String and Soundboard

Our model of blackbody radiation has an acoustic analog in the form of a string interacting with a soundboard, with the following key components (as total or for each frequency in a spectral decomposition):

  • internal string energy
  • string amplitude
  • soundboard amplitude
  • force balance between string and soundboard = wave equation
  • small damping in the interaction expressed by the wave equation.

The small damping is an important feature of the model since it creates an “optimal” interaction between the string and the soundboard in the sense that the tone generated by the soundboard will be both loud and have long sustain. With zero damping the interaction will small and the tone weak, and with too big damping the sustain will be short. In radiation the damping is small.

The effect of the small damping is that the string velocity is out-of-phase with the forcing, which is the key to optimal interaction between string and soundboard.

Radiative equilibrium corresponds to acoustic equilibrium, which can be interpreted in two different ways:

  1. The vibration of the soundboard is sustained by the vibration of the string = outgoing acoustic waves.
  2. The vibration of the string is sustained by the vibration of the soundboard = incoming acoustic waves.

The same equilibrium can thus be interpreted as incoming or outgoing, which can be expressed as absorptivity = emissivity, which thus is built into the model.

This argument applies to frequencies below cut-off with only radiative damping being active. For frequencies above cut-off the interaction between string and soundboard is different with internal heating of the string, and emissivity in general different from absorptivity.

There is also non-equilibrium dynamics below cut-off with e.g. the string energy being transferred to the soundboard as the string amplitude decreases.

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<h_1.

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.