tokens: to the dotted line in Figure 5 . The expected change of gas-phase metallicity due to this process can also be estimated using the toy model worked out in Appendix A . On the one hand, the drop of metallicity after accreting a gas mass Δ M a scales with the accreted gas mass aswhere M g 0 stands for the mass of gas already present in the object and C is a time-dependent parameter that is of the order of 1 after the accretion event (see Equation ( 26 ), with Z in ≪ Z g Z g 0 ). On the other hand, M a / τ in , and thereforeThe logarithmic derivative has been evaluated assuming Δ M a ∼ M g 0 and C ∼ 1. Its actual value can go all the way from −1 (when Δ M a ≫ M g 0 ) to 0 (when Δ M a ≪ M g 0 ). 4. Is the Correlation between Size and Gas-phase Metallicity a Result of Differences in the Depth of the Gravitational Potential of the Galaxies? The short answer to the above question is no . The long answer is elaborated in this section showing that the escape velocity and the metallicity are not correlated in the EAGLE galaxies. In order to parameterize the depth of the gravitational potential, we use the escape velocity at the half-mass radius,with Φ( R ⋆ ) the gravitational potential at R ⋆ . We model Φ as a combination of the potential due to DM, Φ DM , and the potential due to stars, Φ ⋆ ,The EAGLE database does not directly provide the depth of the gravitational potential. We infer the DM component from the DM mass and the half-mass radius of the DM assuming that the density drops with radius following a NFW profile (Navarro et al. 1996 ). The procedure is sketched in Appendix B . The stellar contribution is also inferred from the stellar mass and the half-mass stellar radius, this time assuming that the stellar density drops exponentially with radius. We follow the work by Smith et al. ( 2015 ). We assume the galaxy disks to be thin (scale height of 0.2 kpc), with the escape velocity computed in the plane of the disk. The results are rather insensitive to these assumptions since the potential is dominated by the DM component. This is also the reason why the gas mass is not included in the computation of v esc , because it represents only a minute fraction of the total mass. Figure 7 is similar to Figure 2 , but this time the symbols are color-coded with the escape velocity at the half-mass stellar radius. The spread in gas-phase metallicity is uncorrelated with the escape velocity, which is mostly set by the stellar mass. For a given stellar mass the galaxies have the same escape velocity irrespectively of the metallicity of their gas (cf. Figures 2 and 7 ). This fact is even clearer in Figure 8 . It shows Z g versus v esc for the full set of EAGLE galaxies (top left), as well as for narrow bins in galaxy mass (Δ log M ⋆ ∼ 0.3). Even though the full set shows a global trend for the gas-phase metallicity to increase with the escape velocity, this is just a construct (or a mirage) resulting from the mass–metallicity relation and the
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