Grey vs Black Body

What is a blackbody? What is a greybody? Here are answers:

As a model of blackbody radiation I consider in Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation a wave equation expressing force balance:

• $U_{tt} - U_{xx} - \gamma U_{ttt} - \delta^2U_{xxt} = f$

where the subindices indicate differentiation with respect to space $x$ and time $t$, and

1. $U_{tt} - U_{xx}$ is out-of-equlibrium force of a vibrating string with displacement $U$,
2. $- \gamma U_{ttt}$ is Abraham-Lorentz (radiation reaction) force with $\gamma$ a small positive parameter,
3. $- \delta^2U_{xxt}$ is a friction force acting on frequencies larger than the cut-off frequency $\frac{T}{h}$ and then contributing to internal heating,
4. $\delta =\frac{h}{T}$ is a smallest coordination length with $h$ a measure of finite precision,
5. $T$ is the common energy/temperature of each frequency of the vibrating string,
6. $f$ is exterior forcing.

This model with $\delta =0$ is essentially the starting point also for Planck in his classical proof completed by resorting to statistics of quanta: A system of resonators $U$ in resonance with an exterior forcing $f$.

The model is specified by the parameters $\gamma$ and $h$. It is shown in Universality of Blackbody Radiation that all blackbodies can be assumed to have the same value of the radiation coefficient $\gamma$ and the cut-off ($h$),  given as the values of a chosen reference blackbody.

Stationary periodic solutions $U$ satisfy the energy balance

• $R + H = F$ with
• $R=\int\gamma U_{tt}^2dxdt$
• $H = \int \delta^2 U_{xt}^2dxdt$
• $F=\int f^2dxdt$

which expresses that all incident radiation $F$ is absorbed and is either re-emitted as radiation $R$ or stored as internal energy from heating $H$ with a switch from $R$ to $H$ at the cut-off frequency. We here assume that all frequencies have the same temperature $T$ defined as $T =\int U_{\nu ,t}^2dxdt$ where $U_{\nu}$ is the amplitude of frequency $\nu$.

We now consider a body $\bar B$ defined by $\bar\gamma$ and $\bar h$ with temperature $\bar T$ calibrated so that $\bar T = T$ in radiative equilibrium with a reference blackbody $B$.

Energy balance (below cut-off)  can be expressed as

• $\bar\gamma \bar T =\bar\alpha\gamma T$

where $\bar\alpha$ is a coefficient of absorptivity of $\bar B$, assuming both bodies follow Planck’s Law with radiation per unit frequency for $B$ given by $\gamma T\nu^2$.

We ask a blackbody to have maximal emissivity = absorptivity and we thus must have $\bar\alpha\le 1$ and $\bar\gamma\le\gamma$ reflecting that a blackbody is has maximal $\gamma$ and cut-off (minimal $h$).

A body $\bar B$ with $\bar\gamma <\bar\gamma$ will then be termed  greybody defined by the coefficient of absorptivity

• $\bar\alpha =\frac{\bar\gamma}{\gamma}<1$

and will have a coefficient of emissivity $\bar\epsilon =\bar\alpha$.

A greybody $\bar B$ thus interacts thorough a reduced force $\bar f=\sqrt{\bar\alpha} f$ with a blackbody $B$ with full force $f$. We thus obtain a connection through the factor $\sqrt{\bar\alpha}$ between force interaction and absorptivity.

The spectrum of a greybody is dominated by the spectrum of a blackbody, here expressed as the coefficient $\bar\alpha =\bar\epsilon$. 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.