Mass loss and final remnants

A steady spherical wind obeys mass conservation through every radius if there are no additional mass sources. Comparing its terminal momentum flux with radiative luminosity divided by light speed defines a useful dimensionless momentum number. A value above unity is not automatically an energy violation: photons can interact more than once and transfer momentum through an optically thick or multiline flow. Whether a particular wind is driven requires a force calculation using its opacity, velocity field, ionization, and geometry.

Energy conservation imposes a different necessary condition. Radiation-powered material must gain enough energy both to escape the gravitational potential and to reach its terminal speed. Neglecting base speed, enthalpy changes, and additional energy sources gives a mechanical luminosity to compare with luminosity entering the wind base. If the demanded mechanical power exceeds that supply, the assumed steady escaping solution cannot be maintained. Satisfying the energy bound is necessary but not sufficient: it does not prove the required local radiative acceleration or a stable mass flux exists.

For a slowly rotating spherical star, an isotropic nonmagnetic wind leaving a rigidly rotating surface carries specific axial angular momentum proportional to the square of the cylindrical launch radius. Averaging the squared sine of colatitude over a sphere supplies a factor of two thirds. This gives an instantaneous angular-momentum loss rate in terms of surface radius and angular velocity. The result assumes uniform mass flux per solid angle and no additional torque once matter has left the surface.

Magnetic stresses can maintain coupling to a larger effective lever arm, while rapid rotation changes surface shape and latitude-dependent flux. Those cases require a different torque model. Even the elementary torque does not alone determine the future angular velocity, because the stellar moment of inertia changes with mass, radius and internal redistribution. A small-step loss fraction can be estimated from the initial torque, but holding the initial angular velocity and radius fixed throughout a large evolutionary change would be inconsistent. Structural spherical symmetry also requires centrifugal acceleration to remain small relative to gravity.

A star that sheds its envelope without proceeding through all advanced burning stages can leave a cooling white dwarf supported predominantly by electron degeneracy. The cold nonrelativistic mass–radius approximation eventually changes as electrons become relativistic; finite temperature, composition, interactions and captures matter for quantitative limits. An envelope lost early through binary interaction can leave a different core composition than isolated single-star evolution would produce. A stellar wind estimate alone therefore does not select whether the remaining white dwarf is helium-rich, carbon–oxygen-rich, or another composition.

More advanced cores can undergo collapse, with neutron-star or black-hole outcomes affected by dense-matter physics, explosion dynamics, neutrino transport, rotation and fallback. Initial mass is informative but is not a universal one-to-one remnant label across metallicities, binary histories and mass-loss prescriptions. We assert no numerical observational threshold here. A useful bookkeeping separation is to evolve core growth and envelope stripping together, then ask which support and nuclear processes remain available. The loss of a visible envelope and the collapse of a compact core are physically distinct events with different timescales.