= Solution
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 $v_{\rm rel}$, sound speed $c_s$, and density $\rho_w$. Gravity focuses gas from the accretion radius
$$
R_{\rm acc}\sim\frac{2GM_2}{v_{\rm rel}^2+c_s^2},
$$
giving
$$
\boxed{\dot M_{\rm BHL}\sim
\frac{4\pi G^2M_2^2\rho_w}
{(v_{\rm rel}^2+c_s^2)^{3/2}}}.
$$
For a roughly spherical donor wind, $\rho_w\simeq\dot M_w/(4\pi a^2v_w)$ and $v_{\rm rel}$ 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:
* At short periods, the $5M_\odot$ primary fills its Roche lobe while its envelope is still mainly radiative. If transfer remains stable, rapid mass exchange reverses the mass ratio and produces an <Algol binary>, with an evolved low-mass donor orbiting a rejuvenated main-sequence accretor. Extremely short systems may instead enter contact or merge.
* At intermediate periods, the primary fills its Roche lobe after developing a deep convective giant envelope. The extreme mass ratio makes transfer dynamically unstable, causing a <common envelope>. Successful envelope ejection leaves a close white-dwarf--main-sequence binary; later <magnetic braking of a binary star> or <gravitational-wave emission from a binary system> brings the $0.7M_\odot$ star into contact, producing a <cataclysmic variable>.
* At long periods, the primary reaches the thermally pulsing <Asymptotic giant branch> without filling its Roche lobe. It pollutes the companion through <Bondi–Hoyle accretion> or focused wind transfer and becomes a white dwarf. The polluted $0.7M_\odot$ secondary is observed later as a <barium star> or extrinsic S-type star.
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>.
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