Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== If energy is absorbed and stored on the shell (not immediately re-radiated), mass grows according to E=Mc2E = M c^2E=Mc2. For an absorber that captures power PabsP_{\rm abs}Pabs (watts), the instantaneous mass-growth rate is ===

  dMdt=Pabsc2  .\boxed{\;\frac{dM}{dt}=\frac{P_{\rm abs}}{c^2}\; }.dtdM=c2Pabs.
Hence the time to accumulate a target rest mass MMM is

  t  =  Mc2Pabs  .\boxed{\; t \;=\; \frac{M c^2}{P_{\rm abs}} \; } .t=PabsMc2.
Two useful forms for PabsP_{\rm abs}Pabs:

(A) directional / collimated flux FFF (W·m⁻²) intercepted by an effective cross-section σeff\sigma_{\rm eff}σeff:

Pabs  =  α F σeff,P_{\rm abs} \;=\; \alpha\, F\, \sigma_{\rm eff},Pabs=αFσeff,
where 0≤α≤10\le\alpha\le10≤α≤1 is the fraction of incident energy actually absorbed and retained (absorption efficiency). For a small sphere the projected geometric cross-section is σgeom=πR2\sigma_{\rm geom}=\pi R^2σgeom=πR2. For point-like free electrons the relevant scattering/interaction area is given by the energy-dependent cross-section (e.g. Thomson σT\sigma_TσT at low photon energy).

(B) isotropic blackbody / ambient field (temperature TTT):
a perfectly absorbing sphere in an isotropic blackbody bath receives power

Pabs  =  σSB T4 πR2,P_{\rm abs} \;=\; \sigma_{\rm SB}\, T^4 \,\pi R^2,Pabs=σSBT4πR2,
because the projected area πR2\pi R^2πR2 captures the net isotropic radiative flux (σSB\sigma_{\rm SB}σSB is Stefan–Boltzmann constant).