𝛼 elements, further indicating that rare-CCSNeare the likely r -process progenitors rather than NSM.There is doubt on whether the s -process and r -process alonecan explain the chemical evolution of all neutron-capture isotopes.Skúladóttir et al. (2020) could only model the Sculptor star g-982 with significant i -process (intermediate process) contributions. Fur-thermore, Sculptor has low [Y/Ba] and [La/Ba] ratios which can onlybe explained with the i -process. Other processes may be needed toexplain supersolar [Y/Ba] at [Ba/H] < − r -process, Skúladóttir et al. 2020). Similar sug-gestions were made by Reichert et al. (2020). Despite these studies,we don’t account for these sources in our model as their contributionshave not been well-established. We present a model that describes an average isotopic history ofthe Sculptor dwarf spheroidal galaxy by considering contributionsfrom massive stars, Type 1a SNe, and neutron-capture processes (SeeSection 1 for reasons for choosing Sculptor). The initial state of ourmodel is taken to be the isotopic composition resulting from BBN.Although the solar abundance pattern was used as the final isotopiccomposition for West & Heger (2013), the same composition can-not be used for Sculptor. We computed the late stage evolution forSculptor using the OMEGA GCE code (Côté et al. 2017) with aninitial galaxy mass of 7.8x10 M (cid:12) , ratio between inflow and out-flow rate of 1.02, ratio between outflow rate and SFR of 8, numberof NSM per stellar mass of 2.0x10 − , mass ejected from NSM of1.2x10 − M (cid:12) , and computed across 10,000 timesteps. We also usedthe OMEGA GCE code to estimate the decomposition of Sculptor’slate stage evolution into various processes. AGB and massive con-tributions were computed using a weighted average from Karakas(2010) and Limongi & Chieffi (2018) with a rotational velocity of300 m s − and an average rotational velocity determined by Prantzoset al. (2018). Additionally, we use the Sculptor SFH from de Boeret al. (2012a), Type 1a SNe yields from Leung & Nomoto (2020),and a Salpeter IMF (See Section 4.1.2 for justification on using theSalpeter IMF). We extracted late stage evolution yields from theGCE model at the timestep corresponding to [Fe/H] = − ∼ −
50. The weak s -process plateau is comparatively less definedfor Sculptor yields. The Sculptor yields omit p-isotope contributionsfrom photo-disintegration events, which were not included in theOMEGA model.Each isotope in our model is assigned a unique function based onthe astrophysical processes that produce it, and the function scalesusing a normalized dimensionless parameter 𝜉 . As such, the isotopichistory of Sculptor at 𝜉 = 0 gives the BBN composition, and 𝜉 = 1 gives the late stage evolution abundance pattern. This choiceof parameterization assumes the relation between 𝜉 and Z/ 𝑍 (cid:12) ismonotonic and log( 𝜉 ) ∝ [Z]. For more information on the generalspecifics of this parameterization, see Section 3 of West & Heger(2013). In this section, we determine the functional forms of eachastrophysical process that was described in Section 2. MNRAS000
50. The weak s -process plateau is comparatively less definedfor Sculptor yields. The Sculptor yields omit p-isotope contributionsfrom photo-disintegration events, which were not included in theOMEGA model.Each isotope in our model is assigned a unique function based onthe astrophysical processes that produce it, and the function scalesusing a normalized dimensionless parameter 𝜉 . As such, the isotopichistory of Sculptor at 𝜉 = 0 gives the BBN composition, and 𝜉 = 1 gives the late stage evolution abundance pattern. This choiceof parameterization assumes the relation between 𝜉 and Z/ 𝑍 (cid:12) ismonotonic and log( 𝜉 ) ∝ [Z]. For more information on the generalspecifics of this parameterization, see Section 3 of West & Heger(2013). In this section, we determine the functional forms of eachastrophysical process that was described in Section 2. MNRAS000 , 1–20 (2021) he Galactic Isotopic Decomposition for the Sculptor dSph Figure 3.
Isotopic yields for the sun (Lodders 2019) and the late stage evolution of Sculptor computed using OMEGA (Côté et al. 2017). Purple stars are theratio between the MW and Sculptor abundances in log-space.
We use massive star yields from Heger & Woosley (2010). Similarto West & Heger (2013)’s simulation, we use a standard supernovaexplosion energy of 1.2x10 erg, a Salpeter IMF, a mixing value of0.02512, and stellar masses between 10M (cid:12) − (cid:12) .While West & Heger (2013) used massive star yields from Heger &Woosley (2010) for the MW, the same simulation may not be accuratefor modelling Sculptor. Therefore, we validate our choice for usingHeger & Woosley (2010) for massive star yields. Kirby et al. (2019)compared massive star theoretical yields of 3 𝛼 elements (Mg, Si, andCa) and 3 iron-peak elements (Cr, Co, and Ni) from simulations byNomoto et al. (2006) and Heger & Woosley (2010) to their inferredyields in five different dSphs. They determined that the inferredyields of the 𝛼 and iron-peak elements generally agreed with bothsimulations for numerous dSphs. We briefly outline their findingson the Sculptor dSph below (for more information, see Figure 7 ofKirby et al. 2019): • The [Mg/Fe] value agrees with Nomoto et al. (2006), but this isexpected as the [Mg/Fe] ratio was chosen as the prior and the [Mg/Fe]yields from Nomoto et al. (2006) normalized to quantify the fractionof elements produced in Type 1a SNe • The [Si/Fe] value generally agrees with both simulations • The [Ca/Fe] value falls closer to Nomoto et al. (2006) • The [Cr/Fe] ratio was in poor agreement with both simulations • The [Co/Fe] value exceeds both simulations but is closer toHeger & Woosley (2010) • The [Ni/Fe] ratio is closer to Heger & Woosley (2010)It is interesting to note that for the Sculptor dSph, the simulationby Nomoto et al. (2006) seems to present more accurate results forthe 𝛼 elements while the simulation by Heger & Woosley (2010)performs better for the iron-peak elements. Both perform equallywell for different elemental ratios; however, we decided to use the simulation by Heger & Woosley (2010) since we normalize the yieldsto the Fe abundance.In the following section, we give a brief discussion regarding theSalpeter IMF as our input choice. The initial mass function (IMF) describes the initial mass conditionsthat are necessary to evolve a galactic system. Numerous IMFs havebeen proposed over the last few decades (Salpeter 1955; Scalo 1986;Kroupa et al. 1993; Scalo 1998; Kroupa 2001; Chabrier 2003), andvarious chemical evolution models of dSphs have used these assortedIMFs for their different intentions. The Salpeter IMF has had rea-sonable success in past chemical evolution models for dSphs. Mostnotably, Lanfranchi & Matteucci (2003) used three distinct IMFs:the Salpeter, a flat Salpeter ( 𝛾 = − 𝛾 = 1.7 for m > M (cid:12) from Chabrier(2003). The Salpeter IMF produced reasonable results and modifyingthe model to the Chabrier IMF only produced miniscule secondaryeffects. Vincenzo et al. (2014) adopted the Salpeter IMF for all theirmodels and also tested an IMF with a steeper slope but found neg-ligible differences. The Salpeter IMF has also been successful indescribing observations in other studies (Wyse et al. 1999; Recchiet al. 2001; Kirby et al. 2019).It must be noted, however, that there is some degree of uncertaintywith using specific IMFs for galaxies other than the MW (Matteucci2008). Indeed, some studies have not had success with the SalpeterIMF. In particular, Calura & Menci (2009) found better agreementwith an IMF slope of 𝛾 = 1 for numerous dSphs. Additionally, thethree component IMF from Kroupa et al. (1993) has also been usedsuccessfully by other groups (Fenner et al. 2006; Kirby et al. 2011;Homma & Murayama 2014). MNRAS , 1–20 (2021) Pandey & West
Geha et al. (2013) investigated various IMFs for ultra-faint dwarf(UFD) galaxies and found that they exhibit a shallower IMF (relativeto the Salpeter and Kroupa IMFs) than the MW at a stellar massrange 0.52 − (cid:12) . They also found a correlation between theIMF power law slope and the galactic velocity dispersion and metal-licity (see Figure 5 of Geha et al. 2013). They attributed the IMFslope to correlate more strongly with metallicity, particularly the 𝛼 element abundance, rather than galactic velocity dispersion. Perhapsthis correlation could also be applied to dSphs since they exhibitlower metallicity distribution functions (see Figure 3 of Lanfranchi& Matteucci 2003 and Figure 9 of Homma & Murayama 2014), anearlier [ 𝛼 /Fe] “knee” (see Section 3.2), and galactic velocity disper-sion, but given that the correlation was only applied to a handfulof UFD galaxies and considering the uncertainties associated withobserved data, we await further study before applying it from UFDsto dSphs in the current model.Due to the uncertainties associated with developing a “universal”IMF, we simply apply the Salpeter IMF due to its reasonable successwith past chemical evolution models. We accept that there may beother IMFs that provide better results, but currently see no firmconsensus about the preferred IMF of choice for dSphs.The Salpeter IMF is considered to be constant with space and timeand takes the form of a power law (Salpeter 1955), 𝜙 ( m ) = 𝜙 · m −( + 𝛾 ) (3)where 𝜙 is the derivative of the number of stars with respect to mass, 𝜙 is an arbitrary, initial constant, and 𝛾 is the Salpeter slope of theIMF which is taken to be 𝛾 = 1.35. The abundances that were determined by the massive star simulationdescribed in Section 4.1.1 were normalized to the Fe abundanceat [Fe/H] = −
3. We assume that isotopic contributions from Type1a SNe are negligible at this metallicity (see Section 3.2) and asa result, each abundance was multiplied by the factor Fe 𝑓 /X sim 𝑖 ,where Fe 𝑓 corresponds to the final Fe composition of Sculptor ascomputed by OMEGA and X sim 𝑖 is the yield of isotope 𝑖 from themassive star simulation. These abundances at [Fe/H] = − ( 𝑋 ∗ 𝑖 ( 𝜉 )) = 𝑚 𝑖 ( log ( 𝜉 ) − log ( 𝜉 low )) + log ( 𝑋 sim 𝑖 ) (4)where X ∗ 𝑖 is the massive abundance of isotope i as a function ofthe parameter 𝜉 , X sim 𝑖 is the yield of isotope i from the massive starsimulation, and log( 𝜉 low ) = − − 𝑚 𝑖 ≡ log ( 𝑋 massive 𝑖, 𝑓 ) − log ( 𝑋 sim 𝑖 ) log ( 𝜉 𝑓 ) − log ( 𝜉 low ) (5)where X massive 𝑖, 𝑓 is the massive star contribution to the late stageevolution abundance and log( 𝜉 𝑓 ) = 0, which corresponds to the finalcomposition of Sculptor as computed by OMEGA. Using the fixed point [Fe/H] = −
3, Equation 4 is interpolated from BBN abundances(which is 0 for all isotopes governed by massive stars) to abundancesduring Sculptor’s late stage evolution, resulting in a functional formthat outputs the isotopic abundance from massive stars across theentire metallicity range.
The theoretical Type 1a SNe yields and the dominant Type 1a progen-itor that we choose should accurately model dSphs. Kirby et al. (2019)suggested that sub-Chandrasekhar mass white dwarfs are the mostType 1a SNe progenitor in dSphs. They used previous dSph abun-dances of 𝛼 elements and iron-peak elements to evolve [Mg/Fe] andquantify the fraction of Fe produced in Type 1a SNe to CCSNe. Usingthis fraction, they were able to evolve other elemental ratios using asimple model which adopted a constant value before the Type 1a on-set (the abundance of each element due to CCSNe) and sloped valuesafter the onset depending on the Type 1a contribution to the element,and they compared their model to several theoretical nucleosyn-thetic yields of Type 1a SNe, including deflagration-to-detonationtransition (DDT), pure deflagration (def), and sub-Chandrasekhardetonations (sub). Whereas the theoretical yields fit the data wellfor most of the elements, only the sub category fit the data well for[Ni/Fe]. Therefore, sub was determined as the most dominant mech-anism. This is further supported by McWilliam et al. (2018), whofound evidence of a sub-Chandrasekhar Type 1a SNe COS 171 in theUrsa Minor dwarf galaxy, and also Theler et al. (2020), who foundcompatible results which suggested that the production of Ni wassubsolar compared to Fe, reinforcing the idea that double degener-ate sub-Chandrasekhar-mass white dwarfs are the dominant Type 1aprogenitor in dSphs.The three sub models compared in Kirby et al. (2019) were Leung& Nomoto (2020), Shen et al. (2018), and Bravo et al. (2019). Leung& Nomoto (2020) employed a double detonation simulation on C/Owhite dwarfs, Shen et al. (2018) simulated spherically symmetricbare C/O white dwarf detonations, and Bravo et al. (2019) presented1D simulations for pure central detonations of white dwarfs. [Ni/Fe]yields especially matched the sub models of Shen et al. (2018) andLeung & Nomoto (2020) with masses of 1 − (cid:12) . We choseLeung & Nomoto (2020) Type 1a SNe yields for our model, as Figure6 of Kirby et al. (2019) shows that the standard model of Leung &Nomoto (2020) fits Sculptor reasonably well for all elemental ratios.We used their benchmark model 110 − − −
50 (X), correspondingto a white dwarf mass = 1.1 M (cid:12) , M He = 0.1 M (cid:12) , and Z = 0.02.While the choice of the Type 1a progenitor is consistent withSculptor’s observational data, it is important to note that Type 1aprogenitors likely vary based on metallicity and SFH. Kirby et al.(2019) suggest that while sub-Chandrasekhar mass white dwarfs maybe the dominant progenitors of Type 1a SNe at low metallicities,near-Chandrasekhar mass white dwarfs may be more dominant athigher metallicities. Furthermore, the [Ni/Fe] behavior cannot begeneralized for the total population of Type 1a SNe because [Ni/Fe]evolves differently in galaxies with more extended star formation,such as MW, Fornax, and Leo 1. In such galaxies, it is predictedthat they could host Type 1a SNe that are delayed (Kirby et al.2019). If single-degenerate near-Chandrasekhar mass white dwarfsare delayed compared to sub-Chandrasekhar mass white dwarfs, itis possible that galaxies with a greater SFH like the MW, Fornax,and Leo 1 have single-degenerate near-Chandrasekhar mass whitedwarfs as their dominant Type 1a progenitor. MNRAS000
50 (X), correspondingto a white dwarf mass = 1.1 M (cid:12) , M He = 0.1 M (cid:12) , and Z = 0.02.While the choice of the Type 1a progenitor is consistent withSculptor’s observational data, it is important to note that Type 1aprogenitors likely vary based on metallicity and SFH. Kirby et al.(2019) suggest that while sub-Chandrasekhar mass white dwarfs maybe the dominant progenitors of Type 1a SNe at low metallicities,near-Chandrasekhar mass white dwarfs may be more dominant athigher metallicities. Furthermore, the [Ni/Fe] behavior cannot begeneralized for the total population of Type 1a SNe because [Ni/Fe]evolves differently in galaxies with more extended star formation,such as MW, Fornax, and Leo 1. In such galaxies, it is predictedthat they could host Type 1a SNe that are delayed (Kirby et al.2019). If single-degenerate near-Chandrasekhar mass white dwarfsare delayed compared to sub-Chandrasekhar mass white dwarfs, itis possible that galaxies with a greater SFH like the MW, Fornax,and Leo 1 have single-degenerate near-Chandrasekhar mass whitedwarfs as their dominant Type 1a progenitor. MNRAS000 , 1–20 (2021) he Galactic Isotopic Decomposition for the Sculptor dSph There are three constraints for the Type 1a SNe functional form. Type1a SNe experience a delay before they begin dominating the chemicalenrichment of the galaxy. In the MW, the Type 1a onset is said tooccur at a metallicity of about [Fe/H] = − − ≤ [Fe/H] ≤ − ≈ − 𝑋 Ia 𝑖 ( 𝜉 ) = 𝑋 Ia 𝑖, (cid:12) · 𝜉 · [ tanh ( 𝑎 · 𝜉 − 𝑏 ) + tanh ( 𝑏 )][ tanh ( 𝑎 − 𝑏 ) + tanh ( 𝑏 )] (6)For specific reasoning for choosing the hyperbolic tangent base func-tion, see Section 3.1 of West & Heger (2013). The free parameters 𝑎 and 𝑏 are fit later to observational data. All three components of the s -process (ls, hs, and "strong" compo-nents) are treated together as the s -process which is described as apower law: 𝑋 s 𝑖 ( 𝜉 ) = 𝑋 s 𝑖, (cid:12) · 𝜉 ℎ (7)This approximation follows from lack of observational data to con-strain each separately. As such, results from fitting for the ls and"strong" components are tentative and should be interpreted withcare. In addition, we split the r -process function into two compo-nents: contributions from CCSNe and NSM. The CCSNe componentis described as a power law and the NSM component is describedsimilar to the Type 1a SNe function (See 4.2.2). We define 𝑔 as thefraction that NSM contribute to the late stage evolution abundanceof the r -process isotopes. 𝑋 r 𝑖 ( 𝜉, 𝑔 ) = 𝑔 · 𝑋 NSM 𝑖 ( 𝜉 ) + ( − 𝑔 ) · 𝑋 CCSNe 𝑖 ( 𝜉 ) (8) 𝑋 CCSNe 𝑖 ( 𝜉 ) = 𝑋 r 𝑖, (cid:12) · 𝜉 𝑝 (9) 𝑋 NSM 𝑖 ( 𝜉 ) = 𝑋 r 𝑖, (cid:12) · 𝜉 · [ tanh ( 𝑐 · 𝜉 − 𝑑 ) + tanh ( 𝑑 )][ tanh ( 𝑐 − 𝑑 ) + tanh ( 𝑑 )] (10)The parameters ℎ , 𝑔 , 𝑝 , 𝑐 , and 𝑑 are fit later to data. Power laws havebeen chosen because primary processes produce abundances that arelinear with respect to metallicity while secondary processes produceabundance that are quadratic with respect to metallicity (West &Heger 2013). The free parameter 𝑝 describes the 𝜉 -dependence for aprimary process (the r -process) and the free parameter ℎ describes 𝜉 -dependence for a secondary process ( s -process). Finally, the 𝛾 -process was chosen to have an equal contribution from primary andsecondary seed nuclei: 𝑋 𝛾𝑖 ( 𝜉 ) = 𝑋 𝛾𝑖, (cid:12) · 𝜉 ℎ + 𝑝 + (11)For more specific reasoning behind this averaging for the 𝛾 -process,see section 3.3 of (West & Heger 2013). 𝜈 -Process, andGalactic Cosmic Ray Spallation Since standard GCR spallation is secondary, its functional form issimilar to the s -process scaling and uses the same free parameter ℎ : 𝑋 GCR 𝑖 ( 𝜉 ) = 𝑋 GCR 𝑖, (cid:12) · 𝜉 ℎ (12)Classical novae, the 𝜈 -process, and primary GCR Spallation are allprimary processes and follow the same functional form as the r -process with the same free parameter 𝑝 : 𝑋 𝜈 /Novae/GCR 𝑖 ( 𝜉 ) = 𝑋 𝜈 /Novae/GCR 𝑖, (cid:12) · 𝜉 𝑝 (13)Helium and deuterium abundances are primary and scaled the sameas the r -process. However, they have an offset at 𝜉 = 0 to incorporatetheir BBN abundances, which are described in more detail in Section2.1. He ( 𝜉 ) = ( He (cid:12) − He BBN ) · 𝜉 𝑝 + He BBN (14) He ( 𝜉 ) = ( He (cid:12) − He BBN ) · 𝜉 𝑝 + He BBN (15)D ( 𝜉 ) = ( D (cid:12) − D BBN ) · 𝜉 𝑝 + D BBN (16)Finally, the remaining isotope of hydrogen, H, was scaled by thefollowing: H ( 𝜉 ) = . − 𝜉 · Z (cid:12) − Y ( 𝜉 ) − D ( 𝜉 ) (17)where Y( 𝜉 ) is the mass fraction of helium isotopes, D( 𝜉 ) is the massfraction of deuterium, and Z (cid:12) is the total metallicity of the sun. Weuse Z (cid:12) = 0.0173 (Lodders 2019), which is updated from their valueof Z (cid:12) = 0.0153 (Lodders et al. 2009) used in West & Heger (2013).The H equation simply restates the sum of the hydrogen and heliummass fractions with the mass fraction of all the metals, which is X+ Y + Z = 1. The fitting procedure and an example of using theseisotopic scaling functions is covered in Appendix A.1 in West &Heger (2013). Table 1 describes best-fitting free parameter valuesthat were found by fitting the functions to observable data.
MNRAS , 1–20 (2021) Pandey & West
The isotopic scaling functions were summed into elemental scal-ing functions and stellar abundance data for the Sculptor dSph wasused to tune the free parameters. Kirby et al. (2009) presented stellarabundances of the 𝛼 elements (Mg, Si, Ca, and Ti) as well as Fe abun-dances for almost 400 stellar abundances in Sculptor which were mea-sured using Keck/DEIMOS medium-resolution spectra. Tafelmeyeret al. (2010) presented abundances of iron-peak, neutron-capture,and 𝛼 elements in two metal-poor stars using high spectroscopic res-olution with VLT/UVES. Starkenburg et al. (2013) measured abun-dances of 𝛼 and iron-peak elements in seven low-metallicity stars inSculptor using ESO VLT. Hill et al. (2019) published stellar abun-dances of 99 red-giant branch stars of the 𝛼 elements (O, Mg, Si,Ca, Ti), iron-peak elements (Sc, Cr, Fe, Co, Ni, Zn), and neutron-capture elements (Ba, La, Nd, Eu) in Sculptor which were measuredusing high-resolution VLT/FLAMES spectroscopy. Skúladóttir et al.(2017) measured Zn abundances for approximately 100 red giantbranch stars in Sculptor using ESO VLT/FLAMES/GIRAFFE spec-tra. Kirby et al. (2018) determined abundances of the iron-peak el-ements (Cr, Co, Ni) for approximately 300 red giant stars in Sculp-tor using Keck/DEIMOS medium-resolution spectra. Duggan et al.(2018) measured abundances of [Ba/Fe] for 120 red giant branchstars in Sculptor using medium-resolution spectroscopy. Skúladóttiret al. (2019) used ESO VLT/FLAMES spectre to measure the chem-ical abundances of the neutron-capture elements Y, Ba, La, Nd, andEu in 98 stars. All data is given in units of [X/Fe] for all elements Xand [Fe/H] is used as a proxy for metallicity.The fitting procedure was implemented as follows: the [Fe/H] axis(x-axis) was split into 100 bins in the range − −
1. Similarly,the [X/Fe] axis (y-axis) was split into 100 bins in the range of theelemental data. Every data point was assigned a Gaussian distributionusing their respective observational uncertainties: 𝑓 𝑖 = exp (cid:34) − . · (cid:32) 𝑥 𝑖 − 𝑥 𝜎 𝑥 (cid:33) + − . · (cid:32) 𝑦 𝑖 − 𝑦 𝜎 𝑦 (cid:33) (cid:35) (18)where 𝑓 𝑖 is the value of the distribution at bin ( 𝑥 𝑖 , 𝑦 𝑖 ) on the [Fe/H]and [X/Fe] axes, 𝑥 and 𝑦 are the data point values, and 𝜎 𝑥 and 𝜎 𝑦 are the associated errors for each 𝑥 and 𝑦 . The distribution valueswere then averaged for each bin and a standard deviation was found.A differential evolution algorithm was implemented to minimize the 𝜒 between the average observed curve and the functional curvewhile preserving the expected chemical trends to avoid over-fitting.The free parameter values that resulted in the smallest 𝜒 values forMg, Ba, Eu, and Sr are shown in Table 1. The Type 1a SNe parameters 𝑎 , 𝑏 , and the fraction of Fe (cid:12) attributedto Type 1a SNe, 𝑓 , were found by using [Mg/Fe] data due to the largeabundance of Mg data available. Type 1a SNe and massive stars bothcontribute significantly to the evolution of Mg and Fe (Kirby et al.2019). The parameters were chosen to minimize the 𝜒 betweenthe functional curve of [Mg/Fe] and the observed [Mg/Fe] averagesfound in Sculptor. The best fitting parameters determined by thefitting procedure are 𝑎 = 3.059, 𝑏 = 0.898, and 𝑓 = 0.863 with a 𝜒 of 0.110. The best fitting functional curve and average curve for[Mg/Fe] are shown in Figure 4.The model curve is entirely within the standard deviation of thedata. The drop of the curve to late stage evolution values at [Fe/H] ≈ − − ≈ − − < − > − r -process and s -process Parameters Four parameters are required to describe the r -process functionalform: the primary process exponent ( 𝑝 ), the fraction that NSM con-tribute to r -process isotopes ( 𝑔 ), as well as the scaling and shiftingfactors of the tanh function to describe contributions from NSM ( 𝑐 and 𝑑 , respectively). Eu was chosen for fitting these parameters asit is an r -process peak element with two isotopes, Eu and
Eu,which both have dominant (85 per cent of its solar value, West &Heger 2013) contributions from the r -process. The best fitting pa-rameter values were found to be 𝑝 = 0.391, 𝑔 = 0.303, 𝑐 = 5.343,and 𝑑 = 0.450 with 𝑎 𝜒 = 0.022. The previously found best fittingvalues for 𝑎 , 𝑏 , and 𝑓 (from Section 5.1) were used for Fe. Whilea nominal value of ℎ = 1.5 was used for the s -process parameter inWest & Heger (2013), the late stage evolution of Sculptor computedby OMEGA (Côté et al. 2017) gives no s -process contributions forEu. The best fitting functional curve and average curve for [Eu/Fe]are shown in Figure 5.To determine the best value for the s -process parameter ℎ , Ba waschosen as it has two s-only isotopes, Ba and
Ba, along withthree isotopes,
Ba,
Ba, and
Ba, which have contributionsfrom both the s -process and r -process (West & Heger 2013). The 𝛾 -process contribution from Ba and
Ba were not considered forthe fit since the 𝛾 -process was found to be negligible at all metallic-ities for fitting the ℎ free parameter. The best fitting parameter valuewas found to be ℎ = 1.367 with 𝑎 𝜒 = 0.226. The previously foundbest fitting values for 𝑎 , 𝑏 , 𝑓 were used for Fe, and the best fittingvalue for 𝑝 , 𝑔 , 𝑐 , and 𝑑 were used for the r -process contributions toBa. The best fitting functional curve and average curve for [Ba/Fe]are shown in Figure 6. The complete elemental scaling model is given in Figure 8. While themodel only considers s-process contributions to the s-only isotopes,s-process peaks for Z ≈
38, Z ≈
56, and Z ≈
82 (corresponding toSr, Ba, and Pb, respectively) can be seen. The odd even effect can beseen below Z ≈
10, but this trend becomes unclear at higher atomicnumbers (10 ≤ Z ≤ MNRAS , 1–20 (2021) he Galactic Isotopic Decomposition for the Sculptor dSph Figure 4.
The resulting model for [Mg/Fe] determined after fitting parameters. The dark shadow background shows the Gaussian distributions of stellar dataaccording to the uncertainties. The solid red line is the average curve found by averaging the binned data after applying Gaussian distributions, and the dottedred lines are the resulting standard deviation curves. The model line is depicted by the solid black line.
Figure 5.
The resulting model for [Eu/Fe] determined after fitting parameters. The figure follows the same convention as Figure 4.MNRAS , 1–20 (2021) Pandey & West
Figure 6.
The resulting model for [Ba/Fe] determined after fitting parameters. The figure follows the same convention as Figure 4. yields from a sub-Chandrasekhar model tend to have a higher abun-dance of 𝛼 elements and lower abundances of iron-peak elements(see Figure 6 of Kirby et al. 2019). We do not expect our model tobe reliable for H/He isotopes, since our late stage abundances werecomputed using parameters which are reasonably chosen yet notperfectly known. Given the gas exchange between dSphs and theirenvironments, due in part to the low mass of these systems, H andHe may have significant uncertainties resulting from the chemicalevolution assumptions made. Figure 7 also shows the scaling func-tions of the processes considered in this work and are evaluated bythe best-fitting free parameters determined in Section 5.The best-fitting free parameters that were found in Section 5 offermany useful comparisons between the chemical evolution of theSculptor dSph galaxy and the MW. As mentioned in Section 3.2,iron production begins to dominate at an observed metallicity of[Fe/H] ≈ − 𝑏 , theshifting factor for the Type 1a tanh function, is less for Sculptor( 𝑏 = 0.898) than for the MW ( 𝑏 = 2.722, West & Heger 2013).This suggests that the Type 1a onset occurs earlier in dSphs thanthe MW and supports the conclusions of past chemical evolutionmodels. It is useful to calculate a numerical value for this onset. Wedefine the onset to occur when the Mg abundance from Type 1a SNeis 1 per cent of its Type 1a late stage evolution value. Using theMg abundance from the solar abundance decomposition derived byLodders (2019), we find the onset to occur at a metallicity of [Fe/H]= − − < [Fe/H] < − − − − 𝛼 -knee forMg to be ≈ − ≈ − − ≈ − 𝛼 /Fe] ratio after the Type 1a onset has beenpredicted to be steeper for dSphs than the decrease in the MW. Ourmodel also finds a steeper onset slope for dSphs compared to the MW.The value for the free parameter a, the scaling factor for the Type 1atanh function, for Sculptor ( 𝑎 = 3.059) is less than the value for theMW ( 𝑎 = 5.024, West & Heger 2013). This suggests that the impact ofType 1a SNe occurred over a shorter metallicity range than the MW.Our results contradict some previous studies, however. Kirby et al.(2011) analyzed [ 𝛼 /Fe] knees for numerous dSphs and only foundsignificant knees for the [Ca/Fe] ratio in Sculptor and Ursa Minorbut not for other elemental ratios, such as [Mg/Fe] or [Si/Fe]. This,along with the fact that there are no low metallicity plateaus in the[ 𝛼 /Fe] ratio for any dSph, suggests that Type 1a SNe were dominantmechanisms throughout the entirety of SFHs of all dSphs (Kirby et al.2011). Perhaps this discrepancy arises from different data sets, as ourmodel includes more recent Mg data. Most notably, Mg abundancesfrom Hill et al. (2019) (blue “X” markers from Figure 4) seem toshow the most sloped data at [Fe/H] > − − MNRAS , 1–20 (2021) he Galactic Isotopic Decomposition for the Sculptor dSph Figure 7.
The scaling functions of the model contributions relative to oxygen as functions of metallicity. The “Massive” line shows the scaling of the massivecategory’s contribution to Fe, and is normalized to the late stage evolution contribution from this category only. our data at [Fe/H] < − > < − ) forvarious dSphs and determined a constant value of the ratio of anelement relative to Fe before the onset ([X/Fe] CC ) using the MarkovChain Monte Carlo method. They fixed the initial value of [Fe/H] to − = − = − CC to be +0.56 ± CC value of approximately 0.5 determinedby OMEGA (Figure 4). Our model’s [Mg/Fe] CC value of 0.5 isalso less than the [Mg/Fe] CC value for the MW which plateaus atapproximately 0.57 (West & Heger 2013). This supports the idea thata system with a low [Mg/Fe] CC value at low metallicities undergoesa slow, extended SF (Fenner et al. 2006), since Sculptor’s SFH tookplace for an extended period of time from 14 gyr to 7 gyr ago duringa single formation event and gradually decreased afterwards (de Boeret al. 2012a). Kirby et al. (2011) also observed that the average valueof [ 𝛼 /Fe] drops from +0.4 at [Fe/H] = − − 𝛼 /Fe] value byabout 1 dex at low metallicities. Similarly, Mashonkina et al. (2017)noted that dSphs and the MW exhibit a plateau of [ 𝛼 /Fe] ≈ ≤ − ≈ −
1, whichis above the MW value by 0.4 dex. This supports the conclusionsof Fenner et al. (2006) and other studies that AGB stars are themost likely sites for s -process production in Sculptor due to the large[Ba/Fe] ratios at high metallicities. While the model for Ba from West& Heger (2013) has a value of [Ba/Fe] ≈ −
1, whichsomewhat contradicts this claim, their model slightly decreases after[Fe/H] ≥ −
1, while our model shows no such trend. Therefore, ourmodel does indeed predict higher [Ba/Fe] values, only after [Fe/H]= − r -process to s -process transition can be numerically calculatedby the intersection between the s -process and r -process Ba modellines. We find the intersection to occur at [Fe/H] ≈ − ≈ − − r -process to s -processtransitions. For the MW, West & Heger (2013) found this transitionto occur at [Fe/H] ≈ − s -process transition that is later than that of dSphs.Previous studies have also suggested that AGB stars began dom-inating the chemical evolution of dSphs at the same timescale asType 1a SNe. Our model also predicts this trend; the s -process to r -process transition ([Fe/H] = − − − MNRAS , 1–20 (2021) Pandey & West
Figure 8.
The complete elemental scaling of the model. The abundances are given relative to their MW solar values. The model only considers s -processcontributions to the s -process peak elements. Figure 9.
The resulting model for [Na/Fe] determined after fitting parameters. The figure follows the same convention as Figure 4.MNRAS000
The resulting model for [Na/Fe] determined after fitting parameters. The figure follows the same convention as Figure 4.MNRAS000 , 1–20 (2021) he Galactic Isotopic Decomposition for the Sculptor dSph Figure 10.
The resulting models for [Ni/Fe] determined after fitting parameters. Both sub-Chandrasekhar (subC) and near-Chandrasekhar (nearC) model curvesare shown. The figure follows the same convention as Figure 4.
Figure 11.
The best-fitting line for the Na-Ni correlation in Sculptor comparedto the the best-fitting line in the MW, determined by Nissen & Schuster 1997.The Sculptor best-fitting line was constrained to [Fe/H] ≤ − − ≤ [Ni/Fe] ≤ − to increase after [Fe/H] > − s -process ismore dominant than Type 1a SNe at higher metallicities ([Fe/H] > − s -process is also more dominant at the low metallicityrange ([Fe/H] ≤ -1.6). This also confirms previous suggestions thatcontributions from AGB stars are stronger in Sculptor than the MW.The best-fitting ℎ -value of Sculptor ( ℎ = 1.367) is less than the best-fitting ℎ -value of the MW ( ℎ = 1.509) suggesting a more dominant AGB contribution in Sculptor throughout all metallicities rather thanan abrupt rise of AGB stars as seen in the MW.The Eu model line is super-solar at low metallicities and seemsto plateau at a value of [Eu/Fe] ≈ ≈ r -process is similar in both Sculptorand the MW (Skúladóttir et al. 2020). The [Eu/Fe] ratio begins todecline at [Fe/H] = − −
1, the Type1a onset value of the MW.Surprisingly, the free parameter 𝑝 , the primary process exponent,was found to be much lower ( 𝑝 = 0.391) than the MW value ( 𝑝 =0.938, West & Heger 2013). A lower 𝑝 -value corresponds to a largerdecrease in the [Eu/Fe] ratio as [Fe/H] increases to zero. The freeparameter 𝑔 , the fraction that NSM contributes to r -process isotopes,suggests that NSM only contributes about 30 per cent to the late stageabundance of Eu, which supports previous studies that claim thatCCSNe, rather than NSM, are the dominant r -process progenitorsin dSphs (Tolstoy et al. 2009; Duggan et al. 2018; Skúladóttir et al.2019; Reichert et al. 2020; Skúladóttir et al. 2020).The evolution of Na and Ni in the MW and dSphs have beenremarked upon by previous studies (Nissen & Schuster 1997; Tolstoyet al. 2009; Cohen & Huang 2009; Venn et al. 2012). Figure 9and Figure 10 show the model for Na and Ni, respectively. Ourmodel seems to accurately depict observed Na trends, with [Na/Fe]increasing at low metallicities ([Fe/H] < − ≤ − > − MNRAS , 1–20 (2021) Pandey & West shows this correlation in Sculptor. While Nissen & Schuster (1997)determined the best-fitting line to be [Na/Fe] = 2.577 · [Ni/Fe] + 0.116for the MW, we find the best-fitting line to be [Na/Fe] = 3.663 · [Ni/Fe]+ 0.670 during the early evolution of Sculptor (-0.23 ≤ [Ni/Fe] ≤− 𝜒 = 0.002. We constrain this fit to [Fe/H] ≤ − − ≈ − ≈ − ≈
1) at [Fe/H] = − − < [Fe/H] < −
1. Si shows similar trends, butis overproduced by about 0.3 dex in the same metallicity range. Themodel for Ti is most distinct from observed data. The [Ti/Fe] ratiodecreases at lower metallicities and begins to increase at [Fe/H] ≈− − > − > − − ≤ [Fe/H] ≤− A model has been constructed that describes the average isotopicdecomposition of the Sculptor dwarf spheroidal galaxy for contribu-tions from massive stars, Type 1a SNe, main s -process peak, and the r -process, and the procedure follows the methodology described in Table 1.
Optimized Parameter ValuesParameter Best-fitting Value Description 𝑎 𝑏 𝑓 Fe (cid:12) from Type 1a 𝑝 ℎ hs -process exponent 𝑐 𝑑 𝑔 (cid:12) from NSM West & Heger (2013). Astrophysical processes that dictate the chem-ical evolution of galaxies were discussed and compared between theMW and dSphs. Parametric equations were assigned to these pro-cesses and scaled between BBN and the Sculptor late stage evolutiondecomposition, which was computed using OMEGA at [Fe/H] =-1.25 (Côté et al. 2017). The isotopic scalings were summed to el-emental scalings and compared with Sculptor data to fix the freeparameters of the model. The result is a complete isotopic abun-dance decomposition of Sculptor at any desired metallicity for theprocesses considered.While many assumptions and approximations were made whenconstructing the model, this work still offers many comparisonsbetween dSphs and spiral galaxies, with Sculptor and the MW ascandidates: • The Type 1a onset occurs earlier in Sculptor than the MW. Nu-merically, we define the onset to occur when the Mg abundance fromType 1a SNe is 1 per cent of its Type 1a solar value. This occurs at[Fe/H] = − − − ≈ − • The model predicts that Type 1a SNe contribute ≈
86 per centto the Fe late stage evolution abundance, which is greater thanpredicted values of ≈
70 per cent in the MW (West & Heger 2013).Therefore, Type 1a SNe contribute more to the chemical evolutionof dSphs compared to the MW, in agreement with past chemicalevolution models. • The slope of the [ 𝛼 /Fe] ratio after the Type 1a onset is steeperin Sculptor than the MW, suggesting that the impact of Type 1a SNeoccurred over a shorter metallicity range in Sculptor. • The model predicts that sub-Chandrasekhar Type 1a SNe are thedominant Type 1a progenitor in Sculptor, in agreement with Kirbyet al. (2019). • The model suggests that AGB stars are more dominant inSculptor compared to the MW and are the most likely sites for s -process production. AGB enrichment also seems to occur at similartimescales as Type 1a SNe since the r -process to s -process transition([Fe/H] = − − • The r -process to s -process transition occurs at [Fe/H] ≈ − − < [Fe/H] < − ≈− • The r -process evolution in Sculptor appears to be similar to the MNRAS , 1–20 (2021) he Galactic Isotopic Decomposition for the Sculptor dSph Table 2.
Third-Party Data SourcesDescription Publication Data LinkAbundances of 𝛼 elements for ≈
400 stars Kirby et al. (2009) LinkAbundances of iron-peak, neutron-capture, and 𝛼 elements in two stars Tafelmeyer et al. (2010) Link (Table 7)Abundances of iron-peak and 𝛼 elements in seven stars Starkenburg et al. (2013) Link (Table 6)Abundances of iron-peak, neutron-capture, and 𝛼 elements in 99 stars Hill et al. (2019) LinkAbundances of Zn for 100 stars Skúladóttir et al. (2017) LinkAbundances of iron-peak elements for ≈
300 stars Kirby et al. (2018) LinkAbundances of Ba for 120 stars Duggan et al. (2018) LinkAbundances of neutron-capture elements for 98 stars Skúladóttir et al. (2019) LinkOMEGA GCE code Côté et al. (2017) LinkMassive star theoretical yields Heger & Woosley (2010) LinkType 1a SNe theoretical yields (sub-Chandrasekhar) Leung & Nomoto (2020) Link (Table 7)Type 1a SNe theoretical yields (near-Chandrasekhar) Leung & Nomoto (2018) Link (Table 4)AGB contributions Karakas (2010) LinkCCSNe contributions Limongi & Chieffi (2018) Link evolution in the MW, with similar [Eu/Fe] plateaus at low metallici-ties and a [Eu/Fe] decline at the Type 1a onset. • The model shows that NSM only contribute ≈
30 per cent toEu (cid:12) , supporting previous studies that CCSNe, rather than NSM, arethe most dominant progenitor for r -process production.In addition to offering comparisons between dSph and MW chem-ical evolutions, this model also provides isotopic inputs for future nu-cleosynthesis studies of dSphs. Future works can explore this modelfor other dSphs to provide further insight into the chemical evolutiondifferences between dSphs and spiral galaxies and ultimately furtherour understanding of the galactic hierarchy. DATA AVAILABILITY
The data that support the findings of this study are available in thepublic domain. Table 2 lists the main data sources used in this work.Additional results from this work (e.g., elemental model plots) areavailable from the corresponding author upon request.
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