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An intrinsic S-type star is a thermally pulsing AGB star whose third dredge-up exposes carbon and products of the slow neutron-capture process. An extrinsic S-type star has similar surface pollution but no current internal source; it received the material from a former AGB companion. A barium star is the warmer main-sequence or giant counterpart, recognized particularly through strong barium and other slow-neutron-capture elements.

This interpretation predicts a white dwarf companion, the remnant of the former AGB donor. Barium and extrinsic S-type stars are indeed binaries, often with periods long enough that the donor could not have undergone ordinary Roche-lobe overflow. Their abundance patterns, white-dwarf companions, and wide or eccentric orbits therefore point to wind mass transfer in a binary star.

The Bondi–Hoyle accretion estimate treats the companion as moving through a locally uniform wind with relative speed $v_{\mathrm{rel}}$, sound speed $c_s$, and density ${\rho }_w$. Gravity focuses gas from the accretion radiusgivingFor a roughly spherical donor wind, ${\rho }_w\simeq {\dot{M}}_w/(4\pi a^2{v}_w)$ and $v_{\mathrm{rel}}$ combines wind and orbital velocities. This supplies an order-of-magnitude accreted fraction; wind acceleration, density gradients, orbital deflection, and Wind Roche-lobe overflow can substantially change it.

For initial masses $5M_{\odot }$ and $0.7M_{\odot }$, increasing the initial period gives three broad channels:

The transition periods are set by the primary's maximum radius relative to its Roche lobe, and their exact values depend on mass-transfer efficiency, wind speed, and common-envelope energy formalism.

After substantial envelope loss on the Asymptotic giant branch, a low- or intermediate-mass star's exposed core heats while its radius shrinks, often at approximately constant luminosity during continued shell burning. It may ionize the expelled envelope as a planetary nebula. After the remaining nuclear fuel is depleted, its luminosity decreases and it joins a white dwarf cooling track. Evolutionary tracks on the Hertzsprung-Russell diagram connect the central-star temperature and luminosity with core mass and elapsed time.

In a negligible-mass white dwarf envelope, take enclosed mass and luminosity constant, neglect radiation pressure, and use an ideal gas with the Kramers' opacity law. Dividing hydrostatic pressure support equation by radiative diffusion in a star gives $PdP\propto (M/L)T^{15/2}dT$. Neglecting the surface integration constant produces the displayed pressure and mass density laws. It is a deep-envelope approximation, not a finite-temperature photospheric boundary condition.

A stellar remnant is the object left after a star finishes its principal nuclear-burning evolution, such as a white dwarf, neutron star or black hole. Stellar remnants retain mass while often contributing little optical light, affecting an old stellar population's mass-to-light ratio.

Temperature-sensitive nuclear fusion can heat material faster than cooling and expansion can regulate it, accelerating the burning. In a classical nova, hydrogen accumulated on a white dwarf ignites; initial electron degeneracy pressure helps suppress the usual expansion response. The eruption can eject the envelope without destroying the underlying white dwarf.

A hypothetical explosion in which a degenerate carbon-oxygen core within a single evolved star grows toward the Chandrasekhar mass and ignites explosive carbon burning while retaining an envelope. This differs from accretion onto an already exposed white dwarf in standard Type Ia supernova scenarios. Evolutionary calculations by Doherty et al. explain the strong dependence of this proposed channel on envelope mass loss; their adopted winds remove the carbon-oxygen envelope before this outcome. It is a proposed possibility under different assumptions, rather than an established endpoint for normal $5M_{\odot }$ stars.