Time required to spin up the IMBH from a★ = 0 to near-maximum via controlled accretion — the Phase 1 engineering bottleneck
When mass is accreted from the innermost stable circular orbit (ISCO) of a thin prograde disk, the specific angular momentum deposited per unit mass is L_ISCO. Bardeen (1970) and Thorne (1974) showed that the spin parameter evolves approximately as:
ΔM/M ≈ 1 − √(r_ISCO(a_f)/r_ISCO(a_0))
where r_ISCO is in units of the gravitational radius r_g = GM/c² and is given by the Bardeen-Press-Teukolsky formula:
z₁ = 1 + (1−a²)^(1/3)[(1+a)^(1/3) + (1−a)^(1/3)]
z₂ = √(3a² + z₁²)
r_ISCO(a) = 3 + z₂ − √[(3−z₁)(3+z₁+2z₂)]
For a=0: r_ISCO=6 (Schwarzschild ISCO = 3 Schwarzschild radii). For a=0.998: r_ISCO≈1.237.
This calculator uses the Thorne (1974) result that the fractional mass needed to spin up from a₀ to a_f is proportional to the fractional change in √(r_ISCO). Specifically:
ΔM/M₀ = 1 − √(r_ISCO(a_f)/r_ISCO(a₀))
This is derived from the ratio of ISCO binding energies. The time is then simply ΔM/Ṁ. The Thorne limit a=0.998 arises from photon capture: above this spin, counter-rotating photons from the disk are gravitationally captured by the hole faster than co-rotating photons add angular momentum, creating a natural equilibrium.
Spinning a black hole from a=0 to a=0.998 requires accreting ~54% of the initial mass. From a=0 to a=0.9 requires ~38%; from a=0 to a=0.5 requires ~16%. This is because most of the angular momentum is extracted from the final stages of approach to r_ISCO — the deeper the ISCO, the more angular momentum per unit accreted mass, but also the larger the total mass needed to close the gap from Schwarzschild. For an 8,200 M☉ IMBH at the tidal capture rate of 10⁻³ M☉/yr, reaching a=0.95 would require ~3.5 Myr (ΔM ≈ 3,500 M☉ ÷ 10⁻³ M☉/yr) — short compared to OC's age, but dependent on the accretion rate assumption.
The BZ power before spin-up uses the starting spin a₀ (with a minimum of 0.01 to avoid exactly zero). The BZ power after uses the final mass M_f and target spin a_f. The power ratio illustrates the engineering motivation: a factor of ~30–100× increase in available BZ power after spin-up, driving the MTH Phase 2 infrastructure.
Phase 1 requires billions of years of passive tidal captures — the natural accretion rate is too slow to be deliberately engineered without already having the power output that Phase 2 would provide. The OCS MTH treats this as a hard boundary condition: either the IMBH's natural spin state is already high (possible if it formed from a merger or AGN episode), or Phase 1 is the multi-Gyr waiting period before any engineering is feasible.
Omega Centauri's age is estimated at 12.1 ± 1 Gyr (Baumgardt et al. 2019, MNRAS 482:5138; based on main-sequence turnoff and white dwarf cooling). The OC-age fraction shown is the spin-up time relative to this baseline — values above 1.0 mean the process would take longer than OC has existed.