Kirchhoff’s Law 2

Kirchhoff’s Law states that for a blackbody emissivity = absorptivity. In our wave equation with small damping as a model of blackbody in radiative equilibrium with an exterior forcing, this is built into the model for frequencies below high-frequency cut-off.

This is because for these frequencies the outgoing radiation simply reproduces the part of the incoming radiation which is not reflected. Or put differently: In radiative equilibrium there is below cut-off no distinction between incoming radiation and outgoing radiation as explained in the previous post.

However, if the incoming waves (the forcing) contain frequencies above the present cut-off  of the absorbing body, then the game changes, as these frequencies contribute to the internal energy of the body increasing its temperature and are not (re-)emitted. This is a case with positive absorptivity and zero emissivity of the frequencies above cut-off.

This leads to coefficients of absorptivity depending on temperature, which complicates the picture.

The effect is that different bodies with different cut-off at the same distance to the Sun, can take on different equilibrium temperatures, as evidenced in e.g. Satellite Thermal Control Engineering and What Is the Temperature of Space?

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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.