  We analyze the interiors of HD~219134~b and c, which are among the coolest
super Earths detected thus far. Without using spectroscopic measurements, we
aim at constraining if the possible atmospheres are hydrogen-rich or
hydrogen-poor. In a first step, we employ a full probabilistic Bayesian
inference analysis in order to rigorously quantify the degeneracy of interior
parameters given the data of mass, radius, refractory element abundances,
semi-major axes, and stellar irradiation. We obtain constraints on structure
and composition for core, mantle, ice layer, and atmosphere. In a second step,
we aim to draw conclusions on the nature of possible atmospheres by considering
atmospheric escape. Specifically, we compare the actual possible atmospheres to
a threshold thickness above which a primordial (H$_2$-dominated) atmosphere can
be retained against evaporation over the planet's lifetime. The best
constrained parameters are the individual layer thicknesses. The maximum radius
fraction of possible atmospheres are 0.18 and 0.13 $R$ (radius), for planets b
and c, respectively. These values are significantly smaller than the threshold
thicknesses of primordial atmospheres: 0.28 and 0.19 $R$, respectively. Thus,
the possible atmospheres of planets b and c are unlikely to be H$_2$-dominated.
However, whether possible volatile layers are made of gas or liquid/solid water
cannot be uniquely determined. Our main conclusions are: (1) the possible
atmospheres for planets b and c are enriched and thus possibly secondary in
nature, and (2) both planets may contain a gas layer, whereas the layer of HD
219134 b must be larger. HD 219134 c can be rocky.
