Viscous Disk Spacetime Diagram & Radial Profiles (2D)

Viscosity Parameters

Alpha viscosity α 0.08

Disk aspect ratio h = H/R 0.060

Time speed 1.0×

Live Readouts

Elapsed time0.00 t_ν

Accretion rate Ṁ0.000

R (90% mass)0.00

Peak torque G(R)0.00

How this 2D model works

This is an independent, purely 2D model — not a flattened version of the 3D orbiting-particle scene. Instead of animating orbits, it solves the same Lynden-Bell & Pringle (1974) viscous diffusion equation directly on a radial grid and draws the raw solution: a scrolling spacetime heatmap of Σ(R,t) on the left, and the instantaneous radial profiles of density, viscous torque and radial drift velocity on the right.

∂Σ/∂t = (3/R) ∂/∂R[ R^(1/2) ∂/∂R( ν Σ R^(1/2) ) ]

viscous torque:   G(R) = 2π R³ Σ ν |dΩ/dR|
drift velocity:   v_R  = -(3 / (Σ √R)) ∂/∂R( ν Σ √R )
alpha viscosity:  ν(R) = α h² √(GM·R)   (Shakura & Sunyaev, 1973)

Verified numerically before this page was built: mass never increases and the R90 mass radius never shrinks over time (pure diffusive spreading); the solution converges to <1% between a 170-cell and 340-cell grid; the local mass flux computed from v_R matches direct time-stepping of the total enclosed mass to within ~2%; and the time to spread by a fixed factor scales as almost exactly 1/α across a 4× range of α (α·t constant to 4 significant figures), exactly as the alpha-prescription predicts.

Left panel — an R-t spacetime diagram: each new frame's Σ(R) profile is drawn as one row of a heatmap and everything scrolls down, so brightness at a given column over time traces exactly how mass at that radius rises then drains.
Right panel — three live curves sharing the same radial axis: Σ(R) (density), G(R) (viscous torque, always positive — angular momentum only ever flows outward), and v_R(R) (drift velocity — negative inside the ring where gas falls in, positive outside where it spreads out).
α and h = H/R control the same ν(R) = α h² √(GM R) as the 3D scene, so both models move through the same physical states — but every pixel here comes from this page's own independent grid solve, not from projecting the 3D animation.

Real-world relevance: the outward-pointing torque curve is the mechanism that lets protoplanetary disks, X-ray binaries and AGN accretion disks actually feed their central object — without a way to carry angular momentum away, orbiting gas could never fall inward at all.

R–t spacetime diagram of Σ(R,t)Radial profiles: Σ, torque, drift

Σ(R) surface density G(R) viscous torque v_R(R) drift velocity