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.

1. cementafriend

Claes, I think you are looking at an ideal situation for a grey body but in practice it does not work like that. Have a look at actual measured absorptivities for some materials such as figure 5-12 in Perry’s Chemical Engineering Handbook. There it is shown the variation of absorpivity of some materials with temperature of the source eg for graphite alpha is 0.6 at 500R and 0.85 at 10,000R while for anodised aluminium alpha is 0.8 at 500R while at at 10,000R it is 0.15.(R is Rankine where 0C=273K=492R). Both emissivity (which relates to the temperature of emitter) and the absorptivity (which relates to the temperature of the source) vary with temperature and are unique for each material. By the way I quote from Perry CEH Page 5-25 “According to Kirchhoff’s law the emissivity and absorptivity of a surface to surroundings at its own temperature are the same for both monochromatic and total radiation. When the temperature of the surface and its surrounds differ the total emissivity and absorptivity of the surface often are found to be different”

2. claesjohnson

You are right. I consider here the equilibrium case, while in non-equilibrium absorptivity may differ from emissivity.