Appendix: Nucleosynthetic yield parameterizations

A SIMPLE CHEMICAL EVOLUTION MODEL FOR SCULPTOR DSPH

4.7 Appendix: Nucleosynthetic yield parameterizations

As described in Chapter 4, rather than select particular model yield sets for our galactic chemical evolution model, we represent the nucleosynthetic yields from supernovae and AGB stars with analytic functions. Parameters of these functions are varied in conjunction with the other model parameters.

The yields from Type Ia SNe are assumed to be independent of progenitor mass and metallicity. We plot the Type Ia SN yields from different models in Figure 4.12, as well as the yields chosen for our model (dotted blue lines). For C, Mg, Si, and Ca, the Type Ia SN yields were selected to be within the range of model yields, although the exact values do not matter because these yields do not significantly impact the final abundance trends. Manganese and nickel are not included in the GCE model fitting, so for our initial model we arbitrarily choose the yields from Leung and Nomoto (2020) for these elements (see Section 4.5.2). The yield of Fe from Type Ia SNe is a free parameter (Fe_Ia); Figure 4.12 shows the best-fit value from our fiducial model (Table 4.3).

For CCSN and AGB yields, we fit the model yield sets (Table 4.2) with combina- tions of exponential and Gaussian functions. We chose functions that qualitatively approximate the shapes of the theoretical yields as functions of progenitor mass (M) and metallicity (Z). In particular, we fit yields that are mass-weighted by a Kroupa et al. (1993) IMF (i.e., multiplied by the IMFdN/dM). The analytic functions can

then be integrated over a range of progenitor stellar masses to compute the total yield of a given element that will be produced by stars in that mass range after an instantaneous 1 M star formation burst. These functions are listed in Table 4.4 and illustrated as dotted blue lines in Figures 4.13 and 4.14 for CCSNe and AGB stars, respectively. The theoretical yield sets are also plotted for comparison (solid and dotted lines). For completeness, Figure 4.13 also illustrates the Li et al. (2014) r-process yields from CCSNe assumed in Section 4.5.3.

Table 4.4: Analytic functions describing IMF-weighted CCSN and AGB yields.

Element CCSN yield (M)

H 10⁻³

255

−1.88

−0.1

He 10⁻³

−1.35

−0.2

C 10⁻⁵

100

−expC_II

Mg 10⁻⁵

normMg_II+13G _M

M,19,6.24

Si 10⁻⁵

2260

−2.83

+0.8

Ca 10⁻⁶

15.4

−1

+normCa_II

40−10⁴_Z^Z

G _M

M,15,3

+0.06

Mn 10⁻⁷

−1.32

−0.25

Fe 10⁻⁵

2722

−2.77

Ni^a 10⁻⁷

8000 _M

−3.2

Ba^b 10⁻¹²

1560 _M

−1.80

+0.14−480G _M

M,5,5.5

Element AGB yield (M)

H 10⁻¹

1.1

−0.9

−0.15

He 10⁻²

−1.07

−0.22

C 10⁻³

normCAGB

1.68−220_Z^Z

G _M

M,2,0.6

Mg 10⁻⁵

400_Z^Z +1.1 _M^M

₀.08−340(Z/Z)

360_Z^Z −1.27

Si 10⁻⁵

800_Z^Z _M^M −0.9

−

80_Z^Z +0.03

Ca 10⁻⁶

800_Z^Z −0.1 _M^M −0.96

−80_Z^Z

Mn 10⁻⁷

1500_Z^Z _M

−0.95

−160_Z^Z

Fe 10⁻⁵

1500_Z^Z _M

−0.95

−160_Z^Z

Ni 10⁻⁶

840_Z^Z

−0.92

−

80_Z^Z +0.04

Ba 10⁻⁸

normBa_AGB

10³_Z^Z +0.2 G _M

M,meanBa_AGB,0.75−100_Z^Z Eu 10⁻¹¹ 3400_Z^Z +0.4

G _M

M,2,0.65

a The CCSN yield function for Ni was chosen to fit the yields from Limongi and Chieffi (2018). Since Ni is not included in fitting our GCE model (Section 4.2.2), this does not affect any of our results; however, we find that the Nomoto et al.

(2013) yields significantly underpredict the observed [Ni/Fe] at low [Fe/H].

b Our fiducial model assumes that zero Ba and Eu are produced by CCSNe. The functions describing CCSN yields listed here are for ther-process; as described in the text, we use the Ba yields from Li et al. (2014) and scale by the universal r-process ratio [Ba/Eu]∼ −0.7 (see Sneden et al. 2008) to obtain the Eu yields.

Note:As described in the text, these yields are weighted by a Kroupa et al. (1993) IMF. In these equations, we use the notationG _M

M, µ, σ

to denote a normalized Gaussian function as a function of mass with meanµand standard deviationσ.

0.5 1.0 1.5 2.0

10−3M H (8.84Mper CCSN)

0.3 0.6 0.9 1.2

10−3M

He (6.22Mper CCSN)

2.0 4.0 6.0 8.0

10−5M C (0.22Mper CCSN)

0.6 1.2 1.8 2.4

10−5M Mg (0.17Mper CCSN)

1.0 2.0 3.0 4.0

10−5M

Si (0.16Mper CCSN)

1.5 3.0 4.5 6.0

10−6M Ca (0.01Mper CCSN)

0.4 0.8 1.2 1.6

10−7M Mn (3.14×10⁻⁴Mper CCSN)

0.5 1.0 1.5 2.0

10−5M

Fe (0.08Mper CCSN)

0.5 1.0 1.5 2.0

10−6M Ni (6.70×10⁻³Mper CCSN)

1.5 3.0 4.5 6.0

10−12M Ba (5.91×10⁻⁸Mper CCSN)

15 20 25 30 35 40

Mass (M) 0.0

0.2 0.4 0.6 0.8

10−12M Eu (7.84×10⁻⁹Mper CCSN)

Nomoto et al. (2013) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Limongi & Chieffi (2018) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Best-fit yields Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

r-process Ba and Eu:

Nomoto et al. (2013) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Limongi & Chieffi (2018) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Best-fit yields Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

r-process Ba and Eu:

Nomoto et al. (2013) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Limongi & Chieffi (2018) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Best-fit yields Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

r-process Ba and Eu:

Nomoto et al. (2013) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Limongi & Chieffi (2018) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Best-fit yields Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

r-process Ba and Eu:

Nomoto et al. (2013) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Limongi & Chieffi (2018) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Best-fit yields Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

r-process Ba and Eu:

Nomoto et al. (2013) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Limongi & Chieffi (2018) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Best-fit yields Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

r-process Ba and Eu:

Nomoto et al. (2013) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Limongi & Chieffi (2018) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Best-fit yields Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

r-process Ba and Eu:

Nomoto et al. (2013) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Limongi & Chieffi (2018) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Best-fit yields Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

r-process Ba and Eu:

Nomoto et al. (2013) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Limongi & Chieffi (2018) Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

Best-fit yields Z=0.0 Z=0.0005 Z=0.001 Z=0.0015 Z=0.002

r-process Ba and Eu:

Nomoto et al. (2013)

Limongi & Chieffi (2018)

Best-fit yields

r-process Ba and Eu:

Li et al. (2014) Best-fit yields Nomoto et al. (2013)

Limongi & Chieffi (2018)

Best-fit yields

r-process Ba and Eu:

Li et al. (2014) Best-fit yields r-process Ba and Eu:

Li et al. (2014) Best-fit yields

Figure 4.13: Nucleosynthetic yields from core-collapse supernovae as a function of initial stellar mass. Colors denote different metallicities; yields have been linearly interpolated to match the same metallicity range. Dashed lines show parameterized yields for illustration. For elements labeled with bold blue text, free parameters are used to fit the GCE model. As described in the text, these yields are weighted by a Kroupa et al. (1993) IMF. To aid in interpretation, we list the average Z = 0.002 yield per CCSN—i.e., the Z = 0.002 best-fit analytic function integrated over the yield sets’ CCSN progenitor mass range 13−40 M and multiplied by a factor of 500 (since roughly 1 CCSN is produced for every 500 M of stars formed)—in the upper right corner of each plot.

0.3 0.6 0.9 1.2

10−1M

H (0.92Mper AGB)

1.0 2.0 3.0 4.0

10−2M

He (0.37Mper AGB)

0.5 1.0 1.5 2.0

10−3M

C (0.01Mper AGB)

0.5 1.0 1.5 2.0

10−5M

Mg (1.43×10⁻⁴Mper AGB)

0.3 0.6 0.9 1.2

10−5M

Si (1.43×10⁻⁴Mper AGB)

0.3 0.6 0.9 1.2

10−6M

Ca (1.30×10⁻⁵Mper AGB)

0.6 1.2 1.8 2.4

10−7M

Mn (2.64×10⁻⁶Mper AGB)

0.6 1.2 1.8 2.4

10−5M

Fe (2.64×10⁻⁴Mper AGB)

0.4 0.8 1.2

10−6M

Ni (1.47×10⁻⁵Mper AGB)

2 4 6

Mass (M) 0.0

0.4 0.8 1.2 1.6

10−8M

Ba (1.37×10⁻⁷Mper AGB)

FRUITY (Cristallos et al. 2015) Z=0.0