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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.

In a circular binary star, the distances from the centre of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing the two orbital angular momenta givesEquivalently, $J=\mu \sqrt{GMa}$ by Kepler third law.

First consider a rapid conservative perturbation. Both $M=M_1+M_2$ and $J$ are fixed, while $d{M}_1=-d{M}_2$. With $q=M_2/M_1$,soThe Roche lobe formula then gives its mass-radius exponentAfter mass loss, dynamical stability of binary mass transfer requires the donor to shrink relative to its lobe. Since $d\log M_2<0$, this means ${\zeta }_{\mathrm{ad}}>{\zeta }_L$, or

For stable secular conservative binary mass transfer, put $m={\dot{M}}_2/M_2$. The angular-momentum and contact conditions areElimination of $\dot{a}/a$ givesHere magnetic braking of a binary star removes orbital angular momentum, while the donor's expansion maintains Roche-lobe overflow.

In a cataclysmic variable, hydrogen-rich material accumulates on a degenerate white dwarf. Degeneracy prevents initial expansion from regulating its temperature, so nuclear ignition produces the thin-shell instability and a classical nova. Nuclear burning of hydrogen to helium releases about $0.007m{c}^2$, whereas the binding energy at a white-dwarf surface is only of order $G{M}_{1}m/R_1$, typically a few $10^{-4}m{c}^2$. Even modest coupling can therefore eject all the newly accreted envelope without disrupting the white dwarf.

Finally suppose every transferred mass element is expelled by isotropic re-emission from a binary star. Then ${\dot{M}}_1=0$, $\dot{M}={\dot{M}}_2$, and expelled matter carries the white dwarf's specific angular momentum $j_1=(M_2/M{)}^2{a}^2\Omega$. HenceOn the other hand, logarithmic differentiation of $J=M_1{M}_2{M}^{-1/2}{G}^{1/2}{a}^{1/2}$ and of the Roche-lobe radius givesEliminating the separation produces