Quantifying Element Importance for Mass Recovery from Population III Supernova Yield Fits

Abstract

Massive Population III stars are currently not observed, but their initial mass function (IMF) can be inferred through stellar archaeology: fitting core-collapse supernova yield models to elemental abundances of low-mass, long-lived metal-poor stars. While prior work demonstrates that yield fitting can recover progenitor properties, it remains unclear which measured elements most control mass recovery quality and what level of IMF precision is achievable for a measured element set. We perform a systematic study of element importance for progenitor mass recovery. Using the Heger & Woosley (2010) yield grid, we generate mock observations, fit the initial mass, and evaluate the typical performance on the fractional mass recovery. Add/remove-one-element experiments and comparisons among different baseline element sets are used to rank elements by importance. We find that the most important elements for accurate mass recovery are C, N, Na, and K, with O, Al, Co, and Ni consistently improving performance when available. Overall, with currently measurable elements from high-resolution spectroscopy, stellar archaeology can deliver practical Population III IMF constraints assuming the core-collapse supernova yield models provide a good representation of stellar evolution in the early universe.

Figures (7)

• Figure 1: Example of mock observation generation and mass recovery for a HW10 model. Top panel: black circles show the original model abundances in $\log \varepsilon$ with $M_{\rm true}=17.4M_{\odot}$, $E_{\rm true}=2.4$B, and $X_{\rm true}=0.0$; blue triangles show one mock observation generated by adding independent Gaussian noise with $\sigma=0.2$ dex to each element (error bars indicate $1\sigma$); orange squares show the best-fit HW10 model returned by a full-grid $\chi^2$ search with $M_{\rm fit}=18.2M_{\odot}$, $E_{\rm fit}=2.4$B, and $X_{\rm fit}=0.251$. In this example, the fit uses the medium coverage baseline defined in Section \ref{['sec 2.2']}. Elements included in the fit are indicated by black element labels and filled markers, while excluded elements are shown with gray labels and open markers. Bottom panel: residuals relative to the original model, shown for the mock observation (blue; Mock$-$Original) and the best-fit model (orange; Best-fit$-$Original). Repeating this procedure over 100 independent noise realizations per model yields the distributions of recovered parameters used in the metric analysis in Section \ref{['Sec 2.3']}.
• Figure 2: Recovery of Pop III star mass from mock observations under three baseline element sets: Low coverage (C, Mg, Ca, Fe), mid coverage (C, Na, Mg, Al, Si, Ca, Ti, Fe, Ni), and high coverage (C, N, O, Na, Mg, Al, Si, K, Ca, Ti, V, Mn, Fe, Co, Ni). Each point shows the mean and standard deviation of the 100 recovered masses $M_{fit}$ obtained from independent mock observations of a single HW10 model with the corresponding $M_{true}$, $E_{true}$$=0.3$, and $X_{true}$$=0$, and noise level $\sigma=0.2$; the gray dashed line denotes the one-to-one relation. The high coverage baseline places nearly all points on the dashed line, indicating near-perfect mass recovery in this process, whereas the low coverage baseline shows substantially larger scatter.
• Figure 3: Demonstration of the metric calculation on the $M_{true}$--$E_{true}$ grid at a fixed mixing $X$. For each HW10 model ($M_{true}$, $E_{true}$), 100 mock observations are generated and fit to recover 100 $M_{fit}$. Points are colored by the fraction of correct recoveries, where a recovery is correct if $\delta = \left | (M_{true} - M_{fit}) / M_{true} \right | < 0.1$: green $76-100\%$, blue $51-75\%$, yellow $26-50\%$, red $0-25\%$. The frame-level metric is $d = (g + 0.5b - 0.5y - r)/N$, where $g, b, y, r$ are counts of points in each color and N is the number of points; $d$ ranges from -1 to 1, with higher values indicating better recovery. Example shown: $X_{true}$ = 0 and the medium coverage baseline element set. Only part of the grid is displayed for clarity.
• Figure 4: Element-by-element perturbations of the baselines. (a) Medium baseline: $\bar{d}$ versus atomic number $Z$ when a single element is modified. The dashed line is the unmodified medium-baseline $\bar{d}$. Purple circle markers indicate the element is removed from the baseline; orange square markers indicate an element is added to the baseline. Greater differences from the dashed line imply a larger effect on the metric. (b-d) Low (red), medium (blue), and high (green) coverage baselines: $\Delta \bar{d} = |\bar{d}_{modified} - \bar{d}_{baseline}|$ versus $Z$. Horizontal dashed lines show the thresholds used to group elements into importance tiers. All results use 100 mocks per model and noise level $\sigma$ = 0.2; $\bar{d}$ is averaged over the 14 mixing values.
• Figure 5: Mapping between the fit quality metric $\bar{d}$ and the fractional mass error $\delta = \lvert(M_{true}-M_{fit})/M_{true}\rvert$. At the $\bar{d}$ attained by ten representative element sets (Fe-Peak, Low, Low+Ni, Even-Z, Mid, Mid+, Mid+N-O-K, High, High+, All), scatter points show all $\delta$ values across the HW10 grid (16,800 models $\times$ 100 mocks $= 1.68\times10^{6}$ points per set). Symbols mark the 50th, 68th, 75th, and 90th percentile thresholds at each $\bar{d}$, and the smooth curves are exponential fits to these thresholds. The 75th-percentile curve is highlighted (thicker line) since it is used to compute $\delta_{75}$ in Figure \ref{['fig:5']}.