tokens: the observed CMD of core He-burning stars. Figure 7. CMDs of Hodge 6 and NGC 1978, the two youngest clusters in our sample. Observations are denoted with grey open triangles, synthetic HB models are overplotted in red. From left to right: Δ Y = 0 models, models with Δ Y > 0 (η R fixed to the best-fitting value used in the left-hand panel), and with Δη R > 0 ( Y fixed to the He-normal value of the left-hand panel). The arrows describe the direction along which variations in Y and mass-loss work, the amplitude being arbitrary here. If we make the assumption that η R must be constant, in agreement with the results for the other clusters in our sample, we would obtain Δ Y ∼0.06 for Hodge 6, and Δ Y ∼0.04 for NGC 1978. But without this assumption, the CMD analysis does not discriminate between a spread in η R or in Y for these two clusters. However, we also note that Hodge 6 has the largest photometric errors of any of the clusters in our sample, adding further uncertainty for this cluster. 5 DISCUSSION We have determined the total initial He abundance spread Δ Y in a sample of intermediate-age, massive LMC and SMC clusters – and the old cluster NGC 121 – by reproducing the shape and colour extension of their HB/RC stars with synthetic HB models. Our derived Δ Y values are shown in Table 2 . The typical error on these estimates of Δ Y is below 0.01. Table 2. Initial helium abundance spread derived from the HB fitting. The values in parenthesis are determined in the assumption that the RGB mass-loss does not vary amongst clusters’ stars, because of a degeneracy between the effects of mass-loss spread and He spread in these clusters (see the text for details). We can compare our results with the estimates by Lagioia et al. ( 2019 ). These authors found spreads of initial He abundances equal to 0.009 ± 0.006, 0.007 ± 0.004, 0.010 ± 0.003, and 0.000 ± 0.004 for NGC 121, NGC 339, NGC 416, and Lindsay 1, respectively. These values are clearly smaller than our results in Table 2 . But as mentioned already in the Introduction section, the method employed by Lagioia et al. ( 2019 ) most likely determines mean abundance spreads amongst the cluster subpopulations, whereas our modelling tends to determine the maximum abundance spread, irrespective of the exact distribution of initial He abundances. This is quite clear by looking at the HB of NGC 121 in Fig. 4 . The bulk of the HB population has ( m F555 W − m F814 W ) > 0.8, consistent with a negligible Δ Y with just a plume of stars extending towards bluer colours and brighter magnitudes, that is, with significantly different initial Y . The values in Table 2 are also plotted in Fig. 8 as a function of the mass of the host cluster. In the same figure, we display also the maximum initial Y spread determined for a sample of Galactic GCs by Milone et al. ( 2018 ). Milone et al. ( 2018 ) found a trend between Δ Y and the mass of the host cluster, that is clearly visible in Fig. 8 and the results for our clusters
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