Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 322 / 3

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_{rel}

c_{s}

ρ_{w}

. Gravity focuses gas from the accretion radius

R_{acc} \sim \frac{2 G M _{2}}{v _{rel}^{2} + c _{s}^{2}},

\dot{M}_{BHL} \sim \frac{4 π G ^{2} M _{2}^{2} ρ _{w}}{( v _{rel}^{2} + c _{s}^{2} ) ^{3/2}} .

For a roughly spherical donor wind,

ρ_{w} ≃ \dot{M}_{w} / (4 π a^{2} v_{w})

and

v_{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

5 M_{⊙}

and

0.7 M_{⊙}

, increasing the initial period gives three broad channels:

At short periods, the $5 M_{⊙}$ 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 $0.7 M_{⊙}$ 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 $0.7 M_{⊙}$ 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.

There are no discussions about this article yet.

There are currently no matching articles.