Black-hole shadow
Compare a Kerr black hole’s shadow and equatorial ISCO radii for a chosen spin and viewing angle.
About this tool
Kerr shadow and ISCO
The shadow is the analytical critical curve of an uncharged Kerr black hole in vacuum, viewed from far away against a background light source. It is not a simulated accretion image. The spin is a* = Jc/(GM²), here 0–0.998. Inclination is measured from the spin axis: 0° is polar and 90° is equatorial.
Size and distance
All geometric lengths use r_g = GM/c². The mass is expressed in solar masses, using the nominal solar GM = 1.3271244 × 10²⁰ m³/s². The distance D is an angular-diameter distance in Mpc, not a luminosity distance or redshift. There is no cosmological distance calculation. The far-field conversion is angular size = size × r_g/D, in μas. We require D/r_g ≥ 10⁶ as a chosen model boundary, not a physical discontinuity or a guaranteed error bound. AU = 149597870700 m and pc = AU × 648000/π.
What is calculated?
The curve follows the conserved quantities of spherical photon orbits and their asymptotic sky coordinates (Perlick & Tsupko, equations 57–58). Both spin and inclination affect it. The horizontal and vertical diameters are calculated from its extrema; the offset is (α_min + α_max)/2, not a brightness centroid. Numerical roots and the maximum are determined separately from the plotted samples. For zero spin the shadow is a circle of radius 3√3 r_g at every inclination. The polar view is also circular for rotating holes.
The outer horizon is r₊/r_g = 1 + √(1−a*²). The photon and ISCO radii listed here are equatorial Boyer–Lindquist coordinate radii for prograde and retrograde orbits, not proper radial distances. ISCO is the innermost stable circular orbit. Its radius depends on spin and orbital direction, not on the observer’s inclination.
The disk sketch
Solid and dashed curves show two alternative inner disk edges at their actual ISCO radii. The outer edge is illustrative. Projection scales the vertical coordinate by cos(inclination), without enlarging small radii. This sketch does not calculate lensing, Doppler colours, emission, disk thickness or orbital animation. It is separate from the physically calculated shadow boundary. The numerical implementation rejects invalid input and non-finite or non-normal positive required results instead of silently replacing values.