MEASURING THE TYPE IA DELAY-TIME DISTRIBUTION WITH GALACTIC ARCHAEOLOGY
3.2 A method for measuring the Type Ia DTD in an individual galaxy
Here I present a method for measuring the time-resolved Type Ia DTD within an individual galaxy. An overview of this method is illustrated in Figure 3.1. Broadly, I use stellar abundances to measure the rate of Type Ia supernovae. I then compare thisobservedrate with anexpected Type Ia supernova rate, obtained by assuming a DTD from a theoretical model. By comparing the difference between the observed and expected Type Ia rates, I am able to determine the best-fit model Type Ia DTD.
Further details are described in the following subsections.
3.2.1 The observed rate of Type Ia SNe
To estimate the observed rate of Type Ia SNe, I first use the stellar metallicity distribution function (MDF) of the galaxy to measure the cumulative number of
Ψ(𝜏)
1. Measure observed Type Ia SN rate from stellar abundances:
𝑁!" $%&
𝑁'' $%& × 𝑁'' $%&
⊗ SFH(𝑡) 2. Compute expected Ia
rate from model DTDs:
[𝛼/Fe] vs. [Fe/H]
MDF
𝑅!", )*+(𝑡)
SFH
𝑅!", &,-(𝑡)
=
= 𝑅!", )*+
MDF
Function of [Fe/H] Function of time Age-metallicity conversion
Figure 3.1: A schematic outlining the method used to compute the Type Ia DTD in a single galaxy. Observed quantities from stellar abundances (italic labels) are converted from functions of stellar metallicity (blue boxes) to functions of time (yellow boxes) using the age-metallicity relation (orange arrows) in order to compute
“observed” and “expected” rates of Type Ia supernovae (yellow boxes with orange outlines).
core-collapse SNe as a function of metallicity.1 The MDF records the number of stars produced (effectively, the star formation history) as a function of [Fe/H]. By assuming a stellar initial mass function, which determines the fraction of high- mass stars formed in a burst of star formation, the cumulative MDF can then be converted to the cumulative number of core-collapse SNe as a function of metallicity:
NCC([Fe/H]).
I then measure the cumulative ratio of core-collapse to Type Ia SNe as a function of metallicity using the stellar [Mg/Fe] abundance ratio. Mg is predominantly produced in core-collapse SNe, while iron is produced in both core-collapse and Type Ia SNe.
As a result, the ratio [Mg/Fe] probes the relative nucleosynthetic contribution of both types of supernovae, since CCSNe yield ejecta with a high ratio of Mg to Fe whereas Type Ia SNe yield negligible amounts of Mg.
Kirby et al. (2019) showed that a simple model can be used to more precisely quantify these metallicity contributions. This model assumes that CCSNe are the only nucleosynthetic sources at early times, and that CCSN yields are metallicity- independent—the ratio [Mg/Fe] will therefore be constant as a function of [Fe/H]
1Throughout this chapter, “metallicity” refers exclusively to observed stellar metallicity, traced by [Fe/H]. I use bracket abundances referenced to solar (e.g., [Fe/H] = log10(nFe/nH)∗−log10(nFe/nH)), wherenXis the atomic number density of X. Solar abundances are adopted from Asplund et al. (2009).
for old, metal-poor stars. After some delay-time, Type Ia SNe will begin to explode and produce iron, diluting [Mg/Fe] over time. As a result, above some threshold metallicity, [Mg/Fe] will decrease as a function of [Fe/H]. Kirby et al. (2019) showed that in several dSph systems, this decrease can be modeled as a linear trend.
As shown in Equation 13 of Kirby et al. (2019), this model can be used to estimate the mass ratio of iron produced by Type Ia SNe compared to iron produced by CCSNe:
R= MMFe,Ia
Fe,CC. Using theoretical yields of iron for these two types of supernovae,Rcan be converted to the cumulative number ratio of core-collapse to Type Ia SNe as a function of metallicity:NCC/NIa([Fe/H]).
Finally, I can compute the cumulative number of Type Ia SNe as a function of metallicity:
NIa([Fe/H])= NCC([Fe/H])
NCC
NIa ([Fe/H]). (3.2) The first derivative of this cumulative distribution is then theobservedinstantaneous rate of Type Ia SNe as a function of metallicity:RIa,obs([Fe/H]).
To meaningfully compare with an expected rate of Type Ia SNe,RIa,obs([Fe/H])must be converted to a function of time using the age-metallicity relation. This conversion implicitly assumes that the age-metallicity relation is a strictly monotonic (i.e., one- to-one) function. Although this assumption is fortunately true for most dSphs (see Section 3.3), the intrinsic functional form of the age-metallicity relation is unknown;
converting a function of metallicity to a function of age is therefore more complicated than a simple change of variables. I circumvent this with a numerical approach, by considering that RIa,obs([Fe/H]) is a discrete, binned function. The edges of each metallicity bin are first converted to time. The value in each bin represents the number of Type Ia SNe per bin—to convert this to Type Ia SNe per time bin, I simply divide by the size of each time bin∆t, which yields the rate of Type Ia SNe as a function of time: RIa,obs([Fe/H]).
3.2.2 Comparing with the expected rate of Type Ia SNe
The Type Ia supernova rate can be independently computed by convolving the DTD with the star formation history (SFH) of the galaxy (Equation 3.1). I use this method to estimate anexpectedrate of Type Ia supernovae.
As I describe below, the SFH can be measured directly (from, e.g., photometry or a chemical evolution model). It can also be inferred indirectly from the stellar MDF, which is then converted to a function of time following the same procedure described
in the previous section (i.e., dividing by the time bin size∆t). I then convolve the SFH with a model DTDΨ(τ)to obtain the expected rate of Type Ia SNe RIa,exp(t). Figure 3.2 shows a number of theoretical DTDs from binary population synthesis models for different types of Type Ia SNe. As expected (see Section 3.1), nearly all double-degenerate models (top panel) produce DTDs that roughly scale as τ−1 after a minimum delay-time, consistent with the delay-time primarily being set by a merger timescale. Single-degenerate Type Ia models (bottom panel), on the other hand, produce far more diverse DTDs, though there are some common features: for example, the DTDs are largely concentrated within the rangeτ∼ 0.1−2 Gyr. This is because companion stars within a narrow mass range are needed to donate material to a white dwarf at suitable accretion rates without causing unstable hydrogen burning.
I can test each of these theoretical models by computing RIa,exp(t) from each pre- dicted DTD, then comparing RIa,exp(t)and RIa,obs(t). To account for errors in these rates, I “bootstrap” our calculations by computing the observed and expected rates in 105bootstrap iterations. In each iteration, I randomly perturb the [Fe/H] of each star, assuming that the true values are distributed normally with standard deviations equal to the measurement errors, then compute a new MDF from the perturbed metallicities; the new MDF is then used to computeRIa,exp(t) andRIa,obs(t).
I then compute the mean absolute deviation (MAD; i.e., the mean of the absolute value of RIa,exp− RIa,obs) as a measure of the goodness-of-fit. I use the MAD rather than the more commonly-used root-mean-square error (RMSE); RMSE is not nec- essarily an ideal statistic for this application due to its sensitivity to large deviations (e.g., Chai and Draxler 2014).