Disk luminosity follows energy transport and boundary conditions

Assume a steady, axisymmetric Newtonian disk with nearly circular Keplerian motion, negligible wind mass loss, and zero torque at Rᵢₙ. Suppose heat leaves vertically before it can be advected inward. Integrating the shear heating over height gives the energy emitted from both disk faces. Each face then has half that power per disk-plane area. Define an effective temperature by its frequency-integrated outward flux; a blackbody shape additionally requires suitable thermalization, rather than just a definition of temperature.

The flux is suppressed near the inner edge by the same boundary factor found from angular momentum conservation. Far outside that edge it scales as radius to the minus third power, so effective temperature scales as radius to the minus three quarters. The numerical factor is three times the naive annulus-only orbital-energy argument far from the boundary because the torque redistributes energy. This does not create extra total power: integrating the full boundary-dependent profile restores the global energy budget.

An annulus has area 2πrdr on each face, so its total radiating area is 4πrdr. Integrate the flux from the chosen inner radius to a large outer radius. For an effectively infinite outer disk the luminosity is half the gravitational potential-energy release per unit time at the inner edge. The other half is the orbital kinetic energy retained by material on that orbit. Additional surface boundary-layer emission from a slowly rotating star is a separate component, rather than another face of the disk.

Define gravitational radius r𝓰=GM/c², without the factor of two used for a Schwarzschild horizon radius. The Newtonian efficiency is r𝓰/(2Rᵢₙ). Substituting six gravitational radii into that Newtonian formula gives one twelfth, but that is not the Schwarzschild relativistic thin-disk result. In the ideal zero-torque Schwarzschild model, ignoring photon capture, the binding-energy efficiency is 1−√(8/9), about 0.0572. Strong gravity changes orbital energies and photon propagation; it cannot be repaired merely by choosing an inner radius in the Newtonian expression.

Radiative efficiency means the fraction of supplied rest-mass power emerging as the specified radiation. At low densities, ions and electrons can exchange energy too slowly to radiate the available heat. At high optical depth and rapid inflow, photons can be trapped and advected. Magnetic extraction, winds and irradiation also modify the simple budget. A bright source’s isotropic-equivalent luminosity is not automatically the disk’s angle-integrated emitted luminosity. Inferring accretion rate or spin therefore requires a model for geometry, transfer, inner stress and spectral hardening.

The function lab isolates the fourth-power temperature sensitivity through a declared change of variable. Let its horizontal coordinate x represent (T/T₀)² and its vertical value represent normalized flux. Reset and set both controls to a=1,x=1, predicting one. Reset and set a=1,x=2, predicting four, corresponding to temperature √2 times larger. The instrument itself plots a quadratic; the fourth power arises only through our explicit squared-temperature coordinate. It contains no disk atmosphere or relativistic spectrum.