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Proto-planets embedded in their natal disks acquire hot envelopes as theygrow 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 itsfurther evolution. Our aim is to describe the formation of planets withpolluted envelopes from the moment that impactors begin to sublimate to beyondthe disk's eventual dissipation. We constructed an analytical interiorstructure model, characterized by a hot and uniformly mixed high-Z vapor layersurrounding the core, located below the usual unpolluted radiative-convectiveregions. The evolution of planets with uniformly mixed polluted envelopesfollows four potential phases. Initially, the central core grows directlythrough impacts and rainout until the envelope becomes hot enough to vaporizeand absorb all incoming solids. We find that a planet reaches runaway accretionwhen the sum of its core and vapor mass exceeds a value that we refer to as thecritical metal mass - a criterion that supersedes the traditional critical coremass. It scales positively with both the pollutant's evaporation temperatureand with the planet's core mass. Hence, planets at shorter orbital separationsrequire the accretion of more solids to reach runaway as they accrete lessvolatile materials. If the solids accretion rate dries up, we identify thedecline of the mean molecular weight - dilution - as a mechanism to limit gasaccretion during a polluted planet's embedded cooling phase. When the diskultimately dissipates, the envelope's inner temperature declines and its vaporeventually rains out, augmenting the mass of the core. The energy release thataccompanies this does not result in significant mass-loss, as it only occursafter the planet has substantially contracted.

How planets grow by pebble accretion II: Analytical calculations on the evolution of polluted envelopes