For a computation substrate in a stable prograde circular orbit around OC's spinning IMBH — find the trade-off between time-dilation speedup, thermal radiation, and tidal stress. The MTH "parking position."
Set a target time-dilation factor γ and find the required orbital radius. More dilation = closer orbit = hotter & more tidally stressed. The "optimal" orbit balances computational speedup against physical survivability.
For a particle on a prograde circular geodesic in the equatorial plane of a Kerr black hole (with spin parameter a in units of GM/c²), the proper time per coordinate time is: dτ/dt = r^(3/4) × √(r^(3/2) − 3r^(1/2) + 2a) / (r^(3/2) + a), where r is in units of r_g = GM/c². The inverse, γ = dt/dτ, is the time dilation factor. This formula (from Bardeen et al. 1972) assumes equatorial prograde orbits only; retrograde orbits and off-equatorial motion have different expressions.
The innermost stable circular orbit (ISCO) for a prograde orbit around a Kerr black hole is found via the Bardeen 1972 formula. For a=0 (Schwarzschild): r_ISCO = 6 r_g. For a=0.998 (Thorne limit): r_ISCO ≈ 1.24 r_g. No stable circular orbit exists below r_ISCO — a test particle placed there will plunge into the horizon.
The disk temperature assumes a standard Shakura–Sunyaev thin disk at luminosity f_Edd × L_Edd: T_max ~ 6.3×10⁷ K × (M/M☉)^(−1/4) × f_Edd^(1/4). This is an upper bound — OC's IMBH is currently in a quiescent (sub-ADAF) state with f_Edd ≪ 10⁻⁴, meaning the actual thermal environment is far less hostile.
Tidal acceleration across a 1-meter structure: a_tidal ≈ 2GM/r³ (Newtonian, valid for r ≫ r_g). For OC's IMBH at r = r_ISCO, tidal forces are moderate — much gentler than for a stellar-mass black hole — because r_g ∝ M while r_ISCO ∝ M, making a_tidal ∝ M/M³ = M⁻². Larger IMBH = smaller tidal stress at ISCO.
Bardeen, Press & Teukolsky 1972 (ApJ 178:347) · Lloyd 2000 (Nature 406:1047) · Smart 2012 (Acta Astronautica 78:55) · Kerr 1963 (PRL 11:237)