Demo · MTH Engineering

Spin-Up Economics: Is Phase 1 Viable?

Four tools chain from stellar capture rate to BZ power output — quantifying the Phase 1 engineering timeline and identifying the bottleneck.

4 tools · ~20 min · OCS Phases 1–3
⚙ Choose an accretion regime

Phase 1 spin-up takes very different amounts of time depending on whether the IMBH accretes only from natural tidal captures or from an actively managed feeding programme. Select a scenario to load all four tools with matching parameters.

01
Step 1 · Fuel Supply
The Capture Rate Sets the Clock

Phase 1 of the MTH plan requires spinning up the IMBH via accretion from the ISCO. The fuel comes from stars that wander within the tidal capture radius. In OC’s current core density, the tidal capture rate is estimated at ~10−³ M⊙/yr — roughly one 1 M⊙ star every thousand years. This is the natural rate set by two-body relaxation in OC’s core. Phase 2 involves active stellar deflection to increase this rate by two orders of magnitude, but that requires pre-existing civilisational infrastructure — which itself requires Phase 1 to have been completed first. Natural accretion is therefore the only bootstrapping pathway.

Open Tidal Capture Calculator → mass=8200 Newtonian dynamics
Step payoff
The tidal capture rate is the fundamental clock speed of Phase 1. It is set by OC’s observed core density — a measured quantity — not a free parameter.
02
Step 2 · Mass Accumulation
How Much Mass Is Needed?

Spinning up from a★ = 0 to 0.95 by thin-disc accretion follows the Bardeen (1970) / Thorne (1974) relation M_f/M_0 = √(6/rISCO), with rISCO in units of GM_f/c². At a★ = 0.95, rISCO = 1.937, so M_f/M_0 = 1.760: the hole must accrete ~76% of its initial mass. For a starting mass of 8,200 M⊙ that is ~6,230 M⊙, taking it to ~14,430 M⊙. At 10−3 M⊙/yr this takes 6.2 million years, well under 0.1% of OC’s remaining lifetime. Phase 2 feeding at 0.1 M⊙/yr reduces it to 62,000 years. The IMBH growth history tool shows how mass and spin evolve together along this trajectory, accounting for the changing ISCO radius as spin increases.

Open IMBH Growth History → M=8200 · initial spin=0 · target a★=0.95 Thorne 1974
Step payoff
The mass budget is fixed by the Bardeen–Thorne relation. ~6,230 M⊙ is the price of reaching a★ = 0.95. The timeline question is about how fast that mass can be delivered, and at either rate the answer is short compared with the cluster’s remaining lifetime.
03
Step 3 · The Timeline
From Zero to Kerr-Maximal

The spin-up calculator integrates the Bardeen–Thorne relation. The accreted mass as a fraction of the initial mass is √(6/rISCO(a★)) − 1; the frequently quoted 1 − √(rISCO/6) is the same accretion expressed as a fraction of the final mass, and the two differ by the factor M_f/M_0. At a★ = 0.95 they read 76% and 43% respectively. The Phase 1 bottleneck is the accretion rate rather than the physics. At natural capture rates (Scenario A), Phase 1 requires ~6.2 Myr; at Phase 2 feeding (Scenario B) it compresses to ~62,000 years. The BZ power available at the end of Phase 1 is ~1.5×1038 W at B = 106 T, far above the cost of running active feeding. Both scenarios converge on the same endpoint: a★ = 0.95, M = 14,430 M⊙, Kardashev K ≈ 3.22.

Open Spin-Up Timeline → M=8200 · a0=0 · af=0.95 · &Mdot;=10−³ M⊙/yr Thorne 1974 MTH Phase 1
Step payoff
6.2 Myr is long by human standards and negligible against OC’s ~10 Gyr remaining lifetime, about 0.06% of it. Phase 1 is feasible on natural accretion alone at essentially any point in the cluster’s future.
04
Step 4 · The Payoff
BZ Power at the End of Phase 1

After spin-up to a★ = 0.95 with a final mass of ~14,430 M⊙ (8,200 + 6,230 accreted), the BZ power at B = 106 T is P_BZ ≈ 1.5×1038 W, since P_BZ ∝ M² at fixed spin and field. On the tool's scale K = log10(P)/10 − 0.6 that is Kardashev K ≈ 3.22, past the 4×1037 W Kardashev III threshold. Both scenarios arrive at the same BZ endpoint because they accrete the same total mass to the same final spin, and differ only in how long the journey takes. That output exceeds the energy cost of active feeding by many orders of magnitude, so once started the process can fund its own acceleration.

Open BZ Calculator → mass=14,430 M☉ · spin=0.95 · B=10⁶ T Blandford-Znajek 1977 MTH Phase 2 entry
Step payoff
P_BZ ≈ 1.5×1038 W at the Phase 1 endpoint is a power source large enough to fund everything that follows, and it clears the Kardashev III threshold on the tool's scale. Phase 1 delivers the reservoir that Phase 2 spends.
⚖ Self-Consistent Bootstrap

Natural accretion (Scenario A) makes Phase 1 viable on a few-million-year timescale, roughly 0.06% of OC’s ~10 Gyr remaining lifetime. Phase 2 feeding (Scenario B) compresses that to ~62,000 years but requires prior civilisational infrastructure, which itself requires Phase 1. The dependency is not circular, because the natural-accretion route needs no engineering at all: a civilisation reaching modest Type I technology could start Phase 1 and let the cluster’s own dynamics finish it, then use the resulting BZ power to accelerate Phase 2. The corrected mass budget makes this case stronger than the earlier billion-year figure suggested, not weaker, since the cost is 76% of the initial mass but the clock is short.

The spin-up economics are self-consistent. The BZ power at Phase 1 completion (~4.8×1037 W at the starting mass, ~1.5×1038 W at the final mass, both at B = 106 T) exceeds the power required for Phase 2 active feeding by many orders of magnitude, so the system bootstraps itself. The binding constraint on the MTH plan is the 6,230 M⊙ of fuel, not the time or the energy.

For the reconnection contribution on top of BZ at the Phase 1 endpoint, see Demo — Reconnection vs. BZ. For the full extraction sequence including Phases 2 and 3, see the BH Energy Budget Workflow.

EPISTEMIC TIERS: Established = peer-reviewed physics within the standard formulation. Debated = active disagreement in the published literature. Theoretical = published framework, awaiting decisive observation.