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tokens: for this drop is that the rcb has been pushed closer to the surface of the underlying rock core, and with the envelope more nearly completely radiative, differences between how we treat the radiative transfer become magnified. Extremely gas-poor environments demand that atmospheric density profiles be sufficiently steep to contain a given amount of gas within the planet's Hill sphere, and this steepening pushes the thermal photosphere deeper into the envelope (compare the black dotted curves in the right-hand and second-from-right panels of Fig. 5 ); the rcb is, of course, pushed inwards as well. For ρ out /ρ MMEN = 10 −8 , the thermal photospheres of the dusty and dust-free optically thin models are only about twice the radius of the rocky core, and the corresponding rcb densities and opacities are 2–10 times larger than those of the radiative diffusion models. These larger rcb densities and opacities lower the radiative luminosities, slowing cooling and by extension accretion – hence the especially low GCRs seen in Fig. 4 characterizing ρ out /ρ MMEN = 10 −8 . At even lower nebular densities (data not shown), the rcb can be pushed all the way to the rocky core – i.e. the envelope becomes purely radiative – at which point the model sequence terminates (see also the isothermal endmember models in Lee Chiang 2015 , their fig. 4 and related discussion). Although the models shown in Fig. 4 are technically those of nebulae whose gas densities are fixed from t = 0 to 1 Myr and drop to zero thereafter, they are still reliable guides for nebulae whose gas densities deplete smoothly over time. The very fact that rates of gas accretion change only by factors of ∼3 when the nebular density varies by four orders of magnitude suggests that the detailed time history of nebular depletion introduces, at most, only order-unity effects. This expectation is confirmed in Fig. 6 , which shows how super-Earths amass ∼4–10 per cent-by-mass gas envelopes in a nebula that decays exponentially with an e-folding time-scale of 0.1 Myr. Figure 6. How 5 M ⊕ cores build their envelopes in an exponentially depleting nebula. The gas-to-core mass ratio (GCR ≡ M gas / M core ; solid and dashed curves using the right-hand axis) are computed by first fitting a scaling relationship to our numerical results for GCR versus nebular density (we find GCR ), and then integrating over the time derivative of GCR as the nebular gas depletes with an e-folding time of 0.1 Myr (grey curve using the left-hand axis). We halt envelope growth once ρ out /ρ MMEN falls below 10 −8 since the rate of gas accretion slows considerably (by about an order of magnitude compared to ρ out /ρ MMEN = 10 −7 ; see Fig. 4 ) at this level of depletion. 4 DISCUSSION 4.1 Creating super-puffs beyond 1 au So far in this paper, we have concentrated exclusively on planets located 0.1 au from their central stars. This restriction is due to our use of opacities generated from the model of Ferguson et al. ( 2005 ), which