Unphysical Schwarzschild Model for Radiative Heat Transfer

Schwarzschild formulated in 1906 a mathematical model for radiative heat transfer based on two-way flow of heat energy, which allowed simple analytical solution and thus become popular. Schwarzschild’s model has been used to give support to an idea of “back radiation” with heat transfer from cold to warm as an element of a “greenhouse effect” threatening humanity by global warming from burning of fossil fuel adding CO2 to the atmosphere.

However, Schwarzschild’s model including “back radiation” is unphysical, which questions the existence of a “greenhouse effect” and thus also CO2 alarmism. The unhysical aspect of Schwarzschild’s model is exposed here.

Model of Atmosphere with CO2 shows Small Emissivity

Here is a another argument indicating that the effect of the atmospheric trace gas CO2 on the radiation balance of the Earth is small.

Our model of blackbody radiation consists of collection of oscillators with small damping with equal oscillator internal energy T representing temperature, with oscillator resonance frequencies n varying from 1 to a cut-off set at T and each oscillator radiating

  • E_n = \gamma T n^2

where \gamma is a universal constant, which is Planck’s Law.  Summing over n from 1 to T, we obtain the total radiance

  • E =\sum_n \gamma T n^2 = \sigma T^4

which is Stefan-Boltzmann’s law with \sigma = \gamma /3.  In the case of only one resonance frequency n = T, the radiance would be reduced to

  • e = \gamma T T^2 =\gamma T^3 = 3 E/T

with the reduction factor 1/T.

The radiance of an atmosphere which is fully opaque over the entire spectrum would radiate E, while an atmosphere opaque only for a specific frequency near cut-off T, would radiate e\approx 3E/T with a reduction factor  1/T.

We conclude that the emissivity of transparent atmosphere with a trace gas like CO2 with only a few isolated resonances, would scale like 1/T and thus be small as soon as T is bigger than say 100 K.

We thus find theoretical evidence from a basic model that the emissivity of the Earth’s atmosphere with the trace gas CO2 would be small, and thus that CO2 would have little effect on the Earth’s radiation balance.

We thus find theoretical evidence from a basic model that the emissivity of the Earth’s atmosphere with the trace gas CO2 would be small, and thus that CO2 would have little effect on the Earth’s radiation balance.

Blackbody as Transformer of Radiation

Our blackbody model, which has a high frequency cut-off scaling with temperature, absorbs all incoming radiation (forcing), re-emits frequencies below cut-off and stores frequencies above cut-off as internal energy, which is assumed to be equally distributed over frequencies below cut-off.

The result is that a blackbody subject to forcing with frequencies above cut-off, will heat up with a corresponding increase of cut-off until the highest frequency of the forcing is reached, assuming the blackbody is not somehow cooled. The effect is that the blackbody acts as a transformer of radiation from high to low frequencies by absorbing frequencies above current cut-off while emitting frequencies below cut-off.

The Earth subject to forcing from the Sun acts as such a transformer, see Blackbody: Transformer of Radiation.

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.


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.


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.