Book reference: Ch 31 (Lagrangian Mechanics and ISCO Comparison)
Test file: test_isco_comparison.py
Paper: 31 (Lagrangian)
The Innermost Stable Circular Orbit (ISCO) is the smallest radius at which a test particle can maintain a stable circular orbit around a black hole. Matter inside the ISCO spirals inward, converting orbital energy to radiation.
r_ISCO_GR = 3 * r_s (analytical exact result)
Orbital energy at ISCO: E_ISCO/mc^2 = sqrt(8/9) = 0.9428
Accretion efficiency: eta_GR = 1 - sqrt(8/9) = 5.72%
The ISCO condition from the SSZ effective potential V_eff(r) = D(r)^2 * (1 + L^2/(c^2*r^2)):
Condition 1 (circular orbit): dV_eff/dr = 0
Condition 2 (stability boundary): d^2V_eff/dr^2 = 0
Numerical result (weak field, r_s/R << 1):
r_ISCO_SSZ ≈ 2.95 * r_s (slightly smaller than GR's 3 r_s)
The shift is small: r_ISCO_SSZ / r_ISCO_GR ≈ 0.983 (-1.7%)
| Regime | r_ISCO / r_s | eta (%) | D(r_ISCO) |
|---|---|---|---|
| GR (exact) | 3.000 | 5.72 | 0.816 |
| SSZ (numerical) | 2.95 | 5.85 | 0.821 |
| SSZ strong-field correction | 2.90 | 5.99 | 0.825 |
The accretion efficiency measures what fraction of infalling mass is converted to radiation:
eta = 1 - E_ISCO / (m*c^2)
SSZ gives slightly higher efficiency than GR for Schwarzschild black holes because the ISCO is at slightly lower radius (closer to the horizon), where the specific orbital energy is lower.
eta_SSZ / eta_GR ≈ 1.022 (+2.2% more efficient)
The ISCO shift arises from the modified effective potential. In GR:
V_eff_GR(r) = (1 - r_s/r)(1 + L^2/(c^2*r^2)) * c^2
At r = 3 r_s, the second derivative changes sign (transition from stable to unstable).
In SSZ, D^2(r) replaces (1-r_s/r). Since D^2_SSZ(r) > D^2_GR(r) for all r (SSZ time dilation is smaller than GR), the effective potential has a slightly different shape and the stability transition occurs at r = 2.95 r_s.
For a spinning black hole with spin parameter chi:
GR: r_ISCO(chi=1, prograde) = 0.5 * r_s [maximum spin]
SSZ: r_ISCO_SSZ slightly larger at max spin
The spin dependence of the SSZ ISCO:
r_ISCO_SSZ(chi) = r_ISCO_SSZ(0) * f(chi) where f(0) = 1
f(chi) is approximately the same function as in GR (weak-field limit).
X-ray binaries (black hole + companion star) show thermal emission from their accretion disks. The inner disk radius is approximately the ISCO:
r_inner ≈ r_ISCO
Fitting the disk spectrum gives r_ISCO. If SSZ is correct:
r_ISCO_SSZ / r_ISCO_GR = 0.983 (-1.7%)
This requires disk temperature and flux measurements precise to <2%, which is at the limit of current capabilities but achievable with NICER.
- Orbital Mechanics — ISCO derivation from effective potential
- Black Hole Metric — metric used in V_eff
- Penrose Process — energy above the ISCO
- Rotating Black Holes — Kerr ISCO
- GR vs SSZ Tables — comparison table