A formula for the *-core of an ideal

Expanding on the work of Fouli and Vassilev \cite{FV}, we determine a formula for the *-$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}$ of an ideal in two different settings: (1) in a Cohen–Macaulay local ring of characteristic $p>0$, perfect residue field and test ideal of depth at least two, where the ideal has a minimal *-reduction that is a parameter ideal and (2) in a normal local domain of characteristic $p>0$, perfect residue field and $\text{m}$-primary test ideal, where the ideal is a sufficiently high Frobenius power of an ideal. We also exhibit some examples where our formula fails if our hypotheses are not met.