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In the dominant central Newtonian gravitational potential, conservation of energy for gas falling from rest at infinity gives $u^2/2=G{M}_c/R$, hence $u=a{R}^{-1/2}$ with $a=(2G{M}_c{)}^{1/2}$. The characteristic stellar speed has the same radial scaling. For random stellar directions, the cross term in $|\mathbf{u}-\mathbf{v}{|}^2$ averages to zero, so the typical relative speed is also $V=b{R}^{-1/2}$. With the stated kinetic-energy estimate, an rms convention gives $V^2\simeq 4G{M}_c/R$; another convention changes only the constant $b$.

In the steady spherical inflow approximation, mass conservation gives $A=4\pi R^2\rho u$. ThusThis leading calculation treats the inward gas flux as essentially undepleted over the region used. Substantial distributed removal by stars would make the local flux a function of $R$, an important limitation discussed below.

Insert these scalings in the Bondi--Hoyle--Lyttleton accretion rate. The factors $R^{3/2}$ from $V^{-3}$ and $R^{-3/2}$ from $\rho$ cancel:This is quadratic accretion in a central Keplerian potential. Write its fixed coefficient as $\lambda =\Lambda$. Integrating $d(1/M)/dt=-\lambda$ givesThe cancellation of radius means that orbital variation of $R$ alone does not change this coefficient, provided the assumed steady gas profile, central mass and relative-velocity scaling remain fixed.

Conserve the number of stars along these deterministic mass trajectories. Inverting the map and taking its Jacobian determinant givesThe mass-spectrum transport under quadratic accretion is thereforeFor a narrow but nonzero initial mass interval, the mapping stretches the high end especially strongly. If its endpoints are $m_1<m_2$, their evolved ratio is $(m_2/m_1)(1-\lambda m_{1}t)/(1-\lambda m_{2}t)$, which can become large as $t$ approaches $1/(\lambda m_2)$ from below. In an intermediate high-mass range with $\lambda Mt\gg 1$, mapped back into a portion of the initial interval where $F_0$ changes little,Here $M_{\ast }$ is near $1/(\lambda t)$ on the populated side of the initial interval. Finite initial support gives a finite upper mass cutoff at every time before the earliest blow-up; this is an approximate broad-range power law, not a literal infinite tail of a finite pre-blow-up population. A strictly monochromatic initial population remains monochromatic under identical accretion coefficients and does not generate a spectrum.

The limitations of this accretion model are substantial. The star cluster must remain centrally dominated, and both the central mass and gas supply must remain effectively fixed. Stellar capture depletes the gas flux and eventually invalidates the $R^{-3/2}$ profile. Gas pressure, angular momentum, shocks, radiative feedback and winds can reduce capture relative to the cold Bondi–Hoyle accretion estimate; a broad relative-speed distribution is not faithfully represented by one typical speed, especially because capture scales as $V^{-3}$. A growing star can perturb the surrounding potential, overlap capture regions, change its orbit through momentum loading, or compete with its neighbors for the same gas. Finally, number conservation excludes mergers, new star formation and stellar losses. The formal finite-time mass divergence cannot persist under a finite available gas mass, so the inferred mass-spectrum slope applies only while these approximations remain adequate.

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.