  Proto-planets embedded in their natal disks acquire hot envelopes as they
grow and accrete solids. This ensures that the material they accrete - pebbles,
as well as (small) planetesimals - will vaporize to enrich their atmospheres.
Enrichment modifies an envelope's structure and significantly alters its
further evolution. Our aim is to describe the formation of planets with
polluted envelopes from the moment that impactors begin to sublimate to beyond
the disk's eventual dissipation. We constructed an analytical interior
structure model, characterized by a hot and uniformly mixed high-Z vapor layer
surrounding the core, located below the usual unpolluted radiative-convective
regions. The evolution of planets with uniformly mixed polluted envelopes
follows four potential phases. Initially, the central core grows directly
through impacts and rainout until the envelope becomes hot enough to vaporize
and absorb all incoming solids. We find that a planet reaches runaway accretion
when the sum of its core and vapor mass exceeds a value that we refer to as the
critical metal mass - a criterion that supersedes the traditional critical core
mass. It scales positively with both the pollutant's evaporation temperature
and with the planet's core mass. Hence, planets at shorter orbital separations
require the accretion of more solids to reach runaway as they accrete less
volatile materials. If the solids accretion rate dries up, we identify the
decline of the mean molecular weight - dilution - as a mechanism to limit gas
accretion during a polluted planet's embedded cooling phase. When the disk
ultimately dissipates, the envelope's inner temperature declines and its vapor
eventually rains out, augmenting the mass of the core. The energy release that
accompanies this does not result in significant mass-loss, as it only occurs
after the planet has substantially contracted.
