What is a blackbody? What is a greybody? Here are answers:
where the subindices indicate differentiation with respect to space and time , and
- is out-of-equlibrium force of a vibrating string with displacement ,
- is Abraham-Lorentz (radiation reaction) force with a small positive parameter,
- is a friction force acting on frequencies larger than the cut-off frequency and then contributing to internal heating,
- is a smallest coordination length with a measure of finite precision,
- is the common energy/temperature of each frequency of the vibrating string,
- is exterior forcing.
This model with is essentially the starting point also for Planck in his classical proof completed by resorting to statistics of quanta: A system of resonators in resonance with an exterior forcing .
The model is specified by the parameters and . It is shown in Universality of Blackbody Radiation that all blackbodies can be assumed to have the same value of the radiation coefficient and the cut-off (), given as the values of a chosen reference blackbody.
Stationary periodic solutions satisfy the energy balance
which expresses that all incident radiation is absorbed and is either re-emitted as radiation or stored as internal energy from heating with a switch from to at the cut-off frequency. We here assume that all frequencies have the same temperature defined as where is the amplitude of frequency .
We now consider a body defined by and with temperature calibrated so that in radiative equilibrium with a reference blackbody .
Energy balance (below cut-off) can be expressed as
where is a coefficient of absorptivity of , assuming both bodies follow Planck’s Law with radiation per unit frequency for given by .
We ask a blackbody to have maximal emissivity = absorptivity and we thus must have and reflecting that a blackbody is has maximal and cut-off (minimal ).
A body with will then be termed greybody defined by the coefficient of absorptivity
and will have a coefficient of emissivity .
A greybody thus interacts thorough a reduced force with a blackbody with full force . We thus obtain a connection through the factor between force interaction and absorptivity.
The spectrum of a greybody is dominated by the spectrum of a blackbody, here expressed as the coefficient . A greybody at a given temperature may have a radiation spectrum of a blackbody of lower temperature as illustrated in the above figure.
All blackbodies will thus have the same unique maximal spectrum which dominates the spectrum of a greybody.