Black hole size scales (non-rotating)

r_s = 2GM/c^2 ≈ 2.953 km × (M / M☉). Photon sphere 1.5 r_s; innermost stable circular orbit (ISCO) 3 r_s; the dark shadow seen from afar has radius (3√3/2) r_s ≈ 2.598 r_s. A 10 M☉ hole has r_s ≈ 29.5 km. G is the gravitational constant, c the speed of light, M the mass.

Orbits (N-body)

The Sun, eight planets and the hole are integrated with Newtonian gravity: 4th-order Yoshida symplectic composition of leapfrog, with a time step set by the shortest free-fall or crossing time of any pair (dt = 0.1 min sqrt(r^3 / G(m1+m2))), so close approaches get tiny steps. 160 massless Kuiper-belt test particles (32 to 48 AU) feel the same bodies. Initial planetary orbits come from mean J2000 elements (NASA/JPL, Standish); the Moon is not modelled separately. A planet that comes within 3 r_s of the hole is counted as swallowed.

Osculating orbital elements: a = -mu/(2E), e = sqrt(1 + 2E h^2/mu^2) from the planet's position and velocity relative to the Sun, with E the specific orbital energy and h the specific angular momentum.

Tides

Tidal acceleration across a body of size R at distance d: a_t ≈ 2GM R / d^3. Tidal disruption radius r_t ≈ R (M/m)^(1/3) (R, m: radius and mass of the body). A human is stretched when 2GM L / d^3 exceeds a chosen limit (here 100 m/s^2 across L = 1.8 m, roughly 10 g head-to-foot, an illustrative threshold). For massive enough holes r_t falls below r_s and the body crosses the horizon intact (the "Hills mass").

Lensing and the disk

Light rays in the Schwarzschild geometry bend according to d^2x/dλ^2 = -(3/2) r_s h^2 x / r^5, where h = |x × dx/dλ| is conserved; integrating it reproduces the exact orbit equation u'' + u = (3/2) r_s u^2 (u = 1/r). The view from Earth traces one such ray per pixel (midpoint integrator). The thin disk follows T ∝ r^(-3/4) between 3 and 15 r_s, orbital speed v = c sqrt(r_s / 2(r - r_s)), Doppler factor δ = 1 / [γ(1 - β cosθ)] with intensity ∝ δ^3.5 and gravitational redshift sqrt(1 - r_s/r). Technique after Luminet (1979) and James et al. (2015).

Time dilation

dτ/dt = sqrt(1 - r_s/r) for a static clock at radius r.

What is simplified or uncertain

Planets are point masses; general-relativistic corrections to planetary motion and the hole's own dynamics are ignored (negligible far from the hole). Real tidal break-up depends on internal strength and spin; r_t is an order-of-magnitude guide (a fluid Roche estimate is about 2.4 times larger). A real intruding hole would carry little gas, so the bright disk and jets are illustrative, not predicted. The Earth-view star field is a schematic procedural sky and, for distant holes, is shown at telescope-like magnification (stated on screen). Distances to the hole are compressed to constant screen size in the 3D scene (dramatised); planet sizes are exaggerated. Sources: Schwarzschild 1916; Misner, Thorne & Wheeler, Gravitation; Hills 1975; Yoshida 1990; El-Badry et al. 2023 (Gaia BH1).