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.
Andy: I’m writing here because I don’t want to hijack the discussion onto a topic irrelevant to Roger’s post.
Schwarzschild’s equation for radiation transfer is written in forms for different purposes and to use different units. Above, equations 7 and 8 are mathematically equivalent to what I report as the Schwarzschild’s equation for radiative transfer in my Wikipedia article. In Schwarzschild’s terminology above, “E is the emission of a black body of the temperature of this altitude layer”, which is what all call the Planck Function B_lambda(T) at the local temperature. A and B are the inward and outward radiation intensity, which is I_lambda in my terminology. dh and ds are increments of distance radiation is traveling. Schwarzschild writes Equations 7 and 8 as derivatives, while I have written the corresponding differential equation ready for integrating on both sides of the equals sign:
dA = -a*(E-A)*dh
Schwarzschild doesn’t appear to have rigorously derived Equations 7 and 8 in his 1906 article. For rigorous derivatives, see the freely available Hermann Harde paper, citation 11, in my Wikipedia. My “handwaving” derivation was patched together from Petty’s book.
As I noted elsewhere, my goal in writing the Wikipedia article was to make Schwarzschild equation and its role in calculating forcing for climate change accessible to technically competent people like you.
Another post you might like is here; it was written by myself and Tom Shula and reviewed by Markus.
https://andymaypetrophysicist.com/2025/02/01/energy-and-matter/
Planck’s Law is classically derived by assuming a Boltzmann distribution of excited states at a given temperature T. Such a distribution in a gas is derived by collisional excitation and collisional relaxation of gas molecules and depends ONLY on temperature. An excited state will emit a photon after a characteristic time period (which I’m told is an average of about 1 second for 15 um line of CO2).
Of course, excited states can also be created by absorption of a photon and relaxed by emission of one. When excited states are created and relaxed much faster by collisions than by photons (or some other phenomena?), the fraction of molecules in an excited state capable of emitting a photon is determined only by the temperature. We say those molecules are in Local Thermodynamic Equilibrium (LTE). Planck’s Law, Schwarzschild’s Equation, Kirchhoff’s Law and the Boltzmann Distribution are only valid when LTE exists.
When the local radiation field is strong enough that the fraction of molecules in an excited state is no longer solely determined by collisional excitation and relaxation (and therefore by temperature), then LTE doesn’t exist and these equations are no longer valid. In the Earth’s atmosphere, LTE exists up to about an altitude of about 100 kilometers. Incoming and outgoing radiation isn’t changed appreciably above about 70 km. Therefore Schwarzschild’s Equation can be used to calculate radiation transfer at all relevant altitudes through an atmosphere containing both absorbers and emitters (GHGs).
Above 100 km, IIRC, some scientists refer to a Boltzmann temperature (T_b) that depends on the average kinetic energy in the gas molecules and a Planck Temperature (T_p). Others point to the definition of temperature as being proportional to the mean kinetic temperature of a large group of rapidly colliding molecules, and say temperature isn’t defined above 100 km. If temperature isn’t defined, then one doesn’t need to worry about heat always flowing from hot to cold.
Normally, a substance in LTE needs to be around 5000 K for collisions to create a significant number of excited states capable of emitting visible photons. LEDs, lasers and fluorescent lights are devices not in LTE that create excited states without collisional excitation. They are capable of making emitting visible light without large amounts of heat.
(Some of this discussion comes from Petty’s book. Some is taught in courses on statistical mechanics.)
Hi Frank,
WRT: “We say those molecules are in Local Thermodynamic Equilibrium (LTE). Planck’s Law, Schwarzschild’s Equation, Kirchhoff’s Law and the Boltzmann Distribution are only valid when LTE exists.”
This is true, but not what is done in practice. In small volumes, like the volume around a weather balloon, LTE exists. In larger volumes, such as used in climate models it does not, but the climate model assumes it does.
WRT: “Kirchhoff’s law applies in our troposphere: No energy is lost powering convection. For every parcel of gas moving upward in the atmosphere another one is moving downward. Pages 126-7 of Petty’s book discuss LTE in our atmosphere and Kirchhoff’s Law.”
This is from another comment. It is sort of true, convection does no net work. But that is not the issue. In atmospheric convective conditions the flow close to the surface becomes a net absorber and the high-altitude part becomes a net emitter. That is not compatible with Kirchhoff´s Law. For Kirchhoff´s law to be applicable each layer must absorb as much radiation as it emits and vice versa. In Schwarzschild´s solar atmosphere Kirchhoff´s law becomes a “radiation-energy-conservation-law” and such a law does not exist in molecular spectroscopy. Look at page 53 of PDF I link below by Markus Ott, he explains this in detail. This is closely related to the LTE issue, a small atmospheric volume (say a cubic meter) can be in LTE, but not square km.
https://andymaypetrophysicist.com/wp-content/uploads/2026/02/Dismantling-the-CO2-Hoax-01112021.pdf
Hello Frank,
In a convection current the flow close to the surface is a net absorber of radiation and the part along the upper troposphere is a net emitter of radiation. Kirchhoff´s law requires that each layer absorbs as much radiation as it emits. That is the reason why Schwarzschlid constructs his model in a way that there is no convection. Kirchhoff´s law is not compatible with convection. Constructing the LTE-fiction does not change that. It only distracts from the fact, that Schwarzschlid himself puts great emphasis on the fact that his model has to exclude convection to make a radiative equilibrium possible. In the context of radiation transfer, Kirchhoff´s law becomes a “radiation-energy-conservation-law”. Such a Law does not exist in molecular spectroscopy. Accordingly Schwarzschild´s solutions for the radiation transfer equations are not applicable to convective systems like the troposphere.
Markus and Andy: It’s called LOCAL thermodynamic equilibrium because is doesn’t apply to large volumes of atmosphere. Molecular collisions are a relatively inefficient way of transferring heat through the air; air is a good insulator. Radiation is faster and convection even faster (in the troposphere). That is why there is a linear lapse rate in the troposphere. If there were no convection, temperature would increase exponentially as you approach the surface (and linearly with optical density).
Consider a thought experiment where one adds more and more and more GHG to an atmosphere. The surface warms exponentially as the atmosphere becomes more opaque to thermal IR until the lapse rate becomes unstable to [buoyancy-driven] convection. As you add more and more GHG, heat must be convected higher and higher until you reach an altitude where the atmosphere is thin enough for the flux of outgoing radiation to balance the incoming radiation. That is called “radiative-convective equilibrium” between incoming and outgoing radiation. This concept was advanced by Manabe in the 1960s. Wikipedia has an article on this subject, but Science of Doom has better ones somewhere.
When doing radiative transfer calculations using Schwarzschild’s equation, one need to numerically integrate over distances short enough that the intensity of the incoming radiation and density, temperature, pressure, composition, absorption cross-section of the GHG aren’t changing with distance. (All of these quantities are really functions of altitude, but for clarity we write T rather than T(z) and for the Planck Function P(T, lambda) rather than P(T(z),lambda). So there is effective LTE in each increment we numerically integrate, but not along the entire path (for example from the surface to space).
When I was discussing radiative-convective equilibrium, I should have made it clear that in the troposphere where the lapse rate is controlled by convection, net vertical transfer of heat still occurs. With a linear lapse rate, there is atmosphere is not in radiative equilibrium. If there were no convection, the atmosphere would be in radiative equilibrium and there would be no NET vertical transfer of heat, and the lapse rate is curved.
In the stratosphere, although there is some flow, radiative heat transfer is much slower than convective heat transfer. So temperature can be calculated by radiative transfer calculations. Parts of the stratosphere, however, are significantly warmed by absorption of UV radiation by ozone and oxygen and that distorts the lapse rate.
True, no argument there.
Hello Marcus: Sorry I’m so slow to reply to you in addition to Andy. I didn’t fully grasp your comments about a convection current close to the surface being a net absorber of radiation and one running along the top of the troposphere being a net emitter. You are right, but there is a fundamental principle involved here: the 2LoT. Heat flows (net energy transfer) from warmer to colder and that applies to radiation as well as sensible heat. That principle applies EVERYWHERE in the troposphere, where it is typically colder the higher the altitude, not just the at the bottom or the top of the troposphere, but everywhere between also. Each layer of troposphere usually absorbs more radiation from below (where it is usually warmer) than from above (where it is usually colder). As a result, upwelling radiation slowly weakens on its way from the surface (where is averages 390 W/m2) to space (where it averages 240 W/m2). DLR strengthens from 0 W/m2 at the edge of space to an average of 333 W/m2 at the surface for the same reason.
This behavior arises directly from the Schwarzschild equation:
dI = n*o*( (B(T) – I )*ds
When the intensity of incoming radiation (I) from one direction is stronger than Blackbody emission at the local temperature (B(T)), dI is negative. When the intensity of incoming radiation (I) from another direction is weaker than Blackbody emission at the local temperature (B(T)), dI is positive. In a sense, the 2LoT EMERGES from the behavior of large numbers of molecules and photons obeying the laws of quantum mechanics. (This is taught in statistics mechanics.) In the case of radiative transfer of heat, that law is the Schwarzschild equation (if LTE exists). The behavior of single molecules and photons is irrelevant, because temperature and heat flow are only defined for large groups of rapidly colliding molecules. Molecules have kinetic energy – which changes from collisions about a million times a second – but not a temperature. The temperature of a large group of molecules isn’t changed by those millions of collision – temperature only changes when energy comes in or goes out from the group.
NOW, you are absolutely correct in saying that the temperature gradient in the troposphere (that controls the direction of radiative heat transfer) is created by vertical convection. It arises from both the expansion and cooling of air as it rises and the pressure drops; and from the release of latent heat from the condensation of water vapor. In the Hadley circulation, areas of vertical convection are connected by horizontal convection at the surface and near the top of the troposphere. However, in our atmosphere, that expansion isn’t completely adiabatic, because radiation is constantly bringing energy into and out of each parcel of rising or falling air. We OBSERVE an “adiabatic lapse rate” in the troposphere BECAUSE heat transfer by vertical convection is much faster than heat transfer by radiation (or molecular collisions, ie conduction which would make an isothermal atmosphere). Below the tropopause, our atmosphere is too opaque to thermal IR for radiation to carry away as much heat as incoming SWR delivers; but above the tropopause radiative cooling can balance the heat delivered by the sun. Convection is needed to get heat out of the opaque troposphere to an altitude high enough that it can escape to space fast enough only by radiative cooling.
The stratosphere is complicated by the heat absorbed from incoming solar UV by O2 and O3. Temperature (in the tropics) rises from a low of about 200K at an altitude of 17 km to 270 K at 50 kilometers. We would expect outgoing thermal radiation to strengthen as it passes through the warming stratosphere and it does at the strongest absorbing lines of CO2 and O3. If you go to the online Modtran calculator and “look down” down from 70 km in a cloudless tropical atmosphere, you’ll see the intensity of the outgoing radiation increases in the very center of the CO2 absorption notch (and the O3 at 1040?). If you look down from the top of the troposphere at 17 km, that line in the center of the CO2 notch is gone and the whole CO2 notch is deeper. Radiation transfer in the stratosphere – which warms with altitude – changes the spectrum of outward LWR because of absorption and emission. Those same changes can be observed assuming one is using a detector with enough wavelength resolution.
https://climatemodels.uchicago.edu/modtran/
Andy: Is there any more we should discuss about my article on Schwarzschild’s equation for radiative transfer? Or are you trying to absorb new information from Petty (a professor of meteorology/atmospheric physics, not climate science) and Harde (a physicist and a climate skeptic, who has conveniently derived all of the equations of radiation transfer from first principles going back to Einstein’s equations for emission, stimulated emission and absorption of photons? When I don’t understand theory as well as I wish I did, I like to fall back on experiments in our atmosphere that show the detailed spectra of radiation transfer agree with the predictions of Schwarzschild’s Equation. While I’m here I may leave some comments on emissivity.
Hi Frank, I don’t think this is true:
“I like to fall back on experiments in our atmosphere that show the detailed spectra of radiation transfer agree with the predictions of Schwarzschild’s Equation”
As explained in figure 3 or this post:
https://andymaypetrophysicist.com/2025/02/01/energy-and-matter/
OLR varies linearly with surface temperature, not in a radiative way as predicted by Schwarzschild’s equation.
I don’t have anything to add to the discussion. To me the bottom line is I don’t think Schwarzschild’s equation applies to Earth’s lower atmosphere or any atmosphere where convection is dominant. Once you get high enough (above the troposphere) where convection is minimal it probably works.
Remember, the lapse rate is negative in the troposphere and does not fit what Schwarzschild predicts, that’s why radiative models have to impose a lapse rate.
Above you write: “OLR varies linearly with surface temperature, not in a radiative way as predicted by Schwarzschild’s equation”.
Nevertheless, Koll and Cronin (2018) which you have discussed says exactly the opposite in the first sentence of its abstract:
“Satellite measurements and radiative calculations show that Earth’s outgoing longwave radiation (OLR) is an essentially linear function of surface temperature over a wide range of temperatures (60 K).”
In other words, the observed approximate linearity of OLR with surface temperature can be explained by radiative transfer calculations performed by numerically integrating Schwarzschild’s Equation for Radiative Transfer from the surface to space at different latitudes using the observed temperature and amounts of water vapor and other GHGs in that path.
Remember, unlike climate models that attempt to simulate everything about the atmosphere, the Schwarzschild Equation only calculates the change in radiation along an increment of distance where temperature, pressure and the composition of GHGs are essentially constant inputs taken from observations of the atmosphere. All those changes are numerically integrated. In tropical atmospheres, the upwelling radiation emitted by a 300 K ocean surface is initially more intense than in polar regions where the ocean surface is 273 K. (300/273)^4 is 1.46-fold more initial upward flux from the surface in the tropics than a surface at freezing. However, there are more absorptions and emissions along the path to space in the tropics because there is more water vapor. For upward traveling radiation in the troposphere, the temperature where a photon is emitted is always warmer than at the altitude it is absorbed. When the absorbing altitude emits upward traveling photons, there will be slightly fewer photons than were absorbed. That reduces the upward flux from 460 W/m2 at the surface to 298 W/m2 at space (according to Modtran) and some vertical radiative transfer of heat to the atmosphere from below. According to Modtran, only 235 W/m2 reach space from a a midlatitude surface at 272K in winter emitting 315 W/m2 (according to S-B). The reduction in upward LWR on the way to space in the tropics is 35% and in midlatitude winter is 25%.
If you assume that radiative transfer is the ONLY mechanism of heat transfer in an atmosphere, you can use the Schwarzschild equation to start with any lapse rate and by repeated approximation calculate the shape of the lapse rate in our atmosphere when a steady state has been reached with no net vertical flux of heat. Manabe did so in a graph you have shown. However, you can’t use the Schwarzschild equation without inputing the temperature, pressure and composition atmosphere at all altitudes. There is an online facility for doing so at: https://climatemodels.uchicago.edu/modtran/
If we rewrite the Schwarzschild equation:
dI = n*o*( B(T) – I )*ds
dI is always negative when (B(T) – I) is negative. This is always the case for upwelling radiation in the troposphere where temperature drops with altitude and upwelling radiation was emitted from where it is warmer. Increasing the density of GHGs (n) makes dI more negative, slowing radiative cooling to space (radiative forcing). However, the reduction in upward flux isn’t linear with (n) because the upward distance traveled and temperature change between emission and absorption decreases with increasing density of GHGs (n).
You discuss at your site other more complicated phenomena like clouds that modify the consequences of radiative forcing. When treated as feedbacks (responses to warming, not responses to GHGs), these phenomena modify the consequences of radiative forcing, but can not negate them. We know from watching seasonal changes in OLR and reflected SWR, that climate models do a lousy job of reproducing the feedbacks to season warming that we observe from space. Feedbacks are where I focus my attention as a skeptic, not radiative forcing. https://www.pnas.org/doi/epdf/10.1073/pnas.1216174110
First of all, to me, these two statements ae equivalent and not opposite:
“OLR varies linearly with surface temperature, not in a radiative way as predicted by Schwarzschild’s equation”.
equals:
“Satellite measurements and radiative calculations show that Earth’s outgoing longwave radiation (OLR) is an essentially linear function of surface temperature over a wide range of temperatures (60 K).”
I’m not sure why you think they are opposite. See here for more discussion on that point:
https://andymaypetrophysicist.com/2025/02/01/energy-and-matter/
Second, what you are describing is exactly what I am saying, convection provides the heat transport in the lower atmosphere, not radiative transport. Emissions from the surface are captured in the lower atmosphere by GHGs or clouds, then heat is carried as sensible or latent heat to its emission height where it is emitted to space. This is why the gradient is linear as opposed to curved. Convective heat transfer is linear per Newton’s law, radiative heat transfer is curved per Schwarzschild, as he explicitly says in his 1906 paper.
The abstract says: “Satellite measurements and radiative calculations show that Earth’s outgoing longwave radiation (OLR) is an essentially linear function of surface temperature over a wide range of temperatures (60 K).”
You say: I’m not sure why you think they are opposite. See here for more discussion on that point.” The radiative transfer calculations done by Koll and Cronin (2018) are done using some implementation of Schwarzschild’s Equation. What we observe from space (linearity in OLR with temperature) is what we predict for the ATMOSPHERE WE OBSERVE using Schwarzschild’s equation for radiative transfer. Our atmosphere behaves in the approximately linear fashion is does because a major GHG, water vapor, is condensible and slows radiative cooling to space more where it is warmer on the surface than where it is colder. You can see that in the simple data I got from Modtran comparing a tropical atmosphere with a midwinter midlatitude atmosphere.
I have been confused about emissivity and therefore Kirchhoff’s Law for years, but may have some useful insights. First, the emissivity of a solid metal depends greatly on how smoothly the surface is polished. Emissivity MUST be a phenomena produced at the surface or interface of an object.
The simplest interface to think about is that between air and water. About 5% of visible light is reflected or scattered entering water perpendicular to the surface, resulting an absorptivity of 0.95. By the principle of microscopic reversibility, 5% of visible light leaving water for air will also be reflected or scattered back into the water, meaning the emissivity of water is 0.95. That’s also Kirchhoff’s Law. Note: reflection increases as the angle of approach shallows and so does emissivity at a shallow angles. If the angle is shallow enough (as in fiber optic cables), internal reflection is complete and emissivity is effectively zero. Now let’s consider thermal infrared entering and leaving water. Water emits thermal IR with an emissivity of 0.95 to 0.963 depending on wavelength (temperature), so 3.7-5% must be reflected or scattered back into the water by the surface. And by the principle of microscopic reversibility, the absorptivity of thermal IR by water must be 0.95-0.963. However, thermal infrared is mostly absorbed the first 10 microns of water, so thermal infrared doesn’t “travel” appreciable distances “through” water, but it does pass thousand of water molecules between emission and exiting the surface.
Now let’s consider the interface between a solid and air. It makes perfect sense for thermal IR to be reflected or scattered at the surface between a gas and the solid as it entered the solid. However, it is harder to conceive of thermal infrared as traveling “through” a solid and being reflected or scattered by the surface as it tries to leave the solid. Nevertheless, its thermal emissivity is still equal to its absorptivity. Instead of saying thermal infrared travels “through” a solid, we talk about conduction or phonons. Whatever happens to those phenomena at a surface, the correct amount of energy is still reflected or scattered inward by the surface to make emissivity equal to absorptivity.
Now, does a gas have an emissivity? Of course not. There is no surface or interface between the Earth’s atmosphere and space where light can be reflected or scattered. Nevertheless textbooks still talk about the emissivity of a gas. They say that its emissivity is proportional to how much gas is present. Textbooks also tell us that materials have one of two kinds of properties, “intrinsic” or “extrinsic”. Intrinsic property like density don’t vary with the amount of a substance, but extrinsic properties like mass do. The emissivity of a liquid or solid in an intrinsic property of that material that is independent of mass, but the emissivity of a gas is an extrinsic property that varies with mass???? That can’t be right! Emissivity is being used in two different contexts, an intrinsic one for solids and liquids, but an extrinsic one for gases. We have every reason to feel confused by using the same term for two different things. Gases do NOT have a surface to create emissivity in the same way solids and liquids do.
If I test our ideas about emissivity more strenuously, I can even find apparent contradictions with solids. Gold is malleable enough that it can be pounded into a layer thin enough to see through. Its emissivity certainly changes when it is thin enough to be partially transparent. Emissivity doesn’t just happen at the surface, it depends on how thick the layer of gold is. We deposit a very thin, essentially transparent layer of silver or tin oxide on glass to make low emissivity windows. (and use two panes with argon between to slow down conduction through the glass). The emissivity of this thin layer is not the same as the bulk emissivity of these materials.
Hi Frank,
I feel your pain on gas versus liquid and solid emissivity. When textbooks say “emissivity of a gas” its really the emissivity of a gas *volume* or *column*. It also varies a lot with frequency and altitude.
In the atmosphere we speak of optical depth, and it is an effective value.
In summary, emissivity isn’t purely a surface phenomenon; it emerges from how radiation interacts with the material’s structure, including depth when transparency matters. For solids/liquids, we often approximate it as surface-dominated because they’re opaque, but the full picture includes bulk effects. Optical depth is really the key value.
As I prefer to write Schwarzschild’s Equation:
dI = noB(T)*ds – noI*ds
where the first term is the emission (not emissivity) from an increment of path (ds) and the second term is the absorption along that increment of path ds; and n is it density of absorbers/emitter, o is their absorption cross-section and I is the intensity of the radiation entering that increment of path.
optical depth = nos. Optical depth is the key value, but for emission, not emissivity.
The Schwarzschild equation tells us that as radiation passes through a homogeneous medium of temperature T, absorption and emission cause its intensity to approach “Blackbody Intensity” at a rate (with distance traveled) proportional to the density of absorbers/emitters, and their absorption cross-section. When absorption and emission reach “equilibrium” the radiation has “Blackbody Intensity. Planck’s law is derived by assuming a radiation field where absorption and emission are in equilibrium.
At ScienceofDoom, the host has a post on “Theory and Experiment” showing a small amount of work showing that the Schwarzschild Equation makes correct predictions about radiation traveling in our atmosphere.
https://scienceofdoom.com/2010/11/01/theory-and-experiment-atmospheric-radiation/
Far more has been done, but not presented in the convenient form of overlapping spectra. Climate models use shortcuts to reduce the the amount of calculation needed to perform line-by-line radiative transfer calculations. Intermodal comparison projects have been done to validate or invalidate these shortcuts and numerous experiments have been run comparing observed radiative transfer in the atmosphere to the predictions of the Schwarzschild Equation. There are some minor discrepancies, some caused by the dimerization of water vapor molecules at high humidity near the surface. These molecules create a weak continuum absorption. We don’t have absorption cross-sections for these dimers, but they don’t have a significant impact on OLR.
In a sense, the Schwarzschild equation “can’t be wrong” without also invalidating quantum mechanics, since you can see Harde derive Schwarzschild’s equation from Einstein coefficients characterizing emission, stimulated emission and absorption of photons. However, as a pragmatic matter, the equation does a great job of predicting the radiative transfer we observe in our atmosphere. IMO, Postulating that this equation doesn’t apply in some situations isn’t likely to get anywhere, especially when you aren’t using the standard understanding of Local Thermodynamic Equilibrium.
It isn’t that Schwarzschild is wrong, it is just that it is inapplicable to our lower atmosphere (mainly the troposphere, but also a portion of the lower stratosphere). It is inapplicable because heat transfer in the lower atmosphere is largely done by convection and latent heat transfer to the emission height.
The emission height is above the clouds, which cover about 2/3 of the planet. The actual height (often called the TOA) depends upon local conditions, but when reached more radiation is emitted to space than absorbed.
The convective processes short circuit the radiative processes, which Schwarzschild describes well. Convection changes the lower temperature profiles as shown by Manabe and Strickler in figure 2 here:
https://andymaypetrophysicist.com/2025/02/01/energy-and-matter/
Andy says: “It isn’t that Schwarzschild is wrong, it is just that it is inapplicable to our lower atmosphere (mainly the troposphere, but also a portion of the lower stratosphere). It is inapplicable because heat transfer in the lower atmosphere is largely done by convection and latent heat transfer to the emission height.”
Inapplicable for what? The Schwarzschild Equation can’t predict what the temperature should be anywhere in the troposphere because temperature is an INPUT to, not an OUTPUT from performing radiative transfer calculations using the Schwarzschild Equation. You are confusing radiative transfer calculations with climate models and Weather Forecast models, which do predict temperature. There is a radiative transfer calculation module within climate and weather prediction models that calculates radiative heat transfer between grid cells in those models, but those models also need to simulate convection, condensation and precipitation, which occur on scales vastly smaller than the grid cells they use and therefore must be represented by parameters that are tuned. [Line-by-line] Radiative transfer calculations depend on a large database of measured absorption cross-sections and phenomena that broaden absorption lines that have been carefully measured in the laboratory.
Now, if you postulate an atmosphere where no other phenomena besides radiation transfer energy, you can start with an arbitrary lapse rate – say isothermal – and calculate the radiation transfer along a vertical path through that atmosphere from the surface to space. The change in flux at increment along the path will allow you to calculate a new temperature for that increment. (If dI is negative, the power lost from the vertical flux (more absorption than emission) has gone into warming that increment. If dI is positive the increment has been cooled by more emission than absorption.) If you repeat that process many times, you can approach a lapse rate where a steady-state between absorption and emission exists everywhere from the surface to space. That is how the lapse rate for purely radiative equilibrium was calculated in Manabe and Strickler.
The TOA or “top of the atmosphere” has nothing to do with cloud tops. It is the altitude in our atmosphere above which absorption and emission by gas molecules no longer has a significant impact on the intensity of radiation leaving or entering the atmosphere. That happens about 70 km. When using Schwarzschild’s equation, you need to numerically integrate radiative transfer along a defined path – say from the surface to a particular altitude. If you choose to end at 70 km, you have effectively reached space, the TOA.
TOA has as many definitions as authors that use the abbreviation, which is why I normally don’t use it. Sometimes it is where satellite radiometers measure incoming and outgoing radiation, sometimes it is where net incoming and outgoing fluxes are equal, which is somewhere above the cloud tops. Either way, below the cloud tops convection dominates and the Schwarzschild relationship is invalid. It is stated in his paper clearly.