On the Precision of Full-spectrum Fitting of Stellar Populations. III. Identifying Age Spreads.

In the first paper of this series (Asa’d & Goudfrooij, 2020, hereafter paper I), we investigated the precision of the ages and metallicities of 21,000 mock simple stellar populations (SSPs) determined through full-spectrum fitting in the optical range. We found that for S/N $\geq$ 50, this technique can yield ages of SSPs to an overall precision of $\Delta\,\mbox{log(age/yr)}\sim 0.1$ for ages in the ranges 7.0 $\leq$ log (age/yr) $\leq$ 8.3 and 8.9 $\leq$ log (age/yr) $\leq$ 9.4. For the age ranges of 8.3 $\leq$ log (age/yr) $\leq$ 8.9 and log (age/yr) $\geq$ 9.5, the age uncertainty rises to about $\pm 0.3$ dex. In the second paper of the series (Goudfrooij & Asa’d 2021, hereafter paper II) we studied the influence of star cluster mass through stochastic fluctuations of the number of stars near the top of the stellar mass function, which dominate the flux in certain wavelength regimes depending on the age. Studying several wavelength intervals in the 0.35 – 5.0 $\mu$m range, we found that in general, the spectral shape in the blue optical regime (0.35 – 0.7 $\mu$m) is by far the least impacted by stochastic fluctuations among the wavelength intervals studied. In both previous studies we followed the assumption that star clusters as SSPs, considering all constituent stars to have the same age and metallicity. However, most old globular clusters are now known to host abundance variations of several light elements (mainly He, C, N, O, Na, and Al; see Bastian & Lardo, 2018, for details). This phenomenon is often referred to as Multiple Stellar Populations (MSPs). Interestingly, MSPs have not yet been identified in clusters younger than $\sim$ 2 Gyr (see Martocchia et al., 2018; Asa’d et al., 2020; Cabrera-Ziri et al., 2020; Li et al., 2020). However, if young massive clusters (YMCs) are the young analogs of Globular Clusters (GCs), one might expect to find MSPs in YMCs. The role that age plays in the MSP phenomenon has been difficult to constrain (e.g., Cabrera-Ziri et al., 2020). Additionally, several studies, based on accurate Hubble Space Telescope (HST) photometry and Gaia have revealed that the color-magnitude diagram (CMD) of stellar clusters younger than $\sim$ 2 Gyr exhibit an extended main sequence turn off (eMSTO) which is not due to photometric errors, field-star contamination, differential reddening, or non-interacting binaries (e.g. Bertelli et al., 2003; Mackey & Broby Nielsen, 2007; Milone et al., 2009; Goudfrooij et al., 2009; Cordoni et al., 2018, and references therein). These findings have challenged the traditional picture that the CMDs of star clusters are well described by SSPs, and have triggered huge efforts to understand eMSTOs. The eMSTO phenomenon was initially proposed to be due to multiple stellar formation events, challenging the conventional belief that star clusters are groups of coeval stars. However, alternative scenarios to explain this phenomenon were soon proposed as well. The effects of stellar rotation, first proposed by Bastian & de Mink (2009), is currently one of the most widely accepted scenarios (see, e.g., Dupree et al., 2017; Marino et al., 2018; Bastian et al., 2018; Sun et al., 2019). However, it is still argued that age spreads may be present in clusters (see Goudfrooij et al., 2017, 2018; Gossage et al., 2019; Costa et al., 2019).

Integrated-light spectra of star clusters have proven to be accurate tools for studying extragalactic clusters for which spatially resolved data is not available (Ahumada et al., 2002; Puzia et al., 2005; Puzia et al., 2006; Santos et al., 2006; Palma et al., 2008; Talavera et al., 2010; Cid Fernandes & Gonzalez Delgado, 2010; Asa’d et al., 2013; Asa’d, 2014; Chilingarian & Asa’d, 2018). In this work we aim to address the question: To what extent can integrated-light spectra be used to identify age spreads at different ages?

In order to investigate the role of the size of the age gap on the derived parameters, we create a new set of 59,400 mock cluster spectra with larger age step:

and repeat the analysis aiming to derive Age(SSP₁), Age(SSP₂) and $f$ from the full-spectrum fitting technique. Figure 6 shows that the mean of both Age(SSP₁) and Age(SSP₂) is closer to the real values for mock clusters with Age(SSP₂) - Age(SSP₁) = 0.5 dex. Despite the increase in the age gap, our results show that the full spectrum fitting technique still overestimates Age(SSP₂) consistently for Age(SSP₂) < 8.5. Figure 7 shows the success rate as a function of input age. For Age(SSP₁), the success rate depends on both the mass fraction $f$ and the input age. For mass fraction $f$ = 10, when SSP₂ is dominating, the success rate is very poor, however for mass fraction $f$ > 30 the success rate increases significantly, with drops seen around log (age/year) = 8.0 and log (age/year) > 9.4 due to the similarity of the SEDs in the age ranges where the success numbers are low. When comparing the success rate of both Age(SSP₁) and Age(SSP₂) for mock clusters that have age gap = 0.5 dex with that of mock clusters that have age gap = 0.2 dex, we notice that the numbers are higher for age gap = 0.2. The reason is that by definition the success rate is the number of times the derived age matches the real age to within $\pm$ 0.1 dex, hence smaller age gaps are closer to this precision limit. The mean of the derived age spread for an input age gap of 0.5 is shown in Figure 9. The results are better than for an age gap of 0.2, although the recovery rate of success to within 0.1 dex (lower panel) did not change significantly.

Now we discuss the precision in recovering the mass fraction for the two cases analyzed in the previous sections (i.e., $\Delta$ log (age/yr) = 0.2 and 0.5). Figure 8 shows the mean of the derived mass fraction as a function of the input age. Overall the derived mass fraction is overestimated for low mass fractions and underestimated for high mass fractions. The best results are achieved when the mass fraction is around 50%.

We investigate the accuracy and precision of recovering possible age spreads within a star cluster using the full-spectrum fitting method. We analyze optical spectra of 118,800 mock star clusters with mass fractions from 10% to 90% for two age gaps (0.2 and 0.5 dex) in the age range 6.8 $<$ log (age/yr) $<$ 10.2. Random noise is added to the model spectra to achieve S/N ratios between 50 to 100 per wavelength pixel. We summarize our results in the following points:

A) The results obtained using the full-spectrum fitting method to derive the age of a star cluster that has an age spread of 0.2 dex, depend on the age and the mass fraction of each of the combined SSPs. If one SSP is dominating the spectrum, the full-spectrum fitting technique correctly predicts the age of the dominating SSP. When the contribution of the two SSPs is roughly equal, the full-spectrum fitting technique correctly predicts the average age value of the two SSPs [i.e., ((SSP₁)+(SSP₂))/2]. For all ages and mass fraction values, there is no significant dependence on the S/N value in the range 50 – 100.
B) When using the full-spectrum fitting method to derive combinations of ages with different mass fractions, the mean of the derived Age(SSP₁) matches the real Age(SSP₁) to within 0.1 dex up to ages around log (age/yr) = 9.5. The precision decreases for log (age/yr) $>$ 9.6 for all mass fractions or S/N values. This is because SSP SEDs are very similar in that age range, which makes it difficult to identify their correct age.
C) For young ages (log (age/yr) $\la$ 8.0) the full spectrum fitting technique tends to derive combinations of a young age with a significantly older age. This is because young populations (log (age/yr) $\la$ 7.0) produce significantly more flux than old populations, hence different combinations of young populations with old populations will have very similar $\chi^{2}$ values, causing the large uncertainty for the mean of the derived Age(SSP₁) and Age(SSP₂) as shown in Figure 2.
D) The success rate (defined as the number of times the derived parameter matches the input parameter $\pm$ 0.1 dex) peaks around log (age/year) = 7.2 and 9.0 and drops around 8.6 and $>$ 9.5 due to the similarity of SED shapes for those ages.
E) Increasing the age gap in the mock clusters improve the derived parameters. However, Age(SSP₂) is still overestimated for Age(SSP₂) < 8.5.
F) The derived mass fraction is overestimated for low mass fraction values and underestimated for high mass fraction values despite the age gap size. The best results are derived when the mass fraction is around 50%.

RA thanks the Space Telescope Science Institute for a sabbatical visitorship including travel and subsistence support as well as access to their science cluster computer facilities. The initial analysis of this work was done using the Geospatial Analysis Center (GAC) computers at the American University of Sharjah. We acknowledge the support of GAC and IT staff. This work is based on work supported in part by the FRG17-R-06 and EFRG-18-SET-CAS-74 grants P.I., R. Asa’d from American University of Sharjah and Mohammed Bin Rashid Space Center (MBRSC) grant 201602.SS.AUS P.I. R. Asa’d at the American University of Sharjah.

The spectra of the mock star clusters and model SSPs created for this paper are available upon request from the corresponding author.