The great physicist Richard Feynman adds to three other giants of physics, Maxwell, Clausius, and Carnot, who have explained the "greenhouse effect" is solely a consequence of gravity, atmospheric mass, pressure, density, and heat capacities, and is not due to "trapped radiation" from IR-active or 'greenhouse' gas concentrations.
Only one 33C greenhouse theory can be correct, either the 33C Arrhenius radiative greenhouse theory (the basis of CAGW alarm and climate models) or the 33C Maxwell/Clausius/Carnot/Feynman gravito-thermal greenhouse effect, since if both were true, the surface temperature would be an additional 33C warmer than the present. As we have previously shown, the Arrhenius greenhouse theory confuses the cause (gravito-thermal) with the effect (radiation from greenhouse gases).
In addition, the US Standard Atmosphere, the International Standard Atmosphere, the HS 'greenhouse equation,' Chilingar, et al derive the observed atmospheric temperature profile without use of a single greenhouse gas radiative transfer equation or calculation, and using the same basic atmospheric physics discussed by Feynman in his lecture below. Feynman does not make a single mention of radiation, radiative transfer, greenhouse gases, CO2, nor does he derive any radiative transfer equations to derive the atmospheric temperature profile, and instead utilizes the barometric and statistical mechanics formulas necessary to describe the gravito-thermal greenhouse effect of Maxwell et al (who Feynman quotes extensively below). Feynman demonstrates that an atmosphere comprised solely of the non-greenhouse gases N2 & O2 (99.94% of our atmosphere, but 100% in Feynman's demonstrations) would establish the temperature gradient/"greenhouse effect" observed in the troposphere.
Feynman demonstrates that the conservative force of gravity does indeed do continuous thermodynamic Work upon the atmosphere (a common false argument by those who do not accept the gravito-thermal GHE theory is that gravity allegedly can't do Work upon the atmosphere), and describes gravitational potential energy (PE) accumulated as air parcels rise/expand/cool, which is then exchanged for kinetic energy (KE) as the air parcel descends/compresses/warms, creating the temperature gradient & greenhouse effect.
Another online version here with larger print
Showing posts sorted by relevance for query feynman. Sort by date Show all posts
Showing posts sorted by relevance for query feynman. Sort by date Show all posts
Tuesday, July 28, 2015
Thursday, October 22, 2015
New paper finds Gleissberg cycle of solar activity related to ocean oscillations, land temperatures, & extreme weather
A new paper published in Advances in Space Research, finds,
The authors find,
The paper is coauthored by Joan Feynman (sister of the famous physicist Richard Feynman).
The recent extended, deep minimum of solar variability and the extended minima in the 19th and 20th centuries (1810–1830 and 1900–1920) are consistent with minima of the Centennial Gleissberg Cycle (CGC), a 90–100 year variation of the amplitude of the 11-year sunspot cycle observed on the Sun and at the Earth. The Earth’s climate response to these prolonged low solar radiation inputs involves heat transfer to the deep ocean causing a time lag longer than a decade. The spatial pattern of the climate response, which allows distinguishing the CGC forcing from other climate forcings, is dominated by the Pacific North American pattern (PNA). The CGC minima, sometimes coincidently in combination with volcanic forcing, are associated with severe weather extremes. Thus the 19th century CGC minimum, coexisted with volcanic eruptions, led to especially cold conditions in United States, Canada and Western Europe.
"The recent extended, deep minimum of solar variability and the extended minima in the 19th and 20th centuries (1810–1830 and 1900–1920) are consistent with minima of the Centennial Gleissberg Cycle (CGC), a 90–100 year variation of the amplitude of the 11-year sunspot cycle observed on the Sun and at the Earth. The Earth’s climate response to these prolonged low solar radiation inputs involves heat transfer to the deep ocean causing a time lag longer than a decade."
The authors find,
"The spatial pattern of the climate response [to the Gleissberg solar activity cycle]... is dominated by the Pacific North American pattern (PNA). The Gleissberg minima, sometimes coincidently in combination with volcanic forcing, are associated with severe weather extremes. Thus the 19th century Gleissberg minimum, which coexisted with volcanic eruptions, led to especially cold conditions in United States, Canada and Western Europe."The paper shows clear evidence in the first graph below of a significant, sustained increase of Total Solar Irradiance (TSI) from 1700 to the late 20th century, coincident with the end of the Little Ice Age ~1850 and the global warming observed during the 20th century.
The paper is coauthored by Joan Feynman (sister of the famous physicist Richard Feynman).
 |
| Total Solar Irradiance in top graph shows a significant increase of solar activity since 1700. Second wavelet graph shows periodicity (red areas) corresponding to the 90-100 year Gleissberg cycle of solar activity. Bottom graph shows smoothed Gleissberg cycles since 1700. |
 |
| Second graph solid line shows Total Solar Irradiance correlates with observed land temperatures (dashed line). |
The Earth’s climate at minima of Centennial Gleissberg Cycles
The recent extended, deep minimum of solar variability and the extended minima in the 19th and 20th centuries (1810–1830 and 1900–1920) are consistent with minima of the Centennial Gleissberg Cycle (CGC), a 90–100 year variation of the amplitude of the 11-year sunspot cycle observed on the Sun and at the Earth. The Earth’s climate response to these prolonged low solar radiation inputs involves heat transfer to the deep ocean causing a time lag longer than a decade. The spatial pattern of the climate response, which allows distinguishing the CGC forcing from other climate forcings, is dominated by the Pacific North American pattern (PNA). The CGC minima, sometimes coincidently in combination with volcanic forcing, are associated with severe weather extremes. Thus the 19th century CGC minimum, coexisted with volcanic eruptions, led to especially cold conditions in United States, Canada and Western Europe.
Related:
New paper argues current lull in solar activity is consistent with a Gleissberg Cycle minimum
Wednesday, December 18, 2013
Paper strongly supports the solar/cosmic ray theory of climate
A 1999 paper co-authored by astrophysicist Dr. Joan Feynman [sister of the famous physicist Richard Feynman] is strongly supportive of the Svensmark cosmic ray theory of climate. The paper shows a remarkable correlation of Earth's high-latitude area open to cosmic rays [which may seed cloud formation] and global temperature.
According to the authors, "High energy cosmic rays may influence the formation of clouds and thus impact weather and climate. Due to systematic solar wind changes, the intensity of cosmic rays incident on the magnetopause has decreased markedly during this century. The pattern of cosmic ray precipitation through the magnetosphere to the upper troposphere has also changed. Early in the century, the part of the troposphere open to cosmic rays of all energies was typically confined to a relatively small high-latitude region. As the century progressed the size of this region increased by over 25% and there was a 6.5° equatorward shift in the yearly averaged latitudinal position of the subauroral region in which cloud cover has been shown to be cosmic ray flux dependent. We suggest these changes in cosmic ray intensity and latitude distribution may have influenced climate change during the last 100 years."
Modulation of cosmic ray precipitation related to climate [full paper]
Geophysical Research Letters abstract 1999
J. Feynman, A. Ruzmaikin
High energy cosmic rays may influence the formation of clouds and thus impact weather and climate. Due to systematic solar wind changes, the intensity of cosmic rays incident on the magnetopause has decreased markedly during this century. The pattern of cosmic ray precipitation through the magnetosphere to the upper troposphere has also changed. Early in the century, the part of the troposphere open to cosmic rays of all energies was typically confined to a relatively small high-latitude region. As the century progressed the size of this region increased by over 25% and there was a 6.5° equatorward shift in the yearly averaged latitudinal position of the subauroral region in which cloud cover has been shown to be cosmic ray flux dependent. We suggest these changes in cosmic ray intensity and latitude distribution may have influenced climate change during the last 100 years.
According to the authors, "High energy cosmic rays may influence the formation of clouds and thus impact weather and climate. Due to systematic solar wind changes, the intensity of cosmic rays incident on the magnetopause has decreased markedly during this century. The pattern of cosmic ray precipitation through the magnetosphere to the upper troposphere has also changed. Early in the century, the part of the troposphere open to cosmic rays of all energies was typically confined to a relatively small high-latitude region. As the century progressed the size of this region increased by over 25% and there was a 6.5° equatorward shift in the yearly averaged latitudinal position of the subauroral region in which cloud cover has been shown to be cosmic ray flux dependent. We suggest these changes in cosmic ray intensity and latitude distribution may have influenced climate change during the last 100 years."
Modulation of cosmic ray precipitation related to climate [full paper]
Geophysical Research Letters abstract 1999
J. Feynman, A. Ruzmaikin
High energy cosmic rays may influence the formation of clouds and thus impact weather and climate. Due to systematic solar wind changes, the intensity of cosmic rays incident on the magnetopause has decreased markedly during this century. The pattern of cosmic ray precipitation through the magnetosphere to the upper troposphere has also changed. Early in the century, the part of the troposphere open to cosmic rays of all energies was typically confined to a relatively small high-latitude region. As the century progressed the size of this region increased by over 25% and there was a 6.5° equatorward shift in the yearly averaged latitudinal position of the subauroral region in which cloud cover has been shown to be cosmic ray flux dependent. We suggest these changes in cosmic ray intensity and latitude distribution may have influenced climate change during the last 100 years.
Sunday, October 27, 2013
Huffington Post shows how anthropogenic global warming circumvents the 1st Law of Thermodynamics
A 'physicist'/blogger on the Huffington Post admits that while "The laws of physics dictate that energy is conserved," somehow the Earth/atmosphere system is exempted in the case of global warming, since "more energy is coming in than going out." His claim is directly opposed by satellite measurements, the NASA Earth Observatory site, and many, many other references clearly demonstrating that in accordance with the 1st Law of Thermodynamics, the "flow of energy into the atmosphere must be balanced by an equal flow of energy out of the atmosphere and back to space."
In addition, outgoing infrared radiation to space from greenhouse gases has increased over the past 62 years, instead of decreased as predicted by AGW theory, proving that there is no observational evidence for influence of CO2 on present or past climate.
From the NASA Earth Observatory:
The Earth's Energy Budget II -- Radiation Emitted by the Earth, the Greenhouse Effect, and the Overall Energy Balance
The Amount of Energy the Earth Radiates to Space Equals the Amount Received Received from the Sun, So Energy going Out Equals Energy Coming In
and about 525,000 other references.
Mark Boslough Physicist; Fellow of the Committee for Skeptical Inquiry, Huffington Post
When I was a first-year graduate student at Caltech, my Ph.D. adviser published a paper called "Impact-induced energy partitioning." He asked how an asteroid's energy would change form if it collided with the Earth. He used computer models to estimate what fraction would go into lofting debris, heating, melting, vaporizing rocks, and so on.
This subject was not settled science then, and is still not. One thing is for sure, however. The laws of physics dictate that energy is conserved. If an asteroid is hurtling toward your city, you might not be concerned that scientists are not 100 percent certain about how its energy will be "partitioned."
Global warming is no different.
Another one of my professors was Richard Feynman. In his famous "Feynman Lectures" he had a chapter called "Conservation of Energy" in which he says:
"There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no exception to this law - it is exact so far as we know. The law is called conservation of energy."
He imagined a child with toy blocks that are indestructible and can't be divided. His mother counts the blocks every day and discovers a phenomenal law - no matter what he does with the blocks, there is always the same number at the end of the day.
One day there is a missing block, but she investigates and finds one under the rug. Another day the number comes up two short, but she discovers an open window and two blocks are outside.
Then a surplus occurs, but it turns out a friend came over and left some blocks.
After that, mom locks the window and bans visits, but eventually the count comes up short again. There's a toy box in the room but it's locked and only her son has the key. So she weighs it. The changes in weight correlate exactly to the missing blocks from day to day.
Then the weight stops changing, but the water level in the dirty bathtub starts going up and down in a way that is exactly proportional to the difference in missing blocks.
Feynman even provides the equations that mom uses, and he beats the analogy into our heads. The blocks are like energy, which is neither created nor destroyed.
His punchline is this: Energy has different forms, and there is a formula for each one. "If we total up the formulas for each of these contributions, it will not change except for energy going in and out."
Human-caused global warming is the inevitable consequence of this law of physics, because greenhouse gas pollution is causing more energy to come in than go out.
If the average surface temperature - which is only one way to measure global warming - doesn't go up every year, it's because the "blocks" are being hidden somewhere else, for now.
But energy changes forms, and sloshes back and forth between sub-systems. Ice will continue to melt and sea level will continue to rise as the water warms. A slowdown in one rate is compensated by a speedup in another until the cycle of natural variability reverses.
Scientists know that more energy is coming in than going out. We can measure it and there is no dispute. [we have measured it and energy in = energy out]
Because of carbon pollution, the Earth is gaining energy at the rate of 400,000 Hiroshima atomic bombs every day of every year. And that rate is going up. [nope]
Not knowing exactly how energy is "partitioned" into its various forms at any given time is like not knowing how much melting an asteroid will cause, how small the pieces will be when a boiler explodes, how the victims of a drunk driver will die, or how many minutes it will take for the Titanic to sink.
Uncertainty in exactly how something happens does not translate into uncertainty that it is happening. There is no rational basis for denial of the reality, or the risks, of global warming.
And there is no excuse for ignoring it.
This essay is reprinted from the Albuquerque Journal.
 |
Source: Dr. Noor van Andel: There is no observational evidence for influence of CO2 on present or past climate |
The Atmosphere’s Energy Budget
Just as the incoming and outgoing energy at the Earth’s surface must balance, the flow of energy into the atmosphere must be balanced by an equal flow of energy out of the atmosphere and back to space.
The Earth's Energy Budget II -- Radiation Emitted by the Earth, the Greenhouse Effect, and the Overall Energy Balance
For the entire planet, the radiation energy in equals the radiation energy out
The Amount of Energy the Earth Radiates to Space Equals the Amount Received Received from the Sun, So Energy going Out Equals Energy Coming In
and about 525,000 other references.
Global Warming, Asteroid Impacts, and the Laws of Physics
Mark Boslough Physicist; Fellow of the Committee for Skeptical Inquiry, Huffington PostWhen I was a first-year graduate student at Caltech, my Ph.D. adviser published a paper called "Impact-induced energy partitioning." He asked how an asteroid's energy would change form if it collided with the Earth. He used computer models to estimate what fraction would go into lofting debris, heating, melting, vaporizing rocks, and so on.
This subject was not settled science then, and is still not. One thing is for sure, however. The laws of physics dictate that energy is conserved. If an asteroid is hurtling toward your city, you might not be concerned that scientists are not 100 percent certain about how its energy will be "partitioned."
Global warming is no different.
Another one of my professors was Richard Feynman. In his famous "Feynman Lectures" he had a chapter called "Conservation of Energy" in which he says:
"There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no exception to this law - it is exact so far as we know. The law is called conservation of energy."
He imagined a child with toy blocks that are indestructible and can't be divided. His mother counts the blocks every day and discovers a phenomenal law - no matter what he does with the blocks, there is always the same number at the end of the day.
One day there is a missing block, but she investigates and finds one under the rug. Another day the number comes up two short, but she discovers an open window and two blocks are outside.
Then a surplus occurs, but it turns out a friend came over and left some blocks.
After that, mom locks the window and bans visits, but eventually the count comes up short again. There's a toy box in the room but it's locked and only her son has the key. So she weighs it. The changes in weight correlate exactly to the missing blocks from day to day.
Then the weight stops changing, but the water level in the dirty bathtub starts going up and down in a way that is exactly proportional to the difference in missing blocks.
Feynman even provides the equations that mom uses, and he beats the analogy into our heads. The blocks are like energy, which is neither created nor destroyed.
His punchline is this: Energy has different forms, and there is a formula for each one. "If we total up the formulas for each of these contributions, it will not change except for energy going in and out."
Human-caused global warming is the inevitable consequence of this law of physics, because greenhouse gas pollution is causing more energy to come in than go out.
If the average surface temperature - which is only one way to measure global warming - doesn't go up every year, it's because the "blocks" are being hidden somewhere else, for now.
But energy changes forms, and sloshes back and forth between sub-systems. Ice will continue to melt and sea level will continue to rise as the water warms. A slowdown in one rate is compensated by a speedup in another until the cycle of natural variability reverses.
Scientists know that more energy is coming in than going out. We can measure it and there is no dispute. [we have measured it and energy in = energy out]
Because of carbon pollution, the Earth is gaining energy at the rate of 400,000 Hiroshima atomic bombs every day of every year. And that rate is going up. [nope]
Not knowing exactly how energy is "partitioned" into its various forms at any given time is like not knowing how much melting an asteroid will cause, how small the pieces will be when a boiler explodes, how the victims of a drunk driver will die, or how many minutes it will take for the Titanic to sink.
Uncertainty in exactly how something happens does not translate into uncertainty that it is happening. There is no rational basis for denial of the reality, or the risks, of global warming.
And there is no excuse for ignoring it.
This essay is reprinted from the Albuquerque Journal.
Tuesday, October 27, 2015
Why Tyndall's experiment did not "prove" the theory of anthropogenic global warming
Many warmists cite Tyndall's 1861 experiment as "proof" of the catastrophic anthropogenic global warming theory, but in fact the experiment demonstrated only that CO2 and H2O are IR-active molecules capable of absorbing and emitting infrared radiation, nothing more.
Of course, CO2 does indeed absorb and emit very low-energy ~15 micron infrared radiation, equivalent to a "partial blackbody" at a temperature of 193K (-80C) by Wien's Law. However, radiation from a true or "partial" blackbody cannot warm the much warmer atmosphere (with an "average" temperature of 255K (-18C), equivalent to the equilibrium temperature of Earth with the Sun), nor the even warmer Earth surface at 288K (15C).
Yet the Arrhenius radiative greenhouse theory falsely assumes that "backradiation" from the 193K CO2 "partial blackbody" can warm the Earth surface temperature from the 255K equilibrium temperature with the Sun by 33K up to 288K. This would require a continuous and dominating heat transfer from cold to hot, thus requiring an impossible decrease of entropy, and therefore a gross violation of the Second Law of Thermodynamics (which requires entropy to increase from any transfer of heat).
In contrast, the alternative 33C gravito-thermal greenhouse theory of Poisson, Helmholtz, Maxwell, Boltzmann, Carnot, Clausius, Feynman, US Standard Atmosphere, International Standard Atmosphere, the HS greenhouse equation, et al instead fully explains the 33C 'greenhouse effect' on Earth, as well as on all 7 additional planets for which we have adequate data.
As we can see from the diagram of Tydall's apparatus below, it consists of a horizontal sealed tube containing the gas to be studied. Unlike the actual 100km Earth atmosphere, Tyndall's apparatus does not allow any vertical convective cooling as is found in the real Earth atmosphere. In fact, increased greenhouse gases accelerate convective cooling in the troposphere. Tyndall's apparatus artificially prevents this convective cooling, just like a sealed greenhouse does, but which does not happen in the real atmosphere.
Furthermore, as physicist William Happer points out, the probability of CO2 transferring quanta of energy in the troposphere via collisions instead of emitting a photon is one billion times more likely. This transfer of energy via collisions to the remaining 99.06% of the atmosphere causes acceleration of convective cooling by increasing the adiabatic expansion, rising, and cooling of air parcels. Convection dominates radiative-convective equilibrium in the troposphere by a factor of ~8 times and thus cancels any possible warming effect of the low-energy CO2 backradiation upon the surface.
Further, the presence of IR-active gases in the atmosphere only delays the ultimate passage of IR photons from the surface to space by a few seconds, and is easily reversed and erased during each 12 hour night, and explains why 'greenhouse gases' don't 'trap heat' in the atmosphere.
For these reasons, Tydall's experiment does not in any way prove the Arrhenius radiative greenhouse theory. In contrast, the alternative 33C gravito-thermal greenhouse theory of Poisson, Helmholtz, Maxwell, Boltzmann, Carnot, Clausius, Feynman, US Standard Atmosphere, International Standard Atmosphere, the HS greenhouse equation, et al instead fully explains the 33C 'greenhouse effect' on Earth, as well as on all 7 additional planets for which we have adequate data.
Of course, CO2 does indeed absorb and emit very low-energy ~15 micron infrared radiation, equivalent to a "partial blackbody" at a temperature of 193K (-80C) by Wien's Law. However, radiation from a true or "partial" blackbody cannot warm the much warmer atmosphere (with an "average" temperature of 255K (-18C), equivalent to the equilibrium temperature of Earth with the Sun), nor the even warmer Earth surface at 288K (15C).
Yet the Arrhenius radiative greenhouse theory falsely assumes that "backradiation" from the 193K CO2 "partial blackbody" can warm the Earth surface temperature from the 255K equilibrium temperature with the Sun by 33K up to 288K. This would require a continuous and dominating heat transfer from cold to hot, thus requiring an impossible decrease of entropy, and therefore a gross violation of the Second Law of Thermodynamics (which requires entropy to increase from any transfer of heat).
In contrast, the alternative 33C gravito-thermal greenhouse theory of Poisson, Helmholtz, Maxwell, Boltzmann, Carnot, Clausius, Feynman, US Standard Atmosphere, International Standard Atmosphere, the HS greenhouse equation, et al instead fully explains the 33C 'greenhouse effect' on Earth, as well as on all 7 additional planets for which we have adequate data.
As we can see from the diagram of Tydall's apparatus below, it consists of a horizontal sealed tube containing the gas to be studied. Unlike the actual 100km Earth atmosphere, Tyndall's apparatus does not allow any vertical convective cooling as is found in the real Earth atmosphere. In fact, increased greenhouse gases accelerate convective cooling in the troposphere. Tyndall's apparatus artificially prevents this convective cooling, just like a sealed greenhouse does, but which does not happen in the real atmosphere.
Furthermore, as physicist William Happer points out, the probability of CO2 transferring quanta of energy in the troposphere via collisions instead of emitting a photon is one billion times more likely. This transfer of energy via collisions to the remaining 99.06% of the atmosphere causes acceleration of convective cooling by increasing the adiabatic expansion, rising, and cooling of air parcels. Convection dominates radiative-convective equilibrium in the troposphere by a factor of ~8 times and thus cancels any possible warming effect of the low-energy CO2 backradiation upon the surface.
Further, the presence of IR-active gases in the atmosphere only delays the ultimate passage of IR photons from the surface to space by a few seconds, and is easily reversed and erased during each 12 hour night, and explains why 'greenhouse gases' don't 'trap heat' in the atmosphere.
For these reasons, Tydall's experiment does not in any way prove the Arrhenius radiative greenhouse theory. In contrast, the alternative 33C gravito-thermal greenhouse theory of Poisson, Helmholtz, Maxwell, Boltzmann, Carnot, Clausius, Feynman, US Standard Atmosphere, International Standard Atmosphere, the HS greenhouse equation, et al instead fully explains the 33C 'greenhouse effect' on Earth, as well as on all 7 additional planets for which we have adequate data.
Tyndall's Setup For Measuring Radiant Heat Absorption By Gases (source: Wikipedia)
This illustration dates from 1861 and it is taken from one of John Tyndall's presentations where he describes his setup for measuring the relative radiant-heat absorption of gases and vapors. The galvanometer quantifies the difference in temperature between the left and right sides of the thermopile. The reading on the galvanometer is settable to zero by moving the Heat Screen a bit closer or farther from the lefthand heat source. That is the only role for the heat source on the left. The heat source on the righthand side directs radiant heat into the long brass tube. The long brass tube is highly polished on the inside, which makes it a good reflector (and non-absorber) of the radiant heat inside the tube. Rock-salt (NaCl) is practically transparent to radiant heat, and so plugging the ends of the long brass tube with rock-salt plates allows radiant heat to move freely in and out at the tube endpoints, yet completely blocks the gas within from moving out. To begin the measurements, both heat sources are turned on, the long brass tube is evacuated as much as possible with an air suction pump, the galvanometer is set to zero, and then the gas under study is released into the long brass tube. The galvanometer is looked at again. The extent to which the galvanometer has changed from zero indicates the extent to which the gas has absorbed the radiant heat from the righthand heat source and blocked this heat from radiating to the thermopile through the tube. If a highly polished metal disc is placed in the space between the thermopile and the brass tube it will completely block the radiant heat coming out of the tube from reaching the thermopile, thereby deflecting the galvanometer by the maximum extent possible with respect to blockage in the tube. Thus the system has minimum and maximum readings available, and can express other readings in percentage terms. (The galvanometer's responsiveness was physically nonlinear, but well understood, and mathematically linearizable.)
In one of his public lectures to non-professional audiences Tyndall gave the following indication of instrument sensitivity: "My assistant stands several feet off. I turn the thermopile towards him. The heat from his face, even at this distance, produces a deflection of 90 degrees [on the galvanometer dial]. I turn the instrument towards a distant wall, judged to be a little below the average temperature of the room. The needle descends and passes to the other side of zero, declaring by this negative deflection that the pile feels the chill of the wall." (quote from Six Lectures On Light). To reduce interference from human bodies, the galvanometer was read through a telescope from across the room. The thermopile & galvanometer system was invented by Nobili and Melloni. Melloni measured radiant heat absorption in solids and liquids but didn't have the sensitivity for gases. Tyndall greatly improved the sensitivity of the overall setup (including putting an offsetting heat source on the other side of the thermopile, and putting the gas in a brass tube), and as a result of his superior apparatus he was able to confidently reach conclusions that were quite different from Melloni's concerning radiant heat in gases (book ref below, in chapter I). Air from which water vapor and carbon dioxide had been removed deflected the galvanometer dial by less than 1 degree, in other words a detectable but very small amount (same ref, chapter II). Many other gases and vapors deflected the galvanometer by a large amount -- thousands of times greater than air.
As a check on his system's reliability, Tyndall painted the inside walls of the brass tube with a strong absorber of radiant heat (namely lampblack). This greatly reduced the radiant heat that reached the thermopile when the tube was empty. Nevertheless the percentage absorptions by the different gases and vapors relative to the empty tube were largely and essentially unchanged by this change to the absorption property of the tube's walls. That's excluding a few gases and vapors such as chlorine that must be excluded because they tarnish brass, changing its heat reflectivity. As another test of the reliability of the system, the long brass tube was cut to about a quarter of its original length, and the exact same quantity of gas was released into the shorter tube. Thus the shorter tube will have about four times higher gas density. It was found that the percentage of radiant heat absorbed by or transmitted through the gas relative to the empty-tube state was entirely unchanged by this (even though the two tubes don't have equal empty-tube states). Varying the absolute quantity of the gas in the tube causes corresponding changes in the absorption percentages, but varying the density doesn't matter, nor does the absolute value of the empty-tube reference point.
The emission spectrum of the particular source of heat makes a difference -- sometimes a big difference -- in the amount of radiant heat a gas will absorb, and different gases can respond differently to a change in the source. Tyndall said in 1864, "a long series of experiments enables me to state that probably no two substances at a temperature of 100°C emit heat of the same quality [i.e. of the same spectral profile]. The heat emitted by isinglass, for example, is different from that emitted by lampblack, and the heat emitted by cloth, or paper, differs from both." Looking at an electrically-heated platinum wire, it is obvious to the human eye that the heat's spectral profile depends on whether the wire is heated to dull red, bright orange, or white hot. Some gases were relatively stronger absorbers of the dull-red platinum heat while other gases were relatively stronger absorbers of the white hot platinum heat, he found. For his original and primary benchmark in 1859, he used the heat from 100°C lampblack (akin to a theoretical "blackbody radiator"). Later he got some of his more interesting findings from using other heat sources. E.g., when the source of radiant heat was any one kind of gas, then this heat was strongly absorbed by another body of the same kind of gas, regardless of whether the gas was a weak absorber of broad-spectrum sources. In the illustration above, the radiant heat that is going into the brass tube comes from a pot of simmering water; the heat radiates from the exterior surface of the pot, not from the water, and not from the gas flame that keeps the water at a simmer. An alternative illustration with a modified setup taken from the same book (page 112) is shown below. The main difference is that the heat source is separated from the brass tube by open air, which eliminates the need for circulating cold water cooling at the interface between heat source and brass tube.
Tuesday, September 22, 2015
Lapse Rates for Dummies or Smarties, With & Without Greenhouse Gases
Some commenters have claimed that a theoretical pure Nitrogen (N2) Earth atmosphere without any IR-active 'greenhouse gases' could not have a lapse rate nor a Maxwell et al gravito-thermal greenhouse effect.
However, many prior posts have shown this to be false for a number of reasons, including two posts quoting the Feynman lectures on statistical mechanics of a Boltzmann Distribution pure N2 atmosphere, and the HS post, "Why Greenhouse Gases Don't Affect the Greenhouse Equation or Lapse Rate," which also calculates a pure N2 Boltzmann Distribution for Earth.
We will now use a couple of illustrations for smarties or dummies to understand why the so-called 'greenhouse gas' water vapor cools, not warms, the Earth surface by up to ~25C via changes in heat capacity (Cp) alone (not even including additional cooling from latent heat transfer or clouds). We will also show why a pure N2 atmosphere without any greenhouse gases would have a surface temperature ~25C warmer than the present, due to a much steeper lapse rate.
Recall that the dry adiabatic lapse rate formula is a very simple, linear relationship whereby the change in temperature (dT) with change in height from the surface (dh) is solely dependent upon the gravitational acceleration constant (g) divided by the heat capacity of the atmosphere at constant pressure (Cp):
dT/dh = -g/Cp
And note that change in temperature dT is inversely related to change in heat capacity (Cp). Since water vapor has a much higher heat capacity Cp than air or pure N2, addition of water vapor greatly decreases the lapse rate (dT/dh) by almost one-half (from ~9.8K/km to ~5K/km), thereby cooling, not warming, the surface by up to 25C.
In our hypothetical 1st atmosphere consisting only of N2 plus addition of < 1% water vapor, we assume the addition of water vapor creates a wet adiabatic lapse rate of 5K/km, the same as the wet adiabatic lapse rate on Earth. By calculating the center of mass as the HS Greenhouse Eqn does, and calculating the fixed 255K equilibrium temperature between the Earth and Sun, we can then calculate the entire tropospheric temperature profile from the surface to tropopause, and replicate the 1976 US Standard Atmosphere model:
Thus, we find the net effect of the addition of < 1% 'greenhouse gas' water vapor was to cool, not warm the surface of an N2 atmosphere by up to ~25C.
Thus, the Arrhenius radiative greenhouse theory (which confuses the cause with the effect) is once again demonstrated to be unphysical and falsified, and the Maxwell et al gravito-thermal greenhouse effect once again vindicated. One and only one of these two competing greenhouse theories can be correct, otherwise the observed effect would be double (66C) that observed (33C). The Maxwell et al theory is the only option which does not violate any laws of thermodynamics.
However, many prior posts have shown this to be false for a number of reasons, including two posts quoting the Feynman lectures on statistical mechanics of a Boltzmann Distribution pure N2 atmosphere, and the HS post, "Why Greenhouse Gases Don't Affect the Greenhouse Equation or Lapse Rate," which also calculates a pure N2 Boltzmann Distribution for Earth.
We will now use a couple of illustrations for smarties or dummies to understand why the so-called 'greenhouse gas' water vapor cools, not warms, the Earth surface by up to ~25C via changes in heat capacity (Cp) alone (not even including additional cooling from latent heat transfer or clouds). We will also show why a pure N2 atmosphere without any greenhouse gases would have a surface temperature ~25C warmer than the present, due to a much steeper lapse rate.
Recall that the dry adiabatic lapse rate formula is a very simple, linear relationship whereby the change in temperature (dT) with change in height from the surface (dh) is solely dependent upon the gravitational acceleration constant (g) divided by the heat capacity of the atmosphere at constant pressure (Cp):
dT/dh = -g/Cp
And note that change in temperature dT is inversely related to change in heat capacity (Cp). Since water vapor has a much higher heat capacity Cp than air or pure N2, addition of water vapor greatly decreases the lapse rate (dT/dh) by almost one-half (from ~9.8K/km to ~5K/km), thereby cooling, not warming, the surface by up to 25C.
In our hypothetical 1st atmosphere consisting only of N2 plus addition of < 1% water vapor, we assume the addition of water vapor creates a wet adiabatic lapse rate of 5K/km, the same as the wet adiabatic lapse rate on Earth. By calculating the center of mass as the HS Greenhouse Eqn does, and calculating the fixed 255K equilibrium temperature between the Earth and Sun, we can then calculate the entire tropospheric temperature profile from the surface to tropopause, and replicate the 1976 US Standard Atmosphere model:
 |
| Thought experiment 1 of a N2 atmosphere with < 1% GHG water vapor. Note for easy illustrative purposes only, the actual numbers are rounded slighly, e.g. the actual height of the center of mass is ~5.1km rather than 5.0 km, and the actual dry adiabatic lapse rate is ~9.8K/km, not 10K/km.
Note in the above "greenhouse atmosphere," there is a ~33C "greenhouse effect" from the 255K center of mass to the ~288K surface, as well as an even larger "anti-greenhouse effect" of negative 35K from the 255K center of mass to the ~220K top of troposphere. Thus, gravity has not added any energy to the atmospheric system; gravity has simply redistributed the available energy from the only source the Sun, more towards the surface and less towards the top of the troposphere. That is the gravito-thermal greenhouse effect of Poisson, Maxwell, Clausius, Carnot, Boltzmann, Feynman, US Std Atmosphere, HS greenhouse eqn et al, and has no dependence whatsoever upon IR emission/absorption from greenhouse gases.
Now lets consider a hypothetical Earth atmosphere without any greenhouse gases, consisting solely of pure N2. We again use the dry lapse rate equation above, since obviously N2 is affected by gravity (g) and has a heat capacity (Cp). In this pure N2 Boltzmann distribution, the lapse rate can thus be calculated as ~10K/km, essentially the same as our present atmosphere dry lapse rate (9.8K/km).
For illustrative purposes only, the atmospheric mass of a pure N2 atmosphere is close to that of our present atmosphere, and thus the center of mass is also located near ~5km in the troposphere. However, since the lapse rate is much steeper in a pure N2 atmosphere, the "greenhouse effect" is about 50K from the 255K center of mass to 305K surface, and the "anti-greenhouse effect" is also ~50K from the 255K center of mass to the ~205K top of the troposphere, producing a ~100K temperature gradient from the surface to top of the troposphere:
|
Thus, we find the net effect of the addition of < 1% 'greenhouse gas' water vapor was to cool, not warm the surface of an N2 atmosphere by up to ~25C.
Thus, the Arrhenius radiative greenhouse theory (which confuses the cause with the effect) is once again demonstrated to be unphysical and falsified, and the Maxwell et al gravito-thermal greenhouse effect once again vindicated. One and only one of these two competing greenhouse theories can be correct, otherwise the observed effect would be double (66C) that observed (33C). The Maxwell et al theory is the only option which does not violate any laws of thermodynamics.
Monday, August 4, 2014
New paper argues current lull in solar activity is consistent with a Gleissberg Cycle minimum
A paper published today in the Journal of Geophysical Research Space Physics finds that recent low solar activity "mirrors" extended solar minimums in the 19th & early 20th centuries, as well as other periods over the past 1000 years consistent with the Centennial Gleissberg Cycle of solar activity. Such periods have also been associated with global cooling.
According to the authors
The Centennial Gleissberg Cycle and its Association with Extended Minima
Authors: Joan Feynman [sister of famed physicist Richard Feynman] and A. Ruzmaikin
DOI: 10.1002/2013JA019478
The recent extended minimum of solar and geomagnetic variability (XSM) mirrors the XSMs in the 19th and 20th centuries: 1810–1830 and 1900–1910. Such extended minima also were evident in aurorae reported from 450 AD to 1450 AD. This paper argues that these minima are consistent with minima of the Centennial Gleissberg Cycles (CGC), a 90–100 year variation observed on the Sun, in the solar wind, at the Earth and throughout the Heliosphere. The occurrence of the recent XSM is consistent with the existence of the CGC as a quasi-periodic variation of the solar dynamo. Evidence of CGC's is provided by the multi-century sunspot record, by the almost 150-year record of indexes of geomagnetic activity (1868-present), by 1,000 years of observations of aurorae (from 450 to 1450 AD) and millennial records of radionuclides in ice cores. The aa index of geomagnetic activity carries information about the two components of the solar magnetic field (toroidal and poloidal), one driven by flares and CMEs (related to the toroidal field) the other driven by co-rotating interaction regions in the solar wind (related to the poloidal field). These two components systematically vary in their intensity and relative phase giving us information about centennial changes of the sources of solar dynamo during the recent CGC over the last century. The dipole and quadrupole modes of the solar magnetic field changed in relative amplitude and phase; the quadrupole mode became more important as the XSM was approached. Some implications for the solar dynamo theory are discussed.
"The recent extended minimum of solar and geomagnetic variability (XSM) mirrors the XSMs in the 19th and 20th centuries: 1810–1830 and 1900–1910. Such extended minima also were evident in aurorae reported from 450 AD to 1450 AD. This paper argues that these minima are consistent with minima of the Centennial Gleissberg Cycles (CGC), a 90–100 year variation observed on the Sun, in the solar wind, at the Earth and throughout the Heliosphere. The occurrence of the recent XSM is consistent with the existence of the CGC as a quasi-periodic variation of the solar dynamo. Evidence of CGC's is provided by the multi-century sunspot record, by the almost 150-year record of indexes of geomagnetic activity (1868-present), by 1,000 years of observations of aurorae (from 450 to 1450 AD) and millennial records of radionuclides in ice cores."
If it is true that the current lull in solar activity is "consistent with minima of the Centennial Gleissberg Cycles," and the Gleissberg Cycle is a real solar cycle, the current Gleissberg minimum could last a few decades before solar activity begins to rise again.
Solar physicist Habibullo Abdussamatov predicts the current lull in solar activity will continue until about the middle of the 21st century and lead to a new Little Ice Age within the next 30 years.
 |
| Reconstructed solar geomagnetic aa-index from Lockwood et al 1999. Graph source |
Authors: Joan Feynman [sister of famed physicist Richard Feynman] and A. Ruzmaikin
DOI: 10.1002/2013JA019478
The recent extended minimum of solar and geomagnetic variability (XSM) mirrors the XSMs in the 19th and 20th centuries: 1810–1830 and 1900–1910. Such extended minima also were evident in aurorae reported from 450 AD to 1450 AD. This paper argues that these minima are consistent with minima of the Centennial Gleissberg Cycles (CGC), a 90–100 year variation observed on the Sun, in the solar wind, at the Earth and throughout the Heliosphere. The occurrence of the recent XSM is consistent with the existence of the CGC as a quasi-periodic variation of the solar dynamo. Evidence of CGC's is provided by the multi-century sunspot record, by the almost 150-year record of indexes of geomagnetic activity (1868-present), by 1,000 years of observations of aurorae (from 450 to 1450 AD) and millennial records of radionuclides in ice cores. The aa index of geomagnetic activity carries information about the two components of the solar magnetic field (toroidal and poloidal), one driven by flares and CMEs (related to the toroidal field) the other driven by co-rotating interaction regions in the solar wind (related to the poloidal field). These two components systematically vary in their intensity and relative phase giving us information about centennial changes of the sources of solar dynamo during the recent CGC over the last century. The dipole and quadrupole modes of the solar magnetic field changed in relative amplitude and phase; the quadrupole mode became more important as the XSM was approached. Some implications for the solar dynamo theory are discussed.
Wednesday, October 28, 2015
The Kinetic Theory of Gases explains why the Maxwell et al Gravito-Thermal Greenhouse Effect is Correct
An excellent review of the Kinetic Theory of Gases, similar to Feynman's lecture 40 on the Statistical Mechanics of the Atmosphere, and explains the fundamental basis of the 33C Maxwell et al Gravito-Thermal greenhouse effect (& which also falsifies the Arrhenius radiative-greenhouse effect).
The review also explains the fundamental reasons why the false analogies of inflated tires or static, closed gas cylinders to our atmosphere are incorrect.
Daniel Bernoulli, in 1738, was the first to understand air pressure from a molecular point of view. He drew a picture of a vertical cylinder, closed at the bottom, with a piston at the top, the piston having a weight on it, both piston and weight being supported by the air pressure inside the cylinder. He described what went on inside the cylinder as follows: “let the cavity contain very minute corpuscles, which are driven hither and thither with a very rapid motion; so that these corpuscles, when they strike against the piston and sustain it by their repeated impacts, form an elastic fluid which will expand of itself if the weight is removed or diminished…”
Using Newton’s law in the form force = rate of change of momentum, we see that the particle’s momentum changes by 2mv each time it hits the piston. The time between hits is 2L/v, so the frequency of hits is v/2L per second. This means that if there were no balancing force, by conservation of momentum the particle would cause the momentum of the piston to change by 2mv´v/2L units in each second. This is the rate of change of momentum, and so must be equal to the balancing force, which is therefore F = mv2/L.
Maxwell established that gravity & atmospheric mass create so-called greenhouse effect
Debunking Myths & Strawmen about the Gravito-Thermal Greenhouse Effect & Radiative Greenhouse Effect
1] The Greenhouse Equation
2] How Gravity continuously does Thermodynamic Work on the atmosphere to control pressure & temperature
3] Why Greenhouse Gases Don't Affect the Greenhouse Equation or Lapse Rate (debunks claim that greenhouse gases are necessary for convection or a lapse rate to occur or that greenhouse gas radiative forcing can affect the lapse rate)
4] Quick and dirty explanation of the Greenhouse Equation and theory
5] The Greenhouse Equation predicts 1% change in cloud cover changes global temperature by 1°C
6] Why the atmosphere is in horizontal thermodynamic equilibrium but not vertical equilibrium (debunks claims that the gravito-thermal greenhouse effect assumes thermodynamic equilibrium in all three x, y, and z planes).
7] The Greenhouse Equation predicts temperatures within 0.02°C throughout entire troposphere without radiative forcing from greenhouse gases
8] Why increased water vapor decreases the lapse rate by half to cause surface cooling of up to 25.5C
9] Derivation of the effective radiating height & entire 33°C greenhouse effect without radiative forcing from greenhouse gases
10] Derivation of the entire 33°C greenhouse effect without radiative forcing from greenhouse gases
11] French scientist explains why the greenhouse effect is primarily due to atmospheric mass/gravity/pressure
12] Modeling of the Earth’s Planetary Heat Balance with an Electrical Circuit Analogy
13] Why can't radiation from a cold body make a hot body hotter?
The review also explains the fundamental reasons why the false analogies of inflated tires or static, closed gas cylinders to our atmosphere are incorrect.
Kinetic Theory of Gases: A Brief Review
By Michael Fowler, physicist, U of VA
Bernoulli's Picture
Daniel Bernoulli, in 1738, was the first to understand air pressure from a molecular point of view. He drew a picture of a vertical cylinder, closed at the bottom, with a piston at the top, the piston having a weight on it, both piston and weight being supported by the air pressure inside the cylinder. He described what went on inside the cylinder as follows: “let the cavity contain very minute corpuscles, which are driven hither and thither with a very rapid motion; so that these corpuscles, when they strike against the piston and sustain it by their repeated impacts, form an elastic fluid which will expand of itself if the weight is removed or diminished…”
(An applet is available here.) Sad to report, his insight, although essentially correct, was not widely accepted [yet another example of a false scientific "consensus"]. Most scientists believed that the molecules in a gas stayed more or less in place, repelling each other from a distance, held somehow in the ether. Newton had shown that PV = constant followed if the repulsion were inverse-square. In fact, in the 1820’s an Englishman, John Herapath, derived the relationship between pressure and molecular speed given below, and tried to get it published by the Royal Society. It was rejected by the president, Humphry Davy, who pointed out that equating temperature with motion, as Herapath did, implied that there would be an absolute zero of temperature, an idea Davy was reluctant to accept. And it should be added that no-one had the slightest idea how big atoms and molecules were, although Avogadro had conjectured that equal volumes of different gases at the same temperature and pressure contained equal numbers of molecules—his famous number—neither he nor anyone else knew what that number was, only that it was pretty big.
The Link between Molecular Energy and Pressure
It is not difficult to extend Bernoulli’s picture to a quantitative description, relating the gas pressure to the molecular velocities. As a warm up exercise, let us consider a single perfectly elastic particle, of mass m, bouncing rapidly back and forth at speed v inside a narrow cylinder of length L with a piston at one end, so all motion is along the same line. (For the movie, click here!) What is the force on the piston?
Obviously, the piston doesn’t feel a smooth continuous force, but a series of equally spaced impacts. However, if the piston is much heavier than the particle, this will have the same effect as a smooth force over times long compared with the interval between impacts. So what is the value of the equivalent smooth force?
Using
We now generalize to the case of many particles bouncing around inside a rectangular box, of length L in the x-direction (which is along an edge of the box). The total force on the side of area A perpendicular to the x-direction is just a sum of single particle terms, the relevant velocity being the component of the velocity in the x-direction. The pressure is just the force per unit area, P = F/A. Of course, we don’t know what the velocities of the particles are in an actual gas, but it turns out that we don’t need the details. If we sum N contributions, one from each particle in the box, each contribution proportional to vx2 for that particle, the sum just gives us N times the average value ofvx2. That is to say,
where there are N particles in a box of volume V. Next we note that the particles are equally likely to be moving in any direction, so the average value of vx2 must be the same as that of vy2 or vz2, and since v2 = vx2 + vy2 + vz2, it follows that
This is a surprisingly simple result! The macroscopic pressure of a gas relates directly to the average kinetic energy per molecule. Of course, in the above we have not thought about possible complications caused by interactions between particles, but in fact for gases like air at room temperature these interactions are very small. Furthermore, it is well established experimentally that most gases satisfy the Gas Law over a wide temperature range:
PV = nRT
for n moles of gas, that is, n = N/NA, with NA Avogadro’s number and R the gas constant.
Introducing Boltzmann’s constant k = R/NA, it is easy to check from our result for the pressure and the ideal gas law that the average molecular kinetic energy is proportional to the absolute temperature,
Boltzmann’s constant k = 1.38.10-23 joules/K.
Maxwell finds the Velocity Distribution
By the 1850’s, various difficulties with the existing theories of heat, such as the caloric theory, caused some rethinking, and people took another look at the kinetic theory of Bernoulli, but little real progress was made until Maxwell attacked the problem in 1859. Maxwell worked with Bernoulli’s picture, that the atoms or molecules in a gas were perfectly elastic particles, obeying Newton’s laws, bouncing off each other (and the sides of the container) with straight-line trajectories in between collisions. (Actually, there is some inelasticity in the collisions with the sides—the bouncing molecule can excite or deexcite vibrations in the wall, this is how the gas and container come to thermal equilibrium.) Maxwell realized that it was completely hopeless to try to analyze this system using Newton’s laws, even though it could be done in principle, there were far too many variables to begin writing down equations. On the other hand, a completely detailed description of how each molecule moved was not really needed anyway. What was needed was some understanding of how this microscopic picture connected with the macroscopic properties, which represented averages over huge numbers of molecules.
The relevant microscopic information is not knowledge of the position and velocity of every molecule at every instant of time, but just the distribution function, that is to say, what percentage of the molecules are in a certain part of the container, and what percentage have velocities within a certain range, at each instant of time. For a gas in thermal equilibrium [unlike the 100 km Earth atmosphere, which is not in vertical thermal equilibrium due to gravity], the distribution function is independent of time. Ignoring tiny corrections for gravity [only true for this small container, not the 100km height Earth atmosphere in which gravity-corrections are very large], the gas will be distributed uniformly in the container, so the only unknown is the velocity distribution function.
Velocity Space
What does a velocity distribution function look like? Suppose at some instant in time one particular molecule has velocity 
We can record this information by constructing a three-dimensional velocity space, with axes 
, and putting in a pointP1 representing the molecule’s velocity (the red arrow is of course
):
We can record this information by constructing a three-dimensional velocity space, with axes , and putting in a pointP1 representing the molecule’s velocity (the red arrow is of course):
Now imagine that at that instant we could measure the velocities of all the molecules in a container, and put points P2, P3, P4, … PN in the velocity space. Since N is of order 1021 for 100 ccs of gas, this is not very practical! But we can imagine what the result would be: a cloud of points in velocity space, equally spread in all directions (there’s no reason molecules would prefer to be moving in the x-direction, say, rather than the y-direction) and thinning out on going away from the origin towards higher and higher velocities.
Now, if we could keep monitoring the situation as time passes individual points would move around, as molecules bounced off the walls, or each other, so you might think the cloud would shift around a bit. But there’s a vast number of molecules in any realistic macroscopic situation, and for any reasonably sized container it’s safe to assume that the number of molecules in any small region of velocity space remains pretty much constant. Obviously, this cannot be true for a region of velocity space so tiny that it only contains one or two molecules on average. But it can be shown statistically that if there are N molecules in a particular small volume of velocity space, the fluctuation of the number with time is of order
, so a region containing a million molecules will vary in numbers by about one part in a thousand, a trillion molecule region by one part in a million. Since 100 ccs of air contains of order 1021 molecules, we can in practice divide the region of velocity space occupied by the gas into a billion cells, and still have variation in each cell of order one part in a million!
, so a region containing a million molecules will vary in numbers by about one part in a thousand, a trillion molecule region by one part in a million. Since 100 ccs of air contains of order 1021 molecules, we can in practice divide the region of velocity space occupied by the gas into a billion cells, and still have variation in each cell of order one part in a million!
The bottom line is that for a macroscopic amount of gas, fluctuations in density, both in ordinary space and in velocity space, are for all practical purposes negligible, and we can take the gas to be smoothly distributed in both spaces.
Maxwell’s Symmetry Argument
Maxwell found the velocity distribution function for gas molecules in thermal equilibrium by the following elegant argument based on symmetry.
For a gas of N particles, let the number of particles having velocity in the x-direction between vx and vx + dvx be 
. In other words, 
is the fraction of all the particles having x-direction velocity lying in the interval between vx and vx + dvx. (I’ve written f1 instead of f to help remember this function refers to only one component of the velocity vector.)
. In other words, is the fraction of all the particles having x-direction velocity lying in the interval between vx and vx + dvx. (I’ve written f1 instead of f to help remember this function refers to only one component of the velocity vector.)
If we add the fractions for all possible values of vx, the result must of course be 1:
But there’s nothing special about the x-direction—for gas molecules in a container, at least away from the walls, all directions look the same, so the same function f will give the probability distributions in the other directions too. It follows immediately that the probability for the velocity to lie between vx and vx + dvx, vy and vy + dvy, and vz and vz + dvz must be:
Note that this distribution function, when integrated over all possible values of the three components of velocity, gives the total number of particles to be N, as it should (since integrating over each f1(v)dv gives unity).
Next comes the clever part—since any direction is as good as any other direction, the distribution function must depend only on the total speed of the particle, not on the separate velocity components. Therefore, Maxwell argued, it must be that:
where F is another unknown function. However, it is apparent that the product of the functions on the left is reflected in the sum of variables on the right. It will only come out that way if the variables appear in an exponent in the functions on the left. In fact, it is easy to check that this equation is solved by a function of the form:
This curve is called a Gaussian: it’s centered at the origin, and falls off very rapidly as vx increases. Taking A = B = 1 just to see the shape, we find:
At this point, A and B are arbitrary constants—we shall eventually find their values for an actual sample of gas at a given temperature. Notice that (following Maxwell) we have put a minus sign in the exponent because there must eventually be fewer and fewer particles on going to higher speeds, certainly not a diverging number.
Multiplying together the probability distributions for the three directions gives the distribution in terms of particle speed v, where v2 = vx2 +vy2 + vz2. Since all velocity directions are equally likely, it is clear that the natural distribution function is that giving the number of particles having speed between v and v + dv.
From the graph above, it is clear that the most likely value of vx is zero. If the gas molecules were restricted to one dimension, just moving back and forth on a line, then the most likely value of their speed would also be zero. However, for gas molecules free to move in two or three dimensions, the most likely value of the speed is not zero. It’s easiest to see this in a two-dimensional example. Suppose we plot the points P representing the velocities of molecules in a region near the origin, so the density of points doesn’t vary much over the extent of our plot (we’re staying near the top of the peak in the one-dimensional curve shown above).
Now divide the two-dimensional space into regions corresponding to equal increments in speed:
In the two-dimensional space, 
is a circle, so this division of the plane is into annular regions between circles whose successive radii are 
apart:
is a circle, so this division of the plane is into annular regions between circles whose successive radii are apart:
Each of these annular areas corresponds to the same speed increment . In particular, the green area, between a circle of radius 
and one of radius 
, corresponds to the same speed increment as the small red circle in the middle, which corresponds to speeds between 0 and . Therefore, if the molecular speeds are pretty evenly distributed in this near-the-origin area of the (vx, vy) plane, there will be a lot more molecules with speeds between and than between 0 and —so the most likely speed will not be zero. To find out what it actually is, we have to put this area argument together with the Gaussian fall off in density on going far from the origin. We’ll discuss this shortly.
. In particular, the green area, between a circle of radius and one of radius , corresponds to the same speed increment as the small red circle in the middle, which corresponds to speeds between 0 and . Therefore, if the molecular speeds are pretty evenly distributed in this near-the-origin area of the (vx, vy) plane, there will be a lot more molecules with speeds between and than between 0 and —so the most likely speed will not be zero. To find out what it actually is, we have to put this area argument together with the Gaussian fall off in density on going far from the origin. We’ll discuss this shortly.
The same argument works in three dimensions—it’s just a little more difficult to visualize. Instead of concentric circles, we have concentric spheres. All points lying on a spherical surface centered at the origin correspond to the same speed.
Let us now figure out the distribution of particles as a function of speed. The distribution in the three-dimensional space 
is from Maxwell’s analysis
is from Maxwell’s analysis
To translate this to the number of particles having speed between v and 
we need to figure out how many of those little 
boxes there are corresponding to speeds between v and . In other words, what is the volume of velocity space between the two neighboring spheres, both centered at the origin, the inner one with radius v, the outer one infinitesimally bigger, with radius ? Since dv is so tiny, this volume is just the area of the sphere multiplied by dv: that is, 
.
we need to figure out how many of those little boxes there are corresponding to speeds between v and . In other words, what is the volume of velocity space between the two neighboring spheres, both centered at the origin, the inner one with radius v, the outer one infinitesimally bigger, with radius ? Since dv is so tiny, this volume is just the area of the sphere multiplied by dv: that is, .
Finally, then, the probability distribution as a function of speed is:
Of course, our job isn’t over—we still have these two unknown constants A and B. However, just as for the function 
is the fraction of the molecules corresponding to speeds between v and , and all these fractions taken together must add up to 1.
is the fraction of the molecules corresponding to speeds between v and , and all these fractions taken together must add up to 1.
That is,
We need the standard result 
(a derivation can be found in my 152 Notes on Exponential Integrals), and find:
(a derivation can be found in my 152 Notes on Exponential Integrals), and find:
This means that there is really only one arbitrary variable left: if we can find B, this equation gives us A: that is, 
, and 
is what appears in 
.
, and is what appears in .
Looking at , we notice that B is a measure of how far the distribution spreads from the origin: if B is small, the distribution drops off more slowly—the average particle is more energetic. Recall now that the average kinetic energy of the particles is related to the temperature by . This means that B is related to the inverse temperature.
, we notice that B is a measure of how far the distribution spreads from the origin: if B is small, the distribution drops off more slowly—the average particle is more energetic. Recall now that the average kinetic energy of the particles is related to the temperature by . This means that B is related to the inverse temperature.
In fact, since
is the fraction of particles in the interval dv at v, and those particles have kinetic energy ½mv2, we can use the probability distribution to find the average kinetic energy per particle:
is the fraction of particles in the interval dv at v, and those particles have kinetic energy ½mv2, we can use the probability distribution to find the average kinetic energy per particle:
To do this integral we need another standard result: 
. We find:
. We find:
.Substituting the value for the average kinetic energy in terms of the temperature of the gas,
gives B = m/2kT, so 
.
.
This means the distribution function
where E is the kinetic energy of the molecule.
Note that this function increases parabolically from zero for low speeds, then curves round to reach a maximum and finally decreases exponentially. As the temperature increases, the position of the maximum shifts to the right. The total area under the curve is always one, by definition. For air molecules (say, nitrogen) [i.e. pure N2 without any greenhouse gases] at room temperature the curve is the blue one below. The red one is for an absolute temperature down by a factor of two:
What about Potential Energy?
Maxwell’s analysis solves the problem of finding the statistical velocity distribution of molecules of an ideal gas in a box at a definite temperature T: the relative probability of a molecule having velocity 
is proportional to 
. The position distribution is taken to be uniform: the molecules are assumed to be equally likely to be anywhere in the box.
is proportional to . The position distribution is taken to be uniform: the molecules are assumed to be equally likely to be anywhere in the box.
But how is this distribution affected if in fact there is some kind of potential [like gravity] pulling the molecules to one end of the box? In fact, we’ve already solved this problem, in the discussion earlier on the isothermal atmosphere [in a small box for which gravity-corrections are insignificant]. Consider a really big box, kilometers high, so air will be significantly denser towards the bottom. Assume the temperature is uniform throughout [a thought experiment premise only]. We found under these conditions that with Boyles Law expressed in the form
the atmospheric density varied with height as
Now we know that Boyle’s Law is just the fixed temperature version of the Gas Law 
, and the density
, and the density
with N the total number of molecules and m the molecular mass,
Rearranging,
for n moles of gas, each mole containing Avogadro’s number NA molecules.
Putting this together with the Gas Law,
so
where Boltzmann’s constant 
as discussed previously.
as discussed previously.
The dependence of gas density on height can therefore be written
The important point here is that mgh is the [gravitational] potential energy of the molecule, and the distribution we have found is exactly parallel to Maxwell’s velocity distribution, the [gravitational] potential energy now playing the role that kinetic energy played in that case.
We’re now ready to put together Maxwell’s velocity distribution with this height distribution, to find out how the molecules are distributed in the atmosphere, both in velocity space and in ordinary space. In other words, in a six-dimensional space!
Our result is:
That is, the probability of a molecule having total energy E is proportional to 
.
.
This is the Boltzmann, or Maxwell-Boltzmann, distribution. It turns out to be correct for any type of potential energy [including gravitational potential energy], including that arising from forces between the molecules themselves.
Degrees of Freedom and Equipartition of Energy
By a “degree of freedom” we mean a way in which a molecule is free to move, and thus have energy—in this case, just the x, y, and z directions. Boltzmann reformulated Maxwell’s analysis in terms of degrees of freedom, stating that there was an average energy ½kT in each degree of freedom, to give total average kinetic energy 3.½kT, so the specific heat per molecule is presumable 1.5k, and given that k = R/NA, the specific heat per mole comes out at 1.5R. In fact, this is experimentally confirmed for monatomic gases. However, it is found that diatomic gases can have specific heats of 2.5R and even 3.5R. This is not difficult to understand—these molecules have more degrees of freedom. A dumbbell molecule can rotate about two directions perpendicular to its axis. A diatomic molecule could also vibrate. Such a simple harmonic oscillator motion has both kinetic and potential energy, and it turns out to have total energy kT in thermal equilibrium. Thus, reasonable explanations for the specific heats of various gases can be concocted by assuming a contribution ½k from each degree of freedom. But there are problems. Why shouldn’t the dumbbell rotate about its axis? Why do monatomic atoms not rotate at all? Even more ominously, the specific heat of hydrogen, 2.5R at room temperature, drops to 1.5R at lower temperatures. These problems were not resolved until the advent of quantum mechanics.
Brownian Motion
One of the most convincing demonstrations that gases really are made up of fast moving molecules is Brownian motion, the observed constant jiggling around of tiny particles, such as fragments of ash in smoke. This motion was first noticed by a Scottish botanist, who initially assumed he was looking at living creatures, but then found the same motion in what he knew to be particles of inorganic material. Einstein showed how to use Brownian motion to estimate the size of atoms. For the movie, click here!
Related:
Maxwell established that gravity & atmospheric mass create so-called greenhouse effect
Debunking Myths & Strawmen about the Gravito-Thermal Greenhouse Effect & Radiative Greenhouse Effect
1] The Greenhouse Equation
2] How Gravity continuously does Thermodynamic Work on the atmosphere to control pressure & temperature
3] Why Greenhouse Gases Don't Affect the Greenhouse Equation or Lapse Rate (debunks claim that greenhouse gases are necessary for convection or a lapse rate to occur or that greenhouse gas radiative forcing can affect the lapse rate)
4] Quick and dirty explanation of the Greenhouse Equation and theory
5] The Greenhouse Equation predicts 1% change in cloud cover changes global temperature by 1°C
6] Why the atmosphere is in horizontal thermodynamic equilibrium but not vertical equilibrium (debunks claims that the gravito-thermal greenhouse effect assumes thermodynamic equilibrium in all three x, y, and z planes).
7] The Greenhouse Equation predicts temperatures within 0.02°C throughout entire troposphere without radiative forcing from greenhouse gases
8] Why increased water vapor decreases the lapse rate by half to cause surface cooling of up to 25.5C
9] Derivation of the effective radiating height & entire 33°C greenhouse effect without radiative forcing from greenhouse gases
10] Derivation of the entire 33°C greenhouse effect without radiative forcing from greenhouse gases
11] French scientist explains why the greenhouse effect is primarily due to atmospheric mass/gravity/pressure
12] Modeling of the Earth’s Planetary Heat Balance with an Electrical Circuit Analogy
13] Why can't radiation from a cold body make a hot body hotter?
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