Homologously Collapsing Stellar Cores

We investigate the collapse of nonrotating gas sp~eres with a polytropic.equ~tion of state.: n = 3, corresponding toy = 4/3. Such polytropes provide~ re~sonable approximatiOn t<? collapsmg stellar cores during the early phase before nuclear density IS reached. W~ find a.famtl~ of ~xact homologously collapsing configurations. Homologous collapse of the enti~e core Is possib~e If the pressure at a given density is reduced by up to 3 '7o from the value for a margmally stable static c~re. For a greater pressure reduction, an inner core can collapse homologou~ly, the mass o~ which varies as the 3/2 power of the reduced pressure at the onset of collapse. Lmear perturbat~ons of these homologously collapsing solutions are separable in space and time. Low order radial and nonradial modes are calculated, and it is found that all modes are essentially stable. Subject headings: stars: collapsed stars: interiors stars: supernovae