Kirchhoff’s Law 1

Kirchhoff’s Law of Thermal Radiation: 150 Years starts off by:

  • Kirchhoff’s law is one of the simplest and most misunderstood in thermodynamics.

Let us see what we can say about Kirchhoff’s Radiation Law stating that the emissivity and absorptivity of a radiating body are equal, in the setting of the wave model with damping presented in Computational Blackbody Radiation and Mathematical Physics of Blackbody Radiation:

  • 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} models a vibrating material string with U displacement
  2. - \gamma U_{ttt} is a dissipative term modeling outgoing radiation
  3. - \delta^2U_{xxt} is a dissipative term modeling internal heating by friction
  4. f is the amplitude of the incoming forcing,
  5. T is temperature with T^2=\int\frac{1}{2}(\vert U_t^2+U_x^2)dxdt,
  6. the wave equation expresses a balance of forces,

where \gamma and \delta^2 are certain small damping coefficients defined by spectral decomposition as follows in a model case:

  • \gamma = 0 if the frequency \nu > \frac{1}{\delta}
  • \delta = 0 if the frequency \nu < \frac{1}{\delta},

where \delta = \frac{h}{T} represents a “smallest coordination length” depending on temperature T and h is a fixed smallest mesh size (representing some atomic dimension).

This represents a switch from outgoing radiation to internal heating as the frequency \nu passes the threshold \frac{T}{h}, with the threshold increasing linearly with T.

The idea is that a hotter vibrating string is capable of radiating higher frequencies as coherent outgoing radiation. The switch acts as a band filter with frequencies outside the band being stored as internal heat instead of being radiated: The radiator is then muted and heats up internally instead of delivering outgoing radiating.

A spectral analysis, assuming that all frequencies share a common temperature, shows an energy balance between incoming forcing f measured as

  • F = \int f^2(x,t)\, dxdt

assuming periodicity in space and time and integrating over periods, and (rate of) outgoing radiation $R$ measured by

  • R = \int \gamma U_{tt}^2\, dxdt,

and (rate of) internal energy measured by

  • IE = \int \delta^2U_{xt}^2\, dxdt,

together with the oscillator energy

  • OE =T^2 = \frac{1}{2}\int (U_t^2 + U_x^2)\, dxdt

with the energy balance in stationary state with OE constant taking the form

  • F = \kappa (R + IE)

with \kappa\lessapprox 1 is a constant independent of T, \gamma, \delta and \nu. In other words,

  • incoming energy = \kappa\times outgoing radiation energy for \nu <\frac{1}{\delta}
  • incoming energy = \kappa\times stored internal energy for \nu >\frac{1}{\delta},

which can be viewed as an expression of Kirchhoffs’ law that emissivity equals absorptivity.

The equality results from the independence of the coefficient \kappa of the damping coefficients \gamma and \delta^2, and frequency.

Summary: The energy of damping from outgoing radiation or internal heating is the same even if the damping terms represent different physics (emission and absorption) and have different coefficients (\gamma and \delta^2).

PS: Note that internal heat energy accumulating under (high-frequency) forcing above cut-off eventually will be transformed into low-frequency outgoing radiation, but this transformation is not part of the above model.

Helmholtz Reciprocity 1

Helmholtz Reciprocity states that certain phenomena of propagation of light and interaction between light and matter, are reversible.

Let us study reversibility in our model of blackbody radiation:

  • 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} represents a vibrating string with U displacement
  2. - \gamma U_{ttt} is a dissipative term modeling outgoing radiation = emission
  3. - \delta^2U_{xxt} is a dissipative modeling internal heating = absorption
  4. f is incoming forcing/microwaves,

where \gamma and \delta^2 are positive constants connected to dissipative losses as outgoing radiation = emission and internal heating = absorption.

We see emission represented by -\gamma U_{ttt} and absorption by -\delta^2U_{xxt}. We now ask:

  1. How is the distinction between emission and absorption expressed in this model?
  2. Is Helmholtz Reciprocity valid (emission and absorption are reverse processes)?
  3. Is Kirchhoff’s Radiation Law (emissivity = absorptivity) valid?

Before seeking answers let us recall the basic energy balance between incoming forcing f measured as

  • F = \int f^2(x,t)\, dxdt

assuming periodicity in space and time and integrating over periods, and (rate of) outgoing radiation = emission $R$ measured by

  • R = \int \gamma U_{tt}^2\, dxdt,

the oscillator energy OE measured by

  • OE =\frac{1}{2}\int (U_t^2 + U_x^2)\, dxdt

and (rate of) internal energy = absorption measured by

  • F = R + IE
  • incoming energy = emission + absorption.

The model has a frequency switch switching from emission to absorption as the frequency increases beyond a certain threshold proportional to temperature in accordance with Wien’s displacement law.

 We now return to questions 1 – 3:

Both terms generate dissipative effects when multiplied with U_t as R = \int \gamma U_{tt}^2\, dxdt and IE = \int \delta^2U_{xt}^2\, dxdt, but the terms involve different derivatives with U_{tt} acting only in time and U_{xt} acting also in space.

The absorption U_{xt}^2 represents a smoothing effect in space, which is irreversible and thus cannot be reversed into emission as reversed absorption.

 The emission U_{tt}^2 represents a smoothing effect in time, which is irreversible and thus cannot be reversed into absorption as reversed emission.

 In other words, in the model both absorption and emission are time irreversible and thus cannot be reversed into each other.

 We conclude that the model does not satisfy Helmholtz reciprocity.

 Nevertheless, the model satisfies Kirchhoff’s law as shown in a previous post.

 Conclusion:

The space derivative in U_{xt} models absorption as process of smoothing in space with irreversible transformation of high frequencies in space into low frequencies with a corresponding increase of internal energy as heat energy.

 Absorbed high frequencies can with increasing temperature be rebuilt through (resonance in) the wave equation into high frequency emission.

 Absorption and emission are not reverse processes, but my be transformed into each other through (resonance in) the wave equation and the switch.

We may compare absorption with a catabolic process of destroying (space-time) structure and emission with an anabolic process of building structure, with the wave equation as a transformer.

Universality of Blackbody Radiation 2

To demonstrate universality of blackbody radiation Kirchhoff used a cavity with walls of graphite into which he put different objects of different materials and temperature, waited until the object reached radiative equilibrium with the graphite walls and observed the radiated spectrum through a peep hole. Kirchhoff then observed radiation intensities E(\nu , T)  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 certain given constants. From this observation Kirchhoff declared universality of blackbody radiation.

Kirchhoff observed to his disappointment that removing the graphite destroyed universality, which however returned by adding a small amount of graphite. This allowed Kirchhoff to maintain his declaration of universality of blackbody radiation, as a mystification to coming generations of physicists dreaming of universality as the ultimate expression of deep understanding.

If we now analyze Kirchhoff’s universality from the perspective of the previous post, we understand that the graphite serves as the reference blackbody effectively establishing universality. Again the universality expresses nothing but radiative equilibrium using a blackbody of graphite as reference. The mystery of universality of blackbody radiation is thus revealed as the no-mystery of a form of standardization.

For an introduction to classical work with an empty cavity as an abstract (mystical) universal reference blackbody, see An Analysis of Universality in Blackbody Radiation by P.M. Robitaille.

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.

Introduction

On this blog you find a new approach to the classical problem of blackbody radiation based on finite precision computation, to be compared with the statistics of quanta used by Planck in 1900 which led into quantum mechanics also based on statistics.

The blog starts out now as a collection of posts on blackbody radiation on

and will be complemented by new material in new posts.