Expansion and photon diffusion reshape supernova energy

Consider spherical ejecta of fixed mass M with outer radius R≈vt, once the initial stellar radius is negligible. Homologous expansion means each fluid element maintains approximately constant velocity proportional to radius. For a uniform-density homologous sphere, v is the outer-edge speed and the kinetic energy is 3Mv²/10, obtained by integrating one half velocity squared over mass. Using Mv²/2 instead would describe a different single-speed shell approximation. Declaring the velocity convention prevents an apparently small coefficient from silently changing inferred energy.

Let Eᵢₙₜ be the ejecta’s trapped radiation energy. In a radiation-dominated one-zone model its pressure is Eᵢₙₜ/(3V), so the first law includes expansion work, deposited heating and escaping bolometric luminosity. Heating means energy actually thermalized in the ejecta; radioactive decay energy escaping as gamma rays is not automatically deposited. A shock’s initial internal energy can become kinetic energy through pressure work, and at late freely expanding times the remaining internal energy can be much smaller than the kinetic reservoir.

Use a constant grey opacity κₒₚ in area per mass and uniform density as an illustrative closure. The center-to-edge optical depth is κₒₚρR. A random-walk escape estimate is that optical depth times R/c; its coefficient depends on the density profile and boundary treatment. Because radius grows while density falls as radius cubed, the instantaneous diffusion estimate decreases as one over time. Equating it to the expansion age defines a characteristic light-curve width, with a square-root dependence on mass and opacity.

At that crossing, the optical depth is approximately c/v, not one. For nonrelativistic ejecta this can still be much larger than unity: photons escape efficiently relative to expansion while the ejecta remain optically thick. The actual time of maximum luminosity also depends on the heating history, ionization, recombination and density profile. The familiar statement that luminosity at peak tracks instantaneous deposited heating belongs to particular diffusion models and is approximate for real events. It is not a general identity equating peak radiation with initial kinetic energy.

A bolometric luminosity describes total radiated power, whereas an observed band samples a spectrum shaped by wavelength-dependent opacity and thermalization. A blackbody photospheric estimate can illustrate colors but does not reproduce a line-rich supernova spectrum. Recombination can alter the effective opacity and produce a plateau, while circumstellar interaction can supply continuing shock power. Radioactive heating, magnetar heating and interaction may yield partially degenerate optical curves. Distinguishing them requires velocities, spectra, temporal coverage and consistent deposited-energy accounting, rather than one fitted width.

The heat lab isolates diffusion at fixed geometry. Reset and set both controls to mode one and diffusivity 0.25; Step twice to time one, predicting amplitude exp(−0.25)≈0.779. Reset and set mode one and diffusivity one; Step twice, predicting exp(−1)≈0.368. The instrument’s κ is a diffusivity, not the opacity κₒₚ: physical radiative diffusivity scales inversely with opacity at fixed density. This periodic mode has no expansion, radioactive heating, photosphere or emergent supernova light curve.