We introduce and systematically study the length of the slow stable hybrid star branch as a quantitative measure of the extended stability region that arises in hybrid neutron stars when the hadron-quark phase conversion is slow compared to the radial oscillation timescale. Combining generalized piecewise-polytropic hadronic equations of state of varying stiffness with a constant-speed-of-sound quark-matter model, we construct a large set of hybrid equations of state spanning a broad range of transition pressures, energy-density jumps, and quark-matter speeds of sound. We identify four morphological types for the slow stable branch in the mass-radius plane: waterfall branches that descend monotonically from the hadronic maximum mass, bridges that connect the hadronic branch to a second unconditionally stable hybrid branch, tails that extend briefly beyond the maximum mass of an unconditionally stable hybrid branch, and tail bridges that combine features of the latter two. Their prevalence is governed primarily by the transition pressure and the energy-density jump, while the branch length is also significantly influenced by the stiffness of the hadronic sector and the quark-matter speed of sound. Imposing current astrophysical and microphysical constraints shows that viable long branches are predominantly of a waterfall type, and that stiff hadronic equations of state—strongly disfavored under the rapid-conversion assumption—remain compatible with all current constraints within the slow-conversion framework. In the plane of transition baryon density versus density jump, slow stable configurations open a new region of viable parameter space inaccessible under rapid conversions.

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• Gravitational wave sources
• Nuclear matter in neutron stars
• QCD phase transitions
• Quark matter
• Neutron stars & pulsars