Black Hole Horizon, ISCO & Shadow Explorer
Compute event horizon radius, ISCO, Hawking temperature, and EHT shadow size for any black hole mass and distance.
A non-spinning black hole of mass M has an event horizon at the Schwarzschild radius R_s = 2GM/c² ≈ 2.953 km · (M/M_☉), a photon sphere at 1.5 R_s, an Innermost Stable Circular Orbit (ISCO) at 3 R_s, and a gravitationally lensed shadow of radius √27 GM/c² ≈ 2.60 R_s.
Interactive Physics Calculator
Governing Equation & Worked Benchmark
Schwarzschild Horizon, ISCO, Shadow Radius & Hawking Temperature
R_s = 2GM/c² = 2.953 km (M/M_☉) ; R_ISCO = 3 R_s (a*=0) to 0.5 R_s (a*=1) ; R_shadow = √27 GM/c²Dividing the physical shadow diameter 2 R_shadow ≈ 5.196 R_s by distance D gives the angular shadow diameter θ_sh imaged by the Event Horizon Telescope.
| Benchmark Step | Physical Details & Result |
|---|---|
| Scenario | Sagittarius A* (M = 4.15 × 10⁶ M_☉ at D = 8.178 kpc) and M87* (M = 6.5 × 10⁹ M_☉ at D = 16.8 Mpc). |
| Calculation | For Sgr A*: R_s = 1.226 × 10⁷ km (0.082 AU), R_ISCO(a*=0) = 0.246 AU, and angular shadow diameter θ_sh ≈ 51.8 μas. For M87*: R_s = 1.92 × 10¹⁰ km (128 AU) and θ_sh ≈ 42 μas. |
| Physical Takeaway | Because M87* is ~1,560× more massive and ~2,050× farther away than Sgr A*, both black holes subtend nearly the same ~40–50 microarcsecond shadow on Earth's sky. |
Physical Mechanism & Interpretation
Strong gravitational light bending around a black hole captures any photon whose impact parameter is smaller than b_crit = √27 GM/c² ≈ 2.598 R_s, making the observed dark shadow 2.6× wider than the event horizon itself.
Meanwhile, prograde frame-dragging around a rapidly spinning Kerr black hole (a_* → 1) pulls the Innermost Stable Circular Orbit inward from 6 GM/c² (3 R_s) to 1 GM/c² (0.5 R_s), raising accretion radiative efficiency from 5.7% to 42%.
Model Assumptions & Systematic Uncertainties
The shadow diameter depends only weakly (±4%) on black hole spin a_* and viewing inclination, making it a clean test of the Kerr metric.
Related Concepts & Methods
Scientific Sources & Provenance
- Event Horizon Telescope Collaboration Horizon-Scale Results (2019 ApJL 875 L1 & 2022 ApJL 930 L12) — Event Horizon Telescope Collaboration 2019, ApJL, 875, L1; 2022, ApJL, 930, L12. Archive / Reference ↗
- ESA Gaia Archive — Data Release 3 (DR3 (2022-06-13)) — Gaia Collaboration, Vallenari, A., et al. 2023, A&A, 674, A1 (Gaia Data Release 3). Archive / Reference ↗
- Gravitational-Wave Transient Catalog (GWTC-3) (O1–O3b Cumulative Catalog) — Abbott, R., et al. (LIGO-Virgo-KAGRA) 2023, PRX, 13, 041039 (GWTC-3: Compact Binary Coalescences). Archive / Reference ↗