Accretion requires an angular momentum budget

Use cylindrical radius r around a Newtonian point mass M, neglecting disk self-gravity. A geometrically thin disk has sound speed much smaller than orbital speed, so radial pressure support is a small correction and the orbital angular velocity is approximately Keplerian. The specific angular momentum l=√(GMr) increases outward even though the angular velocity decreases. Thus a parcel moving inward must shed angular momentum. Dissipating orbital energy without a corresponding torque does not solve this constraint.

Let Σ be mass per disk-plane area, integrated vertically, and let vᵣ be negative for inward motion. Define the positive inward mass rate as minus circumference times surface density times radial velocity. An effective kinematic viscosity ν represents a vertically averaged internal stress; it need not be molecular viscosity. In a Keplerian shear, the resulting positive outward torque is proportional to νΣl. Torque has units of angular momentum per time, equivalently energy, but torque transport and local heat generation remain different physical quantities.

In a steady disk without mass loss, the mass rate is independent of radius. Conservation makes the net inward angular momentum flux, advective transport minus the outward torque, constant. If the internal torque vanishes at an inner radius Rᵢₙ, that constant is the mass rate times the angular momentum carried through the inner edge. Substituting the Keplerian torque gives the familiar boundary factor. Far outside the edge, the factor approaches one; close to the edge it cannot be dropped.

The zero-torque condition is a model assumption, rather than a consequence of circular motion. A magnetic connection to a star, appreciable stress through a black-hole inner region, or a disk wind changes the angular momentum balance. Even in the chosen model, extrapolating the thin steady radial-velocity expression all the way to its inner endpoint is unsafe: the assumed slow, nearly circular flow eventually requires an inner transition. A formal divergence from vanishing surface density is not a measured infinite speed.

An often useful convention is ν=αcₛH with vertical scale height H=cₛ/Ω and dimensionless α. We use precisely that convention; alternative stress-based definitions can carry factors such as two thirds. It gives a redistribution estimate tᵥᵢₛc≈r²/ν. Molecular collisions are generally too slow to explain rapidly evolving astrophysical disks. In sufficiently ionized disks, magnetic tension and differential rotation can generate magnetorotational instability and correlated Maxwell and Reynolds stresses. Its operation depends on field strength, conductivity, and available wavelengths, so specifying α does not prove a particular turbulence mechanism.

A simple reservoir comparison isolates the mass budget. In the queue lab, Reset and set both rates to arrival three and service three, then Step twice: the fluid backlog stays four. Reset and set arrival three and service two, then Step twice: it becomes five. Interpret this only as inflow minus outflow into an annulus. The instrument has no angular momentum or stress variable, and its separately labeled stochastic residence-time formula is not a disk accretion time.