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tokens: et al. 2017 ). To decouple the effect of Z and Y , we compute five tracks of mass 1.50 M ⊙ and Z = 0.031 ([Fe/H] = 0.25), but with initial helium Y = 0.25, 0.28, 0.30, 0.35 and 0.40 (with Y = 0.28 as our default value). Fig. 4 , right-hand panel, shows the evolution of ΔΠ 1 with central helium for the five tracks. We note that the effect on ΔΠ 1,max and ΔΠ 1,min grows linearly with Y . However, for a variation of Δ Y = ∼0.02 from the default value ( Y = 0.25 and 0.30), the deviation on ΔΠ 1,max (ΔΠ 1,min ) lies within the observed uncertainty on ΔΠ 1 (∼3–4 s). The deviation we tested is compatible with typical spread on disc populations (Casagrande et al. 2007 ). Nevertheless, the effect becomes substantial for extreme enrichments, for example, in bulge or globular clusters where populations with very different He abundances and the same metallicity may coexist (Renzini et al. 2015 ). Figure 4. Left-hand panel: RC-ΔΠ 1 evolution as a function of central helium mass of an M = 1.50 M ⊙ solar metallicity star with three different mixing schemes and four different 12 C(α, γ) 16 O reaction rates. Right-hand panel: RC-ΔΠ 1 evolution as a function of central helium mass fraction of an M = 1.50 M ⊙ star computed assuming three different combinations of Z and Y . 5 ADDITIONAL UNCERTAINTIES ON ΔΠ<sub><bold>1</bold></sub>: <sup><bold>12</bold></sup>C(α, γ)<sup><bold>16</bold></sup>O NUCLEAR REACTION RATE As shown above, ΔΠ 1 is strongly dependent on assumptions related to core convection and metallicity; however, additional parameters and uncertainties may also have impact on ΔΠ 1 , in particular the 12 C(α, γ) 16 O reaction rate, which, along with triple-α, plays a fundamental role, especially at the end of the HeCB (e.g. see Cassisi et al. 2003 ; Straniero et al. 2003 ; Constantino et al. 2015 ). We compute a series of HeCB evolutionary tracks ( M = 1.5 M ⊙ , solar abundance) in which we adopt four 12 C(α, γ) 16 O reaction rates in conjunction with three mixing schemes: Bare–Schwarzschild model (BS), 1.0 H p step function overshooting (HOV) and 1 H p penetrative convection (HPC). The 12 C(α, γ) 16 O reaction rates considered are the tabulated values given by JINA (Cyburt et al. 2010 ), K02 (Kunz et al. 2002 ), CF88 (Caughlan Fowler 1988 ) and NACRE (Angulo et al. 1999 ), and already made available in mesa . While no difference can be noted at the beginning of the phase, the impact of the different mixing schemes is evident at the maximum period spacing (end of the HeCB), where HOV tracks show a scatter of around 6–7 s between them, compared to only ∼2 s for BS (Fig. 4 ). We therefore expect an uncertainty between 6 and 2 s on the MPC and MOV models. This value is comparable with the observed average ΔΠ 1 uncertainty for clump stars (∼4 s). 6 DISCUSSION AND CONCLUSION The precise measurements of gravity-mode period spacing (ΔΠ 1 ) in thousands of HeCB stars have opened the door to detailed studies of the near-core structure of such stars, which