  Super-Earths and sub-Neptunes are commonly thought to have accreted
hydrogen/helium envelopes, consisting of a few to ten percent of their total
mass, from the primordial gas disk. Subsequently, hydrodynamic escape driven by
core-powered mass-loss and/or photo-evaporation likely stripped much of these
primordial envelopes from the lower-mass and closer-in planets to form the
super-Earth population. In this work we show that after undergoing core-powered
mass-loss, some super-Earths can retain small residual H/He envelopes. This
retention is possible because, for significantly depleted atmospheres, the
density at the radiative-convective boundary drops sufficiently such that thhe
cooling time-scale becomes less than the mass-loss time-scale. The residual
envelope is therefore able to contract, terminating further mass loss. Using
analytic calculations and numerical simulations, we show that the mass of
primordial H/He envelope retained as a fraction of the planet's total mass,
$f_{ret}$, increases with increasing planet mass, $M_{c}$, and decreases with
increasing equilibrium temperature, $T_{eq}$, scaling as $f_{ret} \propto
M_{c}^{3/2} T_{eq}^{-1/2} \exp{[M_{c}^{3/4} T_{eq}^{-1}]}$. $f_{ret}$ varies
from $< 10^{-8}$ to about $10^{-3}$ for typical super-Earth parameters. To
first order, the exact amount of left-over H/He depends on the initial envelope
mass, the planet mass, its equilibrium temperature, and the envelope's opacity.
These residual hydrogen envelopes reduce the atmosphere's mean molecular weight
compared to a purely secondary atmosphere, a signature observable by current
and future facilities. These remnant atmospheres may, however, in many cases be
vulnerable to long-term erosion by photo-evaporation. Any residual hydrogen
envelope likely plays an important role in the long-term physical evolution of
super-Earths, including their geology and geochemistry.
