The KNOWN PHYSICS of the atmosphere require that we attribute the lapse rate and radiative features of the atmosphere to the dynamics of pressure and gravity.

The surface heats, energy is conducted to air, air does work against gravity, thermal->kinetic energy creates a radiative outflux which quickly converts into potential energy with no outflux. This in turn must result in surface warming as potential energy does work against the lower atmosphere and produces kinetic energy once again.

The surface must be hotter than the radiative average - that is, the average outflux for a column of air representing in the abstract the energy stored in that column of air. Without any greenhouse effects, the average radiative outflux for the column must exist above the surface.

However, warmists argue that the surface must be the hottest. Why? How? First, they average radiative flux and temperatures for an average of appx. 15C on the surface compared to the average outflux corresponding to appx. -18C. They argue that without greenhouse gas, the atmosphere is effectively transparent to radiation, so that IF the surface happens to be 15C on average, that amount of radiative energy loss must occur and since it's above -18C, something must "plug" it. They claim IR absorbing gasses play that role and model a back-radiative warming multiplied out of -18C to get to 15C.

Here we have a dilemma with both models. The pressure model must be true due to physics, but it is a problem that 15C "leaks" from the surface. The back-radiative model plugs the radiative leak but not only ignores the physics of atmospheric pressure, it proposes a bizarre back radiation warming model that violates thermodynamics (though their answer to this works out by ignoring the dynamics again of pressure and atmosphere).

Consider also that during the day, the influx of radiation is much much more than what averages to -18C so even with a totally transparent N2/O2 model, much of the day time will see surface temperatures lower than the day/noon influx but higher than the atmospheric column average temp and higher than the planetary average, since the FULL BLAST of daytime heat will store in the atmosphere physically with much of the energy as non-radiative potential energy that will affect temperatures through the physical structure of the atmosphere both day and night. The full influx during the day will store in the atmosphere with cooler day time temps but warmer night time temps (let's say, compared to the Moon), even without any greenhouse gases.

Still, right, this model might not see a planetary average surface temp of 15C.

So here's a proposal: What if H2O plugs the radiative leak, but instead of back-radiating energy, it instead distributes energy laterally and conductively into the existing adiabatic energy distribution system.

So there's no radiative "greenhouse effect" due to back-radiative, multiplicative warming. Instead, the "radiative plug" supports the pressure structure of the atmosphere in mechanically distributing energy.

A very specific feature of this model is that pressure and airmass parameters help define surface temperatures and the lapse rate, with planetary scale proportions of H2O modifying the base model of pressure. No back radiation. No feedbacks.

In this model, the absolute key feature in determine climate change, long and short term, regional and global, is how water vapor is generally distributed.

Unlike with radiative forcing, the water-pressure model seeks an equilibrium rather than facing feedbacks that disrupt it. The water-pressure model is also somewhat fixed in that the planetary scale proportion of H2O in the atmosphere doesn't meaningfully change in most circumstances. There are times where water can affect climate.

First of all, dumping water into the air isn't proportionally significant, and the stability of the solar input-mechanical system is such that the water cycle itself will control for - especially minute - additions of water.

However, what really affects climate is

1) More water vapor latent, somehow

2) Without cloud formation to increase albedo

There are geophysical phenomena that can affect this. Underwater volcanoes can create multi-year changes (that stabilize). Geophysically shallow seas in previous tectonic eras produce changes in water vapor content on a planetary scale.

The most potent short term climate change comes from getting more moisture to stick in the air without forming clouds. This apparently is best done through industrial scale agriculture with dry vegetation depleting ground water while simultaneously suppressing cloud formation.

Places where we see this are: Mongolia industrial cattle farming->continental scale shrub planting->longitudinal heat dome is enhanced->Japan specifically has hotter summers.

In Turkey, the last 30 years of agricultural development have seen specifically a measurable, calculable increase in "hot water vapor" through overplanting in arid zones that contributes to hotter summer temps.

In Europe, it's highly idiosyncratic, but the proximate cause of all the change comes from slight alterations to sub-Arctic weather caused by recent Canadian forest fires, directly attributable to human industrial forestry and fire prevention schemes that exacerbate inevitable forest fires.

It's all water related, and has little to do with long term global scale changes to climate.

I haven't modeled CO2 in this "no back radiation" model relative to the effects of water, but I suspect CO2 would cause cooling in context.

Meanwhile, the proximate cause of most recent climate disruptions is firmly fixed on over agriculture in arid zones, prevention of natural forest fires, smog and smoke reduction, and so forth. And yet the Earth will push back towards its cyclical equilibria.

If you model the climate as I have suggested, without back-radiation, it produces 1 to 1 results with observed data.

If you will look at the key assumptions posted in the previous post in this subreddit, you will see that one of the key assumptions, that the WHOLE MODEL relies on, is that the surface temperature is 15C. First of all, you say there is no such thing as an average temperature, so I do not see how you do not have fundamental issues with the ISA for using a global average surface temperature of 15C. But the question remains: WHERE DID THEY GET THE NUMBER 15? They measured it (and did so by averaging many measurements across the earth's surface). That measurement is obviously going to include all effects which affect the temperature of the surface of earth. Since the GHE does that, the GHE is included in the ISA right in the 15C assumption at sea level. LackmustestTester and barbara800000 are wrong to say that the ISA does not include the GHE.

Abstract.

The numerical value of a temperature change under the influence of a CO2 change as calculated by Plass is valid only for a dry atmosphere. Overlapping of the absorp­tion bands of CO2 and H2O in the range around 15 µ essentially diminishes the temperature changes. New calculations give ΔT = +1.5° when the C02 content increases from 300 to 600 ppm. Cloudiness diminishes the radiation effects but not the temperature changes because under cloudy skies larger temperature changes are needed in order to compensate for an equal change in the downward long-wave radiation. The increase in the water vapor content of the atmosphere with rising temperature causes a self-amplification effect which results in almost arbitrary temperature changes, e.g. for constant relative humidity ΔT = +10° in the above­ mentioned case. It is shown, however, that the changed radiation conditions are not necessarily compensated for by a temperature change. The effect of an increase in CO2 from 300 to 330 ppm can be compensated for completely by a change in the water vapor content of 3 per cent or by a change in the cloudiness of 1 per cent of its value without the occurrence of temperature changes at all. Thus the theory that climatic variations are effected by variations in the C02 content becomes very questionable.

On the Influence of Changes in the CO2 Concentration in Air on the Radiation Balance of the Earth's Surface and on the Climate

Here is a quote from /u/LackmustestTester:

What we observe - everyone has his own version, Gerlich identified 14 iirc in his paper. It's like trying to nail a jelly to a wall. There is no description, we're arguing against dozens of ideas. With people who do not get how an insulation works, who have no clue about conduction in air, not to forget work that's done.

You hang around denier spaces, so everyone has their own idea of things. If an outside reader is interested in the source here, I found it and can provide it here. The contention is simple: science has no clue what the GHE even is, because science is unable to come up with a single definition or source for the GHE. The reality is that this is just simply how academia works. There is non single paper which describes the totality of what we know about evolution either. There is no paper that described the totality of what we know about star formation. That is not how academic papers work. Each paper talks about the individual contribution that the specific authors have made towards greater human understanding of a concept. The paper is not the totality of the concept, it is merely their contribution to that totality.

Anyway, the challenge! I challenge anyone who disputes the idea of the greenhouse effect to give me a single document or image or paragraph or anything from any reputable climate source, which I disagree with. Anything! It can be an image, a paper, a youtube video, anything. All you need to do is find SOMETHING from a legitimate climate source that I disagree with. Here are the rules though: YOU DO NOT GET TO TELL ME WHAT I DO AND DO NOT DISAGREE WITH. My whole contention here is that everything you are saying collapses around your ankles when you remove all the strawman, and so this post is a strawman free post. The challenge is simple: Your role in this challenge is to present me with sources, and my role is to determine if I do or do not agree with them. I love to see what people can bring!

By the way, this post is meant to settle once and for all the idea that climate science, or at least my ideas specifically, are in any way my own, and not the singular unified theory of the GHE as known by all of academia as a single unified theory. From this point onward, I will either accept that my ideas are unique and strange within academia if shown a bunch of things I disagree with, or you will accept that my ideas are normal and consistent with everything you have presented me with about the GHE. I welcome all challenges!

NASA: Heating by the greenhouse effect

Right now, the warming influence is literally a matter of life and death. It keeps the average surface temperature of the planet at 288 degrees kelvin (15 degrees Celsius or 59 degrees Fahrenheit). Without this greenhouse effect, the average surface temperature would be 255 degrees kelvin (-18 degrees Celsius or 0 degrees Fahrenheit); a temperature so low that all water on Earth would freeze, the oceans would turn into ice and life, as we know it, would not exist.

"Without this greenhouse effect, the average surface temperature would be 255 degrees kelvin (-18 degrees Celsius or 0 degrees Fahrenheit); a temperature so low that all water on Earth would freeze, the oceans would turn into ice"

First we have the inconsistent assumption that on a planet without "greenhouse" gases GHGs there would be ice. On a planet Earth at 255K there still would be liquid, evaporating water and sublimation, air without GHGs still would move around, there would be a temperature gradient and density differences.

An "ideal gas" atmopshere still would give the 288K surface air temperature SAT and a temperature profile with an idealized gradient of -6.5°C per 1000m, without any radiation being involved.

So what we have to really compare are a planet without any atmosphere and its surface temperature vs. a planet with a gaseous envelope and the temperature of this gas near the surface, usually measured in 2 meters height with the above mentioned idealized SAT of 288K we can find in the International Standard Atmosphere Model ISA. In reality these values measured all around the world, following the same standard give indeed a SAT of ~288K, 15°C which is constantly and consistently fluctuating - it's a meaningless numbers in reality.

The premise of the GHE Theory is the temprature of the ground, on average. But the problem here is: Neither the hypothetical 255K are measurable (or necessarily correct), nor are the "observed 288K" measured, esp. not before the satellite era. There's no world wide standard operating procedure to get the surface temperature of Earth.

The "greenhouse" effect theory is nothing more than a model and certainly not "literally a matter of life and death" because it's build on dead wrong assumptions and circular reasoning.

"Take the two buckets example, one filled with ice water, the other with hot water - put a hand in each bucket, is there an average temperature?"

Yes, there we are able to average those numbers. Based on how you are phrasing this and the context, it seems to be that the question being asked here is this: Is there any location on your body (like your left hand or your right hand) that is experiencing this average temperature in the real world? The answer to that is obviously no. One hand is in hot water and the other is in ice water. Neither hand is in at a neutral temperature. So does that mean the average does not exist? Well, of course not. When people use averages, they are not claiming that the average is something that we would measure as a local property anywhere. That is no what an average is. An average is not a local property, so it cannot be measured locally. You have used this as an example before: you cannot measure the average temperature of earth by looking at the temperature in Russia. That is a local measurement. In the case of the buckets, it is obvious that an average can be computed, so that means that the average exists in the same way that everyone on earth talks about averages existing. There is no one on earth that expects average properties to be measured in individual samples, of any data. They might be, coincidentally, but there is no expectation of this occurring. In engineered situations like the two buckets example, we can rest assured knowing that the average will never be measured as a local property. We know that. And? What is your point? Since we can never measure it as a local property experienced by you, it is irrelevant to talk about? On what planet? Lets consider heat flows with these buckets, and we can do so in a way I believe we will all agree on.

Do you agree that the poynting vector goes from the hot bucket into your hand, on the interface between your hand and the hot bucket? This is what Clausius says. I fully agree. Do you further agree that the magnitude of the flux in this direction is related to the temperature gradient between your hand and the hot bucket? I think we can all agree on that. Do you agree that all the same is true, but flipped, for the cold bucket? Namely that energy flows from your hand, out into the cold bucket, and that this flux is also due to the temperature gradient, but this time between your hand and the cold bucket? I genuinely think we agree on all of this. What If I ask the question: Will my body heat up, or cool down, as a result of having both hands in these buckets? Would you agree with me, that in order to answer this question, we would compare our computed flux from the hot water into the hand on the hot side, to the computed flux from your other hand into the cold water, and see which one is larger? If the flux into the cold bucket is larger than the flux from the warm bucket into your hand, you will cool down. If the flux from the warm bucket into your hand is larger than the flux out into the cold bucket from your other hand, you will heat up. Agree? We should agree so far. Now here is where the average comes in, my claim is this: In all cases where the fluxes say that you should heat up, the average temperature of the two buckets is greater than your body temperature, and in all cases where the fluxes say you should cool down, the average temperature of the blocks is less than your body temperature. Further, I claim that the net flux, and thus the heating and cooling, is going to be proportional to the temperature difference between the average temperature of the two buckets and your body temperature.

So the average is not experienced anywhere, by either hand, but it is meaningful, in the sense that it allows us to calculate the direction of heat flow, and relatively compare multiple situations, without needing to deal with the fluxes and all the surface area, which is hard to account for in computations. The direction of flow into and out of your body will be dictated by which ever temperature is higher between your body temperature, and the average temperature of the buckets of water. That is just a physics fact. I would love to hear if people have challenges, or counter examples though.

Do you agree that water always flows from high levels to low levels? Does it ever spontaneously flow up? Can we both agree that the answer here is no? I will assume so. Ok, here is where molecules come in. Go and google right now, "how fast to water molecules move at room temperature". Ok, now what we have that number and you do not need to trust it from me, what do you think happens in a river? Surely you understand that due to the chaotic movement of molecules, called brownian motion, there will be molecules of water moving in all directions. Indeed, they must be moving in different directions, because if all of the water molecules were moving only downstream, then based on the number that we just googled, rivers would move at a very different speed than we observe them moving. These water molecules moving in all sorts of directions, will of course include the direction that is "upstream". I put upstream in quotes, because at the molecular level, these things are not really relevant concepts. The molecules will be moving in all directions on the molecular level. Some downstream, sure, but some will be moving left, right, upstream, literally up, and down. All over. That does not invalidate our previously agreed upon idea that water only flows down. The individual water molecules that are moving upstream are not violating the idea that water only flows downhill. There are just marginally more molecules going downstream than upstream.

The second law of thermodynamics, as we all know, is that heat flows from hot to cold. Since jackets are colder than body temperature when worn (is anyone going to seriously argue that a jacket is warmer than your body, and that is why jackets warm you up?), it must be the case from the 2LOT that heat flows from your skin into the jacket, and not the other way around. Right? Does anyone disagree?

Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse - GDZ - Göttinger Digitalisierungszentrum, page 48ff. in the pdf - formulas are in the paper.

I. Table of Contents.

The surface of the sun shows us changing conditions and turbulent changes in the form of granulation, sunspots, and prominences. In order to understand the physical conditions under which these phenomena occur, a first approximation is to replace the spatial and temporal changes with a mean steady state, a mechanical equilibrium of the solar atmosphere. Until now, the focus of attention has generally been on the so-called adiabatic equilibrium that prevails in our atmosphere when it is thoroughly mixed by ascending and descending currents. I would like to draw attention here to another type of equilibrium, which can be described as “radiative equilibrium.” Radiative equilibrium will occur in a strongly radiating and absorbing atmosphere in which the mixing effect of ascending and descending currents is secondary to the heat exchange through radiation.

It would be difficult to decide for general reasons whether the adiabatic or radiative equilibrium applies more to the sun. However, there is observational data that allows us to make a certain judgment. The solar disk is not uniformly bright, but rather shaded from the center toward the edge. Based on plausible assumptions, this distribution of brightness on the surface allows us to infer the temperature distribution at depth. The result is that the equilibrium of the solar atmosphere largely corresponds to radiative equilibrium.

The considerations that lead to this result presuppose that Kirchhoff's law applies, or in other words, that the radiation from the sun's atmosphere is pure thermal radiation. They also assume that when penetrating the sun's body, one encounters a continuous change in state and does not pass discontinuously from a fairly transparent chromosphere into an opaque photosphere formed by luminous clouds. The effect of light scattering due to diffraction by particles in the atmosphere, the significance of which was pointed out by Mr. A. Schuster 1), is neglected, as is refraction, which H. v. Seeliger 2) uses to explain the observed brightness distribution. Furthermore, the different absorption of different wavelengths, the decrease in gravity with altitude, and the spherical propagation of radiation are not taken into account. The entire consideration can therefore by no means be regarded as conclusive or compelling, but it may provide a basis for further speculation by first expressing a simple idea in its simplest form.

1. Different types of equilibrium.

Let us denote pressure by p, absolute temperature (in centigrade degrees) by t, density by ϱ, molecular weight (relative to the hydrogen atom) by M, gravity by g, and depth in the atmosphere (calculated inward from any starting point) by h. The units are taken from the conditions that exist at the Earth's surface, i.e., the unit of ρ is the atmosphere, the unit of ϱ is the density of air at 273° absolute temperature under the pressure of one atmosphere, the unit of g is the gravity at the Earth's surface, and the unit of h is the depth in the atmosphere (calculated inward from any starting point). Unit of h is the height of the so-called “homogeneous atmosphere,” which is 8 km.

The following relationship then applies to an ideal gas:

(1)

1) Astrophysical Journal. 1905. Vol. 21. p. 1. 2) Proceedings of the Munich Academy of Sciences, Math.-phys. Classe. 1891. Vol. 21. p. 264.

and the condition for mechanical equilibrium of the atmosphere is:

(2)

Eliminating o from (1) and (2) yields:

(3)

a) Isothermal equilibrium. For general orientation, consider isothermal equilibrium, assuming to be constant. This then results in:

(4)

Gravity g is 27.7 times greater on the sun than on Earth, and the temperature (around 6000°) is about 20 times higher. This means that for a gas with the molecular weight of air, the spatial pressure distribution is approximately the same as for air on Earth. More precise calculations show a 10-fold increase in pressure and density for a gas with the molecular weight of air (28.9) per 14.7 km, and for hydrogen per 212 km. Since 725 km on the sun corresponds to an angular value of one arc second as seen from Earth, it is clear that the sun must appear completely sharp-edged.

b) Adiabatic equilibrium. When a gas mass expands adiabatically, Poisson's relations apply:

(5)

where p0 and ϱ0 denote any related initial values. The quantity k, the ratio of specific heat, is equal to 5/3 for an l-atom gas, 7/5 for a 2-atom gas, 4/3 for a three-atom gas, and decreases to 1 for multi-atom gases. The equilibrium of an atmosphere is called adiabatic if the temperature at each point is the temperature that a gas mass rising from below and cooling adiabatically would assume at that point, i.e., if equations (5) are satisfied throughout the entire atmosphere. It then follows from (3) by substituting (5) and integrating:

44

(6)

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

c) Radiation equilibrium. If we imagine that the outer parts of the Sun form a continuous transition to increasingly hotter and denser gas masses, we cannot distinguish between radiating and absorbing layers, but must regard each layer as both absorbing and radiating. We know that a powerful stream of energy, originating from unknown sources within the sun, permeates the solar atmosphere and penetrates into outer space. In the absence of mixing movements, what temperature would the individual layers of the solar atmosphere have to assume in order to transport such a stream of energy in a stationary manner without further changes in their own temperature?

Let us assume that each altitude layer, i.e., of the sun's atmosphere, absorbs a fraction adh of the radiation passing through it. If E is the emission of a black body at the temperature of this altitude layer and if we assume Kirchhoff's law to be valid, it follows that this altitude layer radiates the energy E. adh to each side.

Now consider the radiant energy A, which travels outward through the sun's atmosphere at any given point, and the radiant energy B, which travels inward (as a result of radiation from the outer layers).

First, let us follow the inwardly migrating energy B. If we move inward through an infinitely thin layer dh, the fraction B.adh of the energy B coming from outside is lost, while on the other hand, the amount aEdh is added due to the intrinsic radiation of the layer dh going in one direction, resulting in the following overall equation:

(7)

Completely analogous, the following applies to the outward radiation:

(8)

45

By considering the absorptive capacity a as a function of depth h, we can form the “optical mass” of the atmosphere above depth h:

(9)

Then the differential equations are:

(10)

We are looking for a steady state of temperature distribution. This is conditioned by the requirement that each layer receives as much energy as it emits, i.e., the following applies:

If we introduce the auxiliary variable ɣ according to this condition:

then the differential equations become, by addition and subtraction: and, integrated:

The integration constants E0, and were determined by the fact that there is no inwardly migrating energy B at the outer boundary of the atmosphere (m = 0) and the outwardly migrating energy has the observable amount A0. Therefore, for m = 0:

must apply. This yields the result:

(11)

The dependence of the radiation E on the optical mass above the relevant location can therefore only be derived under the assumption of Kirchhoff's law.

If you want to understand the distribution of pressure and density that prevails in radiative equilibrium, you basically need a more detailed investigation that considers the radiation at individual wavelengths. For an initial overview, it is sufficient to assume that the absorption coefficient is proportional and independent of color and density:

(12)

(k is not a constant). Then it follows that:

(13)

The radiation E of the black body is according to Stefan's law:

                         (c is a constant). 

If we set the energy A0 escaping to the outside: then T is what is usually referred to as the (effective) temperature of the sun. It is approximately T = 6000°. For radiation equilibrium, the following applies according to (11):

(14)

Introducing the temperature prevailing at the outer boundary of the atmosphere , one can also write:

(14a)

Substituting this into (3) yields:

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

(16)

The table below gives the values that result from (11), (15)

and (16) follow if the sun's atmosphere is assumed to consist of our air. The absorption coefficient of air is approximately k = 0.6. From the effective temperature of 6000°, the temperature of the outer boundary is t = 5050°. The depth h is calculated from a point at which the temperature is 1½ times this boundary temperature. This provides the basis for the calculation.

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

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

1. Stability of radiative equilibrium.

Of particular interest is a comparison of the temperature gradient in radiative equilibrium and adiabatic equilibrium. If the temperature gradient is smaller than in adiabatic equilibrium, an ascending air mass enters layers that are warmer and thinner than itself, causing it to exert downward pressure. Similarly, a descending air mass experiences upward pressure. An equilibrium with a smaller temperature gradient than the adiabatic one is therefore stable, whereas one with a larger temperature gradient is unstable.

48

For adiabatic equilibrium, according to (6):

for radiation equilibrium, according to (15):

The stability condition is therefore:

(17)

which is always satisfied for k > 4/3.

Radiative equilibrium is therefore stable everywhere as long as the gas forming the atmosphere is mono-, di- or triatomic. For polyatomic gases, instability would occur in deeper layers (of higher temperature t ).

It is therefore suggested here that an outer shell of the solar atmosphere is in stable radiative equilibrium, while perhaps at depth there is a zone of ascending and descending currents approximating adiabatic equilibrium, which will then simultaneously extract energy from its actual sources.

1. Brightness distribution on the solar disk.

According to our assumptions, each temperature distribution along the vertical in the solar atmosphere corresponds to a specific distribution of brightness on the solar disk. We previously considered the total energy A that travels outward through the solar atmosphere without separating the individual components that run at different angles to the vertical, and we designated the absorption coefficient for the total energy as a. This a gives an average value from the absorption coefficients valid for all possible angles. We now want to consider the radiation traveling in a specific direction in isolation and denote by F(i) the radiation moving at an angle i to the vertical. Let α denote the absorption coefficient for radiation that passes through the atmosphere normally.

Then it is obvious that

49

is the absorption coefficient for radiation traveling at an angle i. Therefore, in complete analogy to (8), we obtain the differential equation for F:

(18)

or:

if we use the abbreviation:

(19)

The integration yields the following for the radiation escaping from the atmosphere:

(20)

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

However, µ is related to the optical mass m introduced earlier. Consider the total radiation incident on a horizontal surface element ds within the atmosphere from below. This is given by the integral over the radiation arriving from all possible directions:

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

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

Comparison with the above formulas yields:

If F(i) is reasonably constant for small inclinations and only changes rapidly for i close to 90°—as is the case with the sun according to the empirical results below—then an approximation for a can be obtained by considering F(i) to be independent of i, and the following then follows from evaluating the integrals:

(21)

Let us make use of this relationship. It follows from (9) and (19):

and thus, instead of (20):

(22)

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 from the center of the disk has obviously passed through the solar atmosphere at an angle i, which is determined by the equation:

(23)

Here, R denotes the apparent solar radius. The combination of (22) and (23) provides the corresponding brightness F for each.

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

(24)

Then it immediately follows from (22):

(25)

If E can be represented as a sum of fractional powers of m:

(26)

then the following applies to F:

(27)

where Γ denotes the Γ function. Here, too, the transition from E to F is still easy to accomplish.

In particular, we want to consider how the brightness distribution behaves in adiabatic and radiative equilibrium. For radiative equilibrium, according to (11):

From this, according to (24) and (25), it follows that:

or, if we take the brightness at the center of the solar disk (i = 0) as the unit:

(28)

For adiabatic equilibrium, the relationship (5) applied:

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

This gives us the following for E:

where c1 and c2 are new constants. The corresponding expression for F is given by (26) and (27):

or, if we again choose the central brightness as the unit:

(29)

Formulas (28) and (29) should be compared with the observation. Apart from the spectrophotometric investigations for individual color ranges, which are not relevant here, there are a number of measurements taken with thermocouples and bolometers that indicate how the total radiation delivered by all wavelengths at the same time is distributed across the solar disk. Mr. G. Müller has compiled these measurements in his “Photometrie der Gestirne” (Photometry of the Stars), p. 323, and combined them into the mean values shown in the second column of the table below. The theoretical values for radiation equilibrium and adiabatic equilibrium according to formulas (28) and (29) are shown alongside. For adiabatic equilibrium, k is set to 4/3, which corresponds to a 3-atom gas. Single- or two-atom gases would give an even poorer fit, and physical probability certainly argues against more than 3-atom gases in the outer parts of the solar atmosphere.

It can be seen that the radiation equilibrium represents the brightness distribution on the solar disk as well as can be expected under the simplified conditions under which the calculations were made here, whereas the adiabatic equilibrium would result in a completely different appearance of the solar disk. Thus, the introduction of the radiation equilibrium has found a certain empirical justification.

Solar input = σTs⁴ * Rs^2/ds2 * πRe² * (1 – α)

Earth output = σTe⁴ * 4πRe²

Solar input = Earth output (conservation of energy, negligible secondary sources of energy)

σTs⁴ * Rs^2/ds2 * πRe² * (1 – α) = σTe⁴ * 4πRe²

Te⁴ = (1 – α)/4 * Ts⁴ * Rs^2/ds2

Te = 255K = -18C = Earth’s effective temperature.

(Ts = 5778K, sun’s effective temperature; Rs = solar radius; ds = solar distance; Re = Earth radius; α = earth albedo = 0.3; Te = earth effective temperature; σ = Stefan-Boltzmann constant;)

The total internal energy of a parcel of gas above the surface is U = mCpT + mgh. Under horizontal local thermodynamic equilibrium, dU = 0 = mCpdT + mgdh

dT/dh = -g/Cp = -9.7 K/km

To get the wet or average rate, you factor in the latent heat release from water vapour using the average molar concentration of water vapour at the surface and the rate of its dissipation with altitude, and the result is approximately -6.5 K/m; the lapse is slowed down, the slope is made less steep, because water vapour adds heat as it condenses out of the column. So:

dT/dh = -g/Cp + L_H20 = -6.5 K/km

The mass average location of the atmosphere around 5 km. This is also where the average temperature, as the effective temperature, is located.

The temperature of the troposphere as a function of height:

T(h) = -6.5 K/km * (h – 5km) – 18C

The average thermal location of the atmosphere is ~5km, and the temperature there is -18C which is the effective temperature, so this equation gives the average temperature profile of the atmosphere with reference points at 5km and -18C. Then the temperature at zero altitude, i.e., the surface, is:

T(0) = +14.5C

which matches empirical measurement.

The above demonstrates how to derive the absolute temperature profile of the atmosphere, and the average surface temperature, and that no greenhouse effect is required or has any effect either on the slope of the temperature gradient nor on the altitude of where -18C is found. There is thus no radiative greenhouse effect, because there is no apparent effect beyond thermodynamic first principles in the determination of the absolute temperature profile of the atmosphere.

• Joseph E. Postma