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We revisit secular stability against quasi-radial collapse for rigidlyrotating supermassive stars (SMSs) in general relativity. We suppose that theSMSs are in a nuclear-burning phase and can be modeled by polytropic equationsof state with the polytropic index $n_p$ slightly smaller than $3$. Thestability is determined in terms of the turning-point method. We find a fittingformula of the stability condition for the plausible range of $n_p$ ($2.95 \altn_p \alt 3$) for SMSs. This condition reconfirms that, while non-rotating SMSswith mass $\sim 10^5M_\odot$--$10^6M_\odot$ may undergo ageneral-relativistically induced quasi-radial collapse, rigidly rotating SMSswith a ratio of rotational to gravitational potential energy ($\beta$) of $\sim10^{-2}$ are likely to be stable against collapse unless they are able toaccrete $\sim 5$ times more mass during the (relatively brief) hydrogen-burningphase of their evolution. We discuss implications of our results.

Stability of rigidly rotating supermassive stars against gravitational collapse