By K. Schwarzschild, January 13, 1906.

Translated by Dr. Markus Ott, 2025

In the opinion of Karl Schwarzschild, “Radiation equilibrium will occur in a strongly radiating and absorbing atmosphere, in which the mixing effect of ascending and descending currents [convection] is insignificant compared to heat exchange by radiation.” Ed.

I. Contents.

The surface of the sun shows us in granulations, sunspots and prominences (protuberances) changing conditions and stormy changes. In order to understand the physical conditions under which these phenomena exist, the first approximation is to substitute the spatial and temporal changes with a mean stationary state, a mechanical equilibrium of the solar atmosphere. So far, the so-called adiabatic [without transferring heat between the system and the environment] equilibrium, as it exists in our atmosphere when it is thoroughly mixed by ascending and descending currents [convection], and in the foreground of consideration.

Here I would like to draw attention to another type of equilibrium, which can be described as “radiative equilibrium.” Radiation equilibrium will occur in a strongly radiating and absorbing atmosphere, in which the mixing effect of ascending and descending currents [convection] is insignificant compared to heat exchange by radiation. Whether the adiabatic or the radiative equilibrium applies to the sun would be difficult to decide for general reasons. However, there is observational data, which makes a certain judgment possible. The solar disk is not uniformly bright, but is dimmed from the center towards the edge. Based on this brightness distribution on the surface, it is possible, under plausible assumptions, to draw conclusions about the temperature distribution in the depths of the solar body. The result is that the equilibrium of the solar atmosphere corresponds by and large to the radiative equilibrium.

The considerations which lead to this result presuppose that Kirchhoff’s law* is valid or – in other words- that the radiation of the solar atmosphere is purely thermal radiation. They further presuppose that when penetrating into the solar body, one encounters a continuous change of state and not a discontinuous transition from a rather transparent chromosphere into an opaque photosphere formed from luminous clouds. Neglected is the effect of the scattering of light by diffraction at the particles of the atmosphere, on the importance of which Mr. A. Schuster ¹ has drawn attention to, and the refraction, which was used by H. v. Seeliger ² to explain the observed distribution of brightness. Also, the different absorption of different wavelengths, the decrease in gravity with height and the spherical propagation of the radiation are not taken into account. The whole consideration can therefore by no means be considered conclusive or compelling, but it may provide a starting point for further speculation, by expressing a simple thought initially in its simplest form.

2. Different Types of Equilibrium.

Denote the pressure by p, the absolute temperature (in centesimal degrees) with t, the density with ϱ, the molecular weight (compared to the hydrogen atom) with M, the gravity with g, the depth into the atmosphere (calculated from any starting point inwards) with h. The units are taken from the conditions existing at Earth’s surface. Use the atmosphere (1 atm=101325Pa) as unit of (pressure) p, as unit of ϱ the density of air at 273° absolute temperature at a pressure of one atmosphere, as unit of g the gravity at the surface of the earth and as unit h the altitude of the so called “homogenous atmosphere”, which is 8 km.

The following relationship then applies to an ideal gas:

and the condition of mechanical equilibrium of the atmosphere reads:

The elimination of ϱ from (1) and (2) yields:

a) Isothermal equilibrium. For general orientation consider isothermal equilibrium, assume t to be constant. Then follows:

The gravity g on the sun is 27.7 times greater than on earth, the temperature (approx. 6000°) is about 20 times higher. It follows for a gas of the molecular weight of air approximately the same spatial pressure distribution as for air on earth. If we calculate more precisely, we find an increase in pressure and density by a factor of 10 for a gas of the molecular weight of air (28.9) for every 14.7 km, for hydrogen for every 212 km. Since 725 km on the sun correspond to an angle value of one arc second seen from Earth, it is clear that the sun must look completely sharply defined.

b) Adiabatic equilibrium. If a gaseous mass expands adiabatically, Poisson’s relations apply:

where p₀ and ϱ₀ denote any related initial values. The quantity k, the ratio of the specific heat, for a 1-atomic gas is equal to 5/3, for a 2-atomic gas equal to 7/5, for a triatomic gas 4/3, and decreases for polyatomic gases down to 1. The equilibrium of an atmosphere is called adiabatic, if the temperature prevailing at any point is the same as that of a gaseous mass rising from below and cooling adiabatically would assume at this point, in other words if the equations (5) are fulfilled all-over the atmosphere. It then follows from (3) by inserting (5) and integrating:

The temperature changes linearly with altitude. The temperature gradient is calculated for Earth’s atmosphere as 1° per 100m, for the sun it is 27.7 times greater than on earth. The temperature increase of one degree therefore occurs there for air for every 3.63 m, for hydrogen for every 52 m. The atmosphere has a certain outer limit (t = ϱ = p = 0). The depth of a layer of 6000° temperature below the outer boundary is on the sun is 22 km for air and 300 km for hydrogen.

c) Radiation equilibrium. If one imagines that the outer parts of the sun form a continuous transition to ever hotter and denser gaseous masses, one cannot distinguish between radiating and absorbing layers but must regard each layer as simultaneously absorbing and radiating at the same time. We know that a powerful stream of energy, originating from unknown sources in the interior of the sun, flows through the solar atmosphere and penetrates into the exterior space. What temperature would the individual layers of the solar atmosphere would have to reach to be able to carry such a flow of energy without any further temperature change of their own?

Let us assume that each altitude layer dh of the solar atmosphere absorbs a fraction a dh of the radiation passing through it. If E is the emission of a black body of the temperature of this altitude layer and assuming Kirchhoff’s law as valid, it follows that this altitude layer radiates the energy E.a dh to each side.

Now consider the radiant energy A, which travels through the atmosphere at some point to the outside, and the radiant energy B, which (due to the radiation of the layers further outside) moves inwards.

First follow the energy B that moves inwards. If you move inwards by an infinitely thin layer dh, then of the energy B coming from the outside, the fraction B.a dh is lost, on the other side, the intrinsic radiation of the layer, the amount a E dh is added, over all resulting in:

The same applies to outgoing radiation:

By considering the absorptivity a as a function of the depth h, we form (calculate) the “optical mass” of the atmosphere lying above the depth h :

Then the differential equations are:

We are asking for a stationary state of the temperature distribution. The same is conditioned by the requirement that each layer receives as much energy as it radiates, i.e. that:

Introducing the auxiliary quantity y in accordance with this condition:

The differential equations are transformed by addition and subtraction into:

and integrated:

The integration constants E₀ and y were determined by the fact that at the outer boundary of the atmosphere (m = 0) there is no inward moving energy B present and the outward moving energy has the observed amount A₀ . It must therefore for m = 0:

be. This gives the result:

Only under the assumption that Kirchhoff´s law is valid, the dependence of the radiation E on the optical mass lying above the point in question, can be derived.

If one wants to know the distribution of pressure and density that prevails at radiation equilibrium, one basically needs a more detailed investigation, which considers the radiation in the individual wavelengths. It is sufficient here, for a first overview, to assume that the absorption coefficient is proportional to density and independent of colour:

(k a constant). Then follows:

According to Stefan’s law, the radiation E of the black body is:

If one set for the outgoing energy A₀:

then T is what is usually given as the (effective) temperature of the sun. It is around T = 6000°. For radiative equilibrium then applies according to (11):

By introducing the temperature τ at the outer boundary of the atmosphere it can also be written:

[Text below: “prevailing Temperature” and “you can also write:”]

If we use this in (3), we obtain:

This equation gives the temperature as a function of depth. The corresponding density follows from:

The following table gives the values obtained from (11), (15) and (16), if the solar atmosphere is made up of our air. For the absorption coefficient of the air one has to assume about k = 0.6. From the effective temperature T = 6000° it follows for the temperature of the outer boundary τ = 5050°. The depth h is calculated from a point at which the temperature is 1 1/2 times this boundary temperature. This provides the basis for the calculation.

To obtain the corresponding table for an atmosphere of hydrogen the depth A would have to be multiplied by 14.4, divide the density ϱ by 14.4 on the one hand, and multiply by a factor that indicates by how much the same mass of hydrogen is more transparent than air. The two columns t and m remain unchanged.

It can be seen that the radiation equilibrium, with increasing elevation above the sun, approaches more and more towards the isothermal equilibrium, which corresponds to the limit temperature τ, and, that it, like this, theoretically results in an infinite extension of the atmosphere.

3. Stability of the radiation equilibrium.

Of particular interest is a comparison of the temperature gradient in the case of radiative and adiabatic equilibrium. If the temperature gradient is smaller than in the case of adiabatic equilibrium, an ascending air mass moves into layers that are warmer and thinner than it arrives, it therefore experiences downward pressure. In the same way, a descending air mass experiences upward pressure. An equilibrium with a smaller temperature gradient than the adiabatic one is therefore stable, conversely one with a greater temperature gradient is unstable.

For adiabatic equilibrium according to (6):

for radiative equilibrium according to (15):

Stability condition is therefore:

which is always fulfilled for k > 4/8.

The radiation equilibrium is therefore stable everywhere as long as the gas that forms the atmosphere is monatomic, diatomic or triatomic. For polyatomic gases, instability would occur in deeper layers (of higher temperature t).

The idea is therefore suggested here that an outer shell of the solar atmosphere is in stable radiative equilibrium while perhaps in the depths there is a zone of ascending and descending currents approximating adiabatic equilibrium currents, which will then at the same time ensure the extraction of energy from its actual sources.

4. Brightness distribution on the solar disk.

Each temperature distribution along the vertical in the solar atmosphere corresponds to a certain distribution of brightness on the solar disk.

We had previously considered the total energy A, which travels outwards through the solar atmosphere without separating the individual components, which run under different inclinations against the vertical, and we had denoted the absorption coefficient for the total energy with a. This a gives an average value of the absorption coefficients valid for all possible inclinations.

We now want to consider the radiation traveling in a certain direction in isolation and denote by F(i) just the radiation which moves at an angle i to the vertical. Let α denote the absorption coefficient for radiation, which passes vertically (“normal”) through the atmosphere. Then obviously α/cos i is the absorption coefficient for radiation traveling at angle i. Therefore, in complete analogy to (8), we obtain for F the differential equation:

if you introduce the abbreviation:

The integration gives the radiation emerging from the atmosphere:

F(i) can therefore be calculated as soon as the temperature distribution along the vertical and thus E as a function of µ is known.

Now, however µ is related to the optical mass m introduced earlier. Consider the total radiation that falls on a horizontal surface element ds within the atmosphere from below. This is given by the integral over the radiation coming in from all possible directions:

The absorption suffered by this radiation within the layer dh will be:

The previously used absorption coefficient for the total energy a was defined by the relationship:

The comparison with the above formulae results in:

If F(i) is reasonably constant for small inclinations i and only changes more rapidly for i in the vicinity of 90° – as this is the case with the sun according to the empirical results shown below – an approximation for a is obtained by considering F(i) to be independent of i, thereby then follows by performing the integrals:

We will make use of this relationship. It follows from (9) and (19):

and thus, instead of (20):

Therefore, F(i) is now known, as soon as E is given as a function of the optical mass m. However, the function F(i) also immediately provides the brightness distribution on the solar disk. This is because the radiation that we observe on the solar disk at the apparent distance r from the center of the disk has obviously passed through the solar atmosphere at an angle i, which is determined by the equation:

Where R is the apparent solar radius. The connection of (22) and (23) provides the corresponding brightness F for each r.

The relationship between the radiation distribution in the depth E and the brightness distribution on the surface F becomes very clear, if E can be developed into a power series according to m:

Then it follows immediately from (22):

Can E be represented as a sum of fractional powers of m:

it follows for F:

where T means the tau-function. Here, too, the transition from E to F is still easy to accomplish.

In particular, we want to look at how the brightness distribution presents itself in the case of adiabatic and radiative equilibrium.

For radiative equilibrium, the following applies according to (11):

From this follows according to (24) and (25):

or, if one takes the brightness in the center of the solar disk (i = 0) as the unit:

For adiabatic equilibrium, the relationship (5) was valid:

If we further assume, as above, that the absorption is the same for all colors and proportional to the density, then between p and m have the relationship (13):

Thereby we obtain for E:

where c₁ and c₂ are new constants. The corresponding expression of F is calculated according to (26) and (27):

or, if the central brightness is again chosen as the unit:

The formulae (28) and (29) are to be compared with the observation. Apart from the spectral-photometric investigations for individual color areas, which are not of use in this context, there are a number of measurements that are made with thermopiles and bolometers and which indicate how the total radiation delivered by all wavelengths simultaneously, is distributed over the solar disk. Mr. G. Müller has assembled these measurements in his Photometrie der Gestirne page 323 and combined them into the mean values which are shown in the second column of the table below. The theoretical values for radiative equilibrium and adiabatic equilibrium according to the formulas (28) and (29) are next to them. For the adiabatic equilibrium, k = ⁴/s which corresponds to a 3-atomic gas is chosen. Mono- or diatomic gases would give an even worse match and the physical probability certainly speaks against more than 3-atomic gases in the outer parts of the solar atmosphere.

Measurement/radiative equil./Adiabatic equil.

It can be seen, that the radiative equilibrium depicts the distribution of brightness on the solar disk as well as can be expected under the simplified assumptions, under which the calculations have been made here, that the adiabatic equilibrium would result in a completely different appearance of the solar disk. The introduction of the radiative equilibrium has thus found a certain empirical justification.

[Ed. Note. Thus, Schwarzschild showed that the outer portion of the Sun’s atmosphere conforms to a radiative equilibrium solution. But he specifically excludes any portion of the Sun that is convective. All emphasis in this post is in the original paper.]

The original paper in German can be found here.

The text is lightly edited to make the English more readable by Andy May. Tom Shula also contributed to this post.

*Kirchhoff’s law of thermal radiation states that for an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity. In other words, Schwarzschild’s model requires a loss-free absorption and re-emission process. Under the convective conditions of Earth´s troposphere, such a “conservation of radiative energy” or loss-free conversion of heat back into radiation is not possible. Some of the energy is put to work powering convection and weather and is not recoverable. Therefore Kirchhoff´s law of radiation is not applicable to our troposphere.