The stretched hoop parameter |$\mathcal{H}_{\rm D}$| as a function of |$R/GM$| in the ring initial data. The value |$\mathcal{H}_{\rm D}=0.9754$| is indicated by a horizontal dashed line, and the marginally DTT sphere forms if |$\mathcal{H}_{\rm D}\le 0.9754$| holds. The value |$\mathcal{H}_{\rm D}=1.0$| is also indicated for comparison.