# Universality of Blackbody Radiation 1

One of the mysteries of blackbody radiation is universality with the radiation intensity $E(\nu ,T)$ being independent of the atomic material properties of the blackbody, only depending on temperature $T$ and frequency $\nu$ according to Planck’s Law (with simplified high frequency cut-off)

• $E(\nu ,T) = \gamma T\nu^2$ for $\nu < \frac{T}{h}$,
• $E(\nu , T) = 0$ for $\nu > \frac{T}{h}$,

where $\gamma$ and $h$ are universal positive constants defining the intensity and high frequency cut-off, respectively.

The mystery is that all blackbodies with different atomic properties have the same constants $\gamma$ and $h$. How can that be? The new proof of Planck’s Law suggests the following explanation:

Consider two blackbodies 1 and 2 characterized by $(\gamma_1 ,h_1)$ and $(\gamma_2 ,h_2)$.  Assuming that the two bodies are in radiative equilibrium, we have:

• $\gamma_1T_1 =\gamma_2T_2$ (energy balance),
• $\frac{T_1}{h_1}=\frac{T_2}{h_2}$  (same cut-off),

where $T_1$ is the temperature of 1 and $T_2$ that of 2 using intrinsic temperature scales defined by the atomic properties of 1 and 2.

Let us now decide to choose body 1 as universal reference with a universal temperature scale defined by $T_1$. By energy balance the intrinsic temperature scale of 2 as measured by $T_2$ will then be related to the universal scale $T_1$ by the relation

•  $T_2=\frac{\gamma_1}{\gamma_2}T_2$,

This rescaling of the intrinsic temperature of 2 allows us to effectively assume that $\gamma_2=\gamma_1$ (and then $T_2=T_1$), which gives universality to $\gamma_1$ .

The same cut-off then requires

• $\gamma_2h_2 =\gamma_1h_1$,

that is with $\gamma_2=\gamma_1$, we also require $h_2=h_1$ thus giving also $h_1$ universality.

The above gives universality to ideal blackbodies satisfying the above requirements.

We are then led to suggest to defining non-ideal blackbodies or greybodies by a modified energy balance:

• $\gamma_2T_2 = \alpha_2\gamma_1T_1$

with $\alpha_2<1$ a coefficient of absorptivity, or by smaller cutoff:

• $\frac{T_2}{h_2}<\frac{T_1}{h_1}$.

A greybody 2 can then be characterized by $\gamma_2<\gamma_1$ or $h_2>h_1$ (assuming that $T_2=T_1$ by normalization), that is the reference body 1 is characterized by maximal $\gamma_1$ and minimal $h_1$.

We have thus achieved universality by choosing a specific blackbody as universal reference allowing the reference to define the temperature of different blackbodies through radiative equilibrium. Universality will then be a reflection of universal radiative equilibrium and the use of a specific reference thermometer in the form of an ideal blackbody.

We conclude that blackbody universality is not something mystical, but simply an expression of postulated radiative balance and use of a reference thermometer.

With now universality established for ideal blackbodies defined by $(\gamma_1, h_1)$, the world of greybodies 2 can then be described as suggested above, by $\gamma_2<\gamma_1$ or $h_2.

In particular, the observed fact that different bodies at the same distance to the Sun can assume different temperatures in radiative balance with the Sun, can be understood as an effect of different cut-off by the following argument:

If $\gamma_2=\gamma_1$ then 2 will have the same temperature scale as 1, while the heat absorbed if $h_2>h_1$ will vary with the cut-off, and thus the equilibrium temperature may differ between different bodies 2 with different coefficients $h_2$.

Compare with the following information from a NASA manual identified by Alan Siddons:

• Most people associated with the space effort occasionally must answer the question “What is the temperature of space?” The answer, of course, is that space has no temperature. Temperature is a measure of the internal energy of matter. Since space itself is the absence of matter, it can have no temperature. However, objects in space do have temperatures, and these temperatures will depend upon the nature of the objects themselves as well as the nature and proximity of other objects in space. A thin gold plate in deep space (beyond the thermal influence of any object other than the Sun) with its surface normal to the direction of the solar radiation, at 1 astronomical unit (the mean radius of the Earth’s orbit) from the Sun, will reach a temperature of 405° F. If the plate is painted white, its temperature will drop to -18° F.

We conclude that ideal blackbodies at the same distance to the Sun will assume the same temperature (depending only on the distance), while different greybodies may assume different temperatures because of different cut-off.

Note that a greybody is commonly defined only by a coefficient of absorptivity (equal to emissivity) smaller than 1, while different cut-off is not taken into account. In this setting the observed different temperatures of different bodies at the same distance to the Sun, comes out as a mystery.

One way to rationalize this mystery is to play with different coefficients of absorptivity and emissivity, which however is confusing since energy balance appears to be violated.