arXiv:0802.4324v1 [physics.ao-ph] 29 Feb 2008
Proof of the Atmospheric Greenhouse Effect
Arthur P. Smith*
American Physical Society, 1 Research Road, Ridge NY, 11961
Arecently advanced argument against the atmosphericgreenhouse effectis refuted. Aplanetwithout aninfrared absorbing atmosphereismathematically constrained tohaveanaveragetemperature
lessthan or equal tothe effective radiating temperature. ObservedparametersforEarthprovethat
without infrared absorption by the atmosphere, the average temperature of Earth’s surface would
be at least 33 K lower than what is observed.
PACS numbers: 92.60.Vb,05.90.+m
I. INTRODUCTION
The results presented here are not new. However the form of presentation is designed to clearly and accurately
respond to recent claims1 that a physics-based analysis can “falsify” the atmospheric greenhouse effect. In fact, the
standard presentation in climatology textbooks2 is accurate in all material respects. The following explores in more
detail certain points that seem to have been cause for confusion.
Firstpresented arethedefinitionsofbasictermsand therelevant equationsforthe flowof energy. Thesituationfor
aplanetwith noinfrared-absorbing atmosphereisthenexamined, and aconstraintonaveragetemperatureisproved.
Several specific models ofplanets with noinfrared-absorbing atmospehere arethen solved,including onepresented
by Gerlich and Tscheuschner1, and it is verified that all satisfy this constraint.
A simple infrared-absorbing atmospheric layer is added to these models, and it is proved that the temperature
constraint is easily violated, as is shown by the observational data for Earth.
II. DEFINITIONS AND BASIC EQUATIONS
Define the incoming irradiance S as the energy per unit area and per unit time arriving at a planet from a stellar
source. Theactual radiation fieldischaracterizedby aspectrumofwavelengthsand(depending onthesizeofthe
star(s) anddistancetotheplanet) a small spreadindirections. S is anintegral overall wavelengthsandpropagation
directions of the radiant specific energy at the distance of the planet from the star. As the planet moves through its
yearly orbit, the value of S will vary, so it is strictly a function S(t)of time.
The energy per unit time arriving at the planet is the product of S(t) with the area the planet subtends in the
planeperpendiculartothe radiantpropagationdirection. For a sphericalplanet of radius r, this area is simply pr2 .
Sototal arriving energy(energyper unittime, orpower) from spaceis
Ein(t)= pr2 S(t) (1)
For agivenlocation x on the planetary surface, the normal to the plane of the local surface makes an angle .(x,t)
with the radiationpropagationdirection of agiven stellar source. Only one side of theplanet willbelit at anygiven
timefromthat source; this canbegenerally indicatedby those angles . from 0 to p/2 radians. Angles from p/2 to p
would be unlit. So local incoming irradiance would be:
s(x,t)= cos(.(x,t))S(t) (2)
Integrating this on a sphericalplanet(including only thelit side) givesthe pr2 factor in equation 1.
Define the albedo a of the planet as the fraction of incoming irradiance that is reflected. a is also a local property
(for Earthmuch is reflected by clouds and ice), so the locally reflected energy is a(x,t)s(x,t). Integration across the
lit side of the planet gives a well defined reflected energy:
Ereflected(t)= S(t)a(x,t)cos(.(x,t))dx (3)
An effective albedo aeff can then be defined by the ratio of reflected to incoming energy across the planet as a
whole:
Ereflected(t)= aeff (t)Ein(t)= pr2 aeff S(t) (4)
Thedifferencebetweenincoming and reflected energy is whattheplanet absorbs(again,per unittime):
Eabsorbed(t)= pr2 (1-aeff (t))S(t) (5)
The fundamental characteristic of a planet in space is that of no material interchanges with its surroundings. The
only substantive way energy can come in is through electromagnetic radiation, and the only way energy can leave
is similarly through the planet’s own electromagnetic emissions. There is a very small correction from gravitational
tidal forces, and a planet also receives a small net energy input from internal radioactive decay, but for planets like
the Earth these are thousands of times smaller than the stellar input.
A planet with no incoming absorbed energy would reach thermodynamic equilibrium with the cosmic microwave
background, with a uniform temperature of about 2 K. Absorption of incoming stellar irradiance results in heating
until a steady state with equalincoming and outgoing energy(measured outsidethe atmosphere, and averaged over
one planetary revolution, or whatever the most important variation in time) is reached. Define T(x,t) as the local
surfacetemperatureof theplanet, and o(x,t)asthelocal emissivity. Thermal radiationfromthe surfaceisthengiven
by the Stefan-Boltzmann law3 :
Eemitted(t)= so(x,t)T(x,t)4 dx (6)
Similar to the effective albedo, an effective emissivity and effective radiative temperature canbe defined as averages
over the planetary surface:
1 Z
Teff (t)4 = T(x,t)4 dx (7)
4pr2
and
1 Z
oeff (t)= o(x,t)T(x,t)4 dx (8)
4pr2 Teff (t)4
Totalradiatedthermal energyfrom the surface can thenbe writtenin terms of the effective temperature and albedo:
Eemitted(t)=4pr2 soeff (t)Teff (t)4 (9)
For a planet with no atmosphere, or with an atmosphere that doesn’t absorb electromagnetic radiation to any
significant degree, all this surface-emitted thermal radiation escapes directly into space, just as all the absorbed
stellar radiation reaches the ground. So the net rate of change in energy of the planet at time t is:
E.
planet(t)= Eabsorbed(t)-Eemitted(t)= pr2 (1-aeff (t))S(t)-4pr2 soeff (t)Teff (t)4 (10)
We will look at relevant constraints associated with an absorbing atmosphere via a simple model later in the
discussion.
Theorbitalprocessesforaplanet(andanyinternalvariabilityinthe star) determinethe variationin S(t); that
combined with rotationandinternaldynamicsgivesvariabilityin aeff (t),oeff (t), andTeff (t). As a resulttheplanet
may experience naturalperiods of warming or cooling as E.
planet(t)goespositive or negative, respectively. On average,
however, over time, this rate of energy change should come very close to zero as long as all the input parameters are
reasonable stable over the long term. If it didn’t average to zero for a long period of time, the energy of the planet
would cumulatively build or decline.
In addition to the effective temperature obtained from averaging temperature to the fourth power, relevant for the
thermal radiation problem, we should also look at the more natural average temperature for the planetary surface:
1 Z
Tave(t)= T(x,t)dx (11)
4pr2
As Gerlich and Tscheuschner note1 in their section 3.7, thanks to H¨older’s inequality, this average temperature
(t)is alwaysless than or equal to the effective thermal radiation temperature Teff (t), soT4 isless than or equal
Taveave
to T4 , and rearranging Eq. 10 gives the following constraint on average temperature.
eff
(t)4 1.
Tavesoeff (t)
((1-aeff (t))S(t)/4-Eplanet(t)/4pr2 ) (12)
III. SOME EXAMPLES
A. Model 1: Nonrotating planet
Firstlet’s look at the simple modelplanet solvedby Gerlich andTscheuschner(section3.7.4)Thisis a non-rotating
planet(oraplanetwith arotationaxisparallel totheincoming radiation) with nointernalheattransportinconstant
local radiative equilibrium so that E.
planet is always zero. The non-rotation removes all time-dependences. Emissivity
is assumed to be 1 everywhere; likewise S and a are uniform. Also the microwave background is ignored so the un-lit
side of the planet is always at absolute zero temperature. From Eq. 2 for the local irradiance s(x)we quickly obtain
the local temperature for the lit side of the planet:
Tmodel1 (x)= {(1-a)cos(.(x))S/s} 1/4 (13)
The average temperature is obtained by integrating over the sphere:
Tave =
1
((1-a)S/s)1/4 . /2
cos(.)1/4 2psin(.)d. =
2((1-a)S/s)1/4 (14)
4p0 5
The effective temperature similarly is given by
T4 =
41
p((1-a)S/s)Z0
/2
cos(.)2psin(.)d. =
41((1-a)S/s) (15)
eff
whichis whatithastobeto ensure(Eq. 10) that E.
planet is zero.
So in this case the ratio Tave/Teff is 2p2/5 or about 0.566, and the planet’s average temperature is indeed well
below the effective temperature in this simple model. Plugging in numbers appropriate for Earth, Teff comes to 255
K and Tave would be 144 K, for this non-rotating atmosphere-free version of the planet.
B. Model 2: Simple rotating planet
To this simple model let us now add rotation, including a local heat capacity effect that accounts for some heat
transport vertically, while still leaving out any transport of heat horizontally from one location to another. Assume
the radiation direction is in the plane of rotation. Define the rotation period D (a dayfor the planet) and a thermal
inertia coefficient4 c with units ofJ/Km2 . On a realplanet, c depends on temperature and on D (a time-orfrequencydependence); physically it represents the product of the volumetric heat capacity and the depth or height to which
the incident heat energy is circulated or conducted during a daily thermal cycle. c then determines the local rate of
change of temperature based on the local version of the net energy equation:
cT.(x,t)= Eabsorbed(x,t)-Eemitted(x,t)=(1-a)Scos(.(x,t))(cos(.(x,t)))-sT(x,t)4 (16)
Represent x by angular coordinates f for longitude and . for latitude. Thanks to the rotation, the position of x
relative to the incoming sunlight changes as if f were steadily incremented at a rate 2pt/D. The sun angle . then is
found from:
cos(.)= cos(f+2pt/D)cos. (17)
Thedriving forcesinthe equation repeat withperiod D, so under steady state conditions the solution(s) of Eq. 16
should also repeat with thatperiod. At anypoint f,. on the surface this solution wouldfollow exactly the same curve
of temperature as a function of time for every longitude f, at the given latitude .. In the following we replace 2p/D
with the symbol . and(1 -a)Scos. with A, and for simplicity set f = -p/2(for all otherpoints onthe surfacethe
solutionisjust shiftedforward orback abitintime). Thenthe stepfunction(cos(f+ .t))becomes equivalent to
a square wave W(.t)which is 1 for .t between 0 and p, 0 for .t between p and 2p, and repeating periodically after
that.
So Eq. 16 is reduced to:
cT.
= Asin(.t)W(.t)-sT4 (18)
Any non-transient solution for T(t) will be periodic in time so that T(D)= T(0). Integrating Eq. 18 over a
planetary day(t =0tot =D) gives:
2A . D
0= T4dt (19)
. -s0
Define an effective radiativetemperatureTeff (.)forlatitude. basedon the averagefourthpower(whether averaged
over time or over longitudes is the same):
1 . D
Teff (.)4 = T4dt (20)
D 0
then rearranging Eq. 19 and substituting in the definitions of . and A gives:
Teff (.)4 =
(1-a)Scos.
(21)
ps
Note that the peak Teff for . =0(the equator) is afactor of1/p1/4 0.75 times the peak temperature on the
non-rotatingplanet(fromthepointdirectly underthe sun).
Once again we can check thatthe rate of changein net energy fortheplanet as a whole(Eq. 10) comesto zeroby
finding the effective radiative temperature for the entire surface:
1 . /2 (1-a)Scos. (1-a)S . /2 (1-a)S
T4 =
4p -/2 ps · 2pcos.d. =
2ps -/2
cos 2 .d. =
4s
(22)
eff
as it has to be, the same as for the non-rotating case in Eq. 15.
While we won’t find a full analytic form for the temperature as a function of latitude and time in this model, we
can learn a bit by examining the time dependence of temperature in Eq. 18 more closely. First, define x = .t and
y(x)= T(.,t)/Teff(.)for a given latitude ., with Teff (.)determined by Eq. 21. Eq. 18 then reduces to:
dy
=
(1-a)Scos.
sin(x)W(x)-
sTeff (.)3
y 4 = .(sin(x)W(x)-
1
y 4 ) (23)
dx c.Teff (.) c. p
where we define the dimensionless parameter . =(1-a)Scos./c .Teff (.)Physically this roughly represents the
ratio of the quantity of incoming energy absorbed in a day to the total heat content of the surface (to a relevant
depth) at the effective radiative temperature. If . is small(heat capacity or rotationfrequency high orlatitude close
to the poles), heating or cooling will occur only slowly, and the temperature will stay close to Teff throughout the
day(y will be close to 1). If . islarge(heat capacity or rotationfrequencylow,latitude closerto the equator) then
heating and cooling are rapid, and the temperature variation is more significant.
We can find an analytic solution for the night side of the planet, where the square wave W(x) =0(x between
(2n -1)p and 2np for integer n). Eq. 23 loses all dependence on x and is easily integrated:
Temperature on a rotating planet
1
0
3p/2 2p
0 p/2 p
0.5
1.5
y (T/Teff )
l = 0.05
l = 0.2
l = 1.0
l = 20
x (wt)
FIG. 1: Temperature relative to the effective radiating temperature (Tef f ) for the simple rotating planet model for various
values of the thermal response parameter . The plots are of temperature against time where 0 is sunrise, is sunset, and 2
is sunrise again.
dy . 4 p
)1/3
dx
= -py )y(x)=(
3.(x -a)
(24)
where a is a constant determined by the initial condition y(p)(the value ofthe temperature when night begins for
x = p):
a = p(1-
1)y(2p)/y(p)=1/(1+3.y(p)3 )1/3 (25)
3.y(p)3 )
whichgivesusthe night-timetemperaturedrop. When . is small and y isnottoolargetostart with,the changeis
small -afractionaldeclineof roughly .y(p)3 . Forlarge values of ., the night-time temperaturedropislimited by this
slow inverse 1/3 power. This makes sense as the rate of temperature decrease must decline sharply as temperature
gets lower and the T4 radiative term drops.
For the daytime, if the temperature starts out low with y << 1, then the y4 term is negligible, at least at first, and
Eq. 23 can be integrated easily enough:
dy
dx .sinxy(x)b-.cosx (26)
)
where the inegration constant b again is set by the initial condition b = y(0)+., which y(p)= y(0)+2., i.e. the
temperature increments by 2. on a sinusoidal curve during the day. Of course when . is large or y(0)starts close to
one, this approximation breaks down.
0.1 1 10 100
0
0.5
1
1.5
p1/4
y (T/Teff)
yave
y 4
ave
ymin
ymax
Mars Venus Mercury Moon
Earth
l
FIG. 2: The average, minimum, and maximum values of the relative temperature y from numerically integrating the equations
for different values of . The numerically computed average value of y 4 is also shown; this should always be exactly 1. Lines
are shown indicating the approximate values of corresponding tothe equators of theterrestrialplanets, whichgives apicture
of their most extreme temperature profiles if they had infrared-transparent atmospheres.
Another approximation is to assume variations in y are small and that we can linearize about some chosen value
y0 . This gives:
dy .y04 .y03
dx .sinx - p -4(y-y0 ) (27)
p
which as alinear ordinary differential equationyields a solution:
x/ 0
0
y(x)=
3
y0 + ße-4y3
+
.
(
4.y3
sinx -cosx) (28)
41+(4.y03 /p)2 p
ß here is another constant of integration to be determined by an appropriate initial condition. The linearization
fails once y deviates significantly from y0, but the result should be generally valid if . is small, and can be used to
generate step-wise solutions for the daytime temperatures under any value of .; numerical intergration of the basic
equation of course can do the same.
Numerical computed solutions for various values of . are shown in Fig. 1; Fig. 2 shows the trends for average y
(T/Teff), averagefourthpower ofy, andthe minimum and maximum yvalues as .increases. As expected, the average
fourthpoweris fixed at1,whiletheaverage y decreases as . increases, eventually approaching the non-rotating value
of 2p2/5 as . !1. Also note the maximum temperature quickly approaches the nonrotating value of y = pp for
large .. The minimum temperature drops slowly as y 1/(3.)1/3 .
Approximate formulas for the average temperature are, for large .:
yave 2p2/5+0.392.-0.279;. !1 (29)
and for small .:
yave 1-0.196.2;. !0 (30)
Note that from Eq. 21 the temperature scale varies with latitude as cos(.)1/4 , while the value of . varies as
cos(.)3/4 . So finally, integrating over the whole planet we have an average temperature value of:
(1-a)S . /2
Tave =()1/4 cos(.)1/4 yave(.(.))cos(.)d. (31)
ps 0
For small . we could substitute the expression from Eq. 30; in any case, we know yave < 1 for all latitudes, so we
have an upper bound on the average temperature of the entire rotating planet Tave by substituting in the numerical
value for the cos(.)5/4 integral:
Tave < 0.69921(
(1-a)S
)1/4 (32)
s
and note that this bound is a little over 1% less than Teff for the entire planet (Eq. 22) which has a constant
(1/4)1/4 =0.7071... instead of 0.69921 in the same expression.
So no matter the rotation rate, no matter the surface heat capacity, the average temperature of the planet in this
rotating example, with only radiative energy flows and no absorbing layer in the atmosphere, is always less than the
effectiveradiating temperature. Forvery slowrotationorlowheatcapacityitcanbe significantlyless; forparameters
intheotherdirectionitcancomeascloseas1%(i.e. up to252K onaplanetlikeEarth).
C. Model 3: Rotating planet with varying albedo
Whilethe variabilityininfrared emissivity is relatively small acrossthe surfaceof a realisticplanet,the albedo can
be significantly different from place to place. One of these involves taking into account the effect of ice, by which
the high latitudes reflect more incoming radiation back into space than equatorial latitudes do. What effect does this
have on effective radiating temperature and total temperature?
We can model this by a slight changein model2, by making the value of a dependent onlatitude .. For examplelet
a = sin2 (.)so it is zero at the equator, and approaches 1 at the poles. This changes nothing in most of the analysis
of the preceding section, until we integrate over latitudes. For Eq. 22 we now have:
T4 =
1 . /2 (1-a)Scos.
2pcos.d. =
S . /2
cos 4 .d. =
3S
(33)
eff
4p -/2 ps · ps 0 16s
Thisjust meansthat our sin2 albedo has the same effect on the total radiation absorbed by the planet as would a
uniform albedo value of 1/4. However, it redistributes that energy, putting more near the equator and less near the
poles. The effect on average temperature across the planet, for this modified version of Eq. 31 is:
Tave =(
S
)1/4 . /2
cos(.)3/4 yave(.(.))cos(.)d. (34)
ps0
which then, putting in the numerical value for the cos7/4 integral, gives the inequality
TABLEI:Relevantparametersfortheplanets. Seehttp://nssdc.gsfc.nasa.gov/planetary/factsheet/. for Eq. 23 (at the
equator, = 0) estimated from thermal inertia, solar day, and the other parameters. It is particularly small for Earth thanks
to rapid rotation and the high heat capacity of water covering most of the surface.
Planet solar constant albedo solar day Tef f Tave Difference
(W/m2 ) (Earth days) (K) (K) (K)
Mercury 9127 0.12 176 434 ? ? 11
Venus 2615 0.75 117 232 737 505 0.7
Earth 1367 0.306 1 255 288 33 0.04
Moon 1367 0.11 29.53 270 253 -17 20
Mars 589 0.25 1.03 210 210 0 0.2
Tave < 0.6206(
S
)1/4 (35)
s
whichis about5%belowthe effectivetemperature(the numerical coefficientis(3/16)1/4 or 0.6580).
IV. INFRARED ABSORPTION IN THE ATMOSPHERE
The examples ofthese simple models show that vertical energy transportfor aplanet with a transparent atmosphere
only smooths out the daily temperature curve, without being able to bring the surface temperature higher than the
effective radiative temperature. The same is true if we were to add in more realistic horizontal energy transport from
larger-scale atmospheric and oceanic circulation -of course getting much more realistic means entering the realm of
more full-scale general circulation models5 , which we have no intention of doing here.
On aplanetwith significantinternal energy sourcesthe effective temperaturefor radiativebalance couldbe exceeded
even with atransparentatmosphere. For example aplanet stilllosing itsinitialheat offormation, or aplanet remote
from its sun with a high enough radioactive content, or on a planet or moon with very large tidal forces, you will
have a net outward flow of energy to space, and may well have an average temperature above the limit. But for the
terrestrial worlds of our solar system, these internal sources of heat are thousands of times too small to have any
noticeable effect on surface temperature.
And yet the observed average surface temperature on Earth and Venus significantly exceeds the effective radiative
temperature set by the incoming solar radiation. This is not observed for the Moon or Mars. What makes Venus and
Earth so different?
Net energyfluxisdeterminedby theradiationthatgetsintospace,notwhatleavesthesurface. Theonly wayfora
planet to be radiatively warmer than the incoming sunlight allows is for some of that thermal radiation to be blocked
from leaving. That means some layer above the surface must be absorbing or reflecting a significant fraction of the
outgoing infrared radiation. I.e. the atmosphere must not be transparent to infrared.
So,let’sadd toourrotatingplanetmodel asimplemodel of thisblocking effect: afraction f (between 0and 1) of
the outgoing radiation Eemitted from the surface is absorbed by a thin layer of the atmosphere. This layer will have
its owntemperaturebutforsimplicity we makethe assumptionthattheheat capacity of the atmosphericlayerislow
so that it remains essentially radiatively balanced through the day, and the specific temperature becomes irrelevant.
That means that this atmospheric layer continuously emits an amount fEemitted equal to what it absorbs from the
·
ground.
Since thermal re-emission is randomly directed, half the radiation from this atmospheric layer will go up, and half
down. Assuming the surface is fully absorbing and the rest of the atmosphere is transparent, total outgoing radiation
fromtheplanet(abovethe atmosphericlayer) isthen:
1
Eout =(1-f)Eemitted +
fEemitted =(1-f/2)Eemitted (36)
2
while absorbed radiation on the surface is now:
1
Eabsorbed(t)= pr2 (1-aeff (t))S(t)+
fEemitted (37)
2
Incoming solar radiation still drives everything -if the solar constant S drops, then so does everything else. But
theeffectof theabsorbinglayeristoreducethe final outgoing energy foragiventemperature,sotheplanetheatsup
until things are back in balance again.
GeneralizingEq. 10wehave net energy change(which canbe calculated either at the surface or above the absorbing
layer of the atmosphere):
1
E.
surface(t)= Eabsorbed(t)-Eemitted(t)= pr2 (1-aeff (t))S(t)+
fEemitted(t)-4pr2 soeff (t)Teff (t)4 (38)
2
= pr2 (1-aeff (t))S(t)-4pr2 soeff (t)(1-f/2)Teff(t)4 (39)
We then end up with essentially the same equations as in the previous section, for example Eq. 16 is the same,
except that effectivelythe solarinput S and thermalinertia c in those equations areincreasedbythefactor1/(1-f/2).
That means the surface effective radiative temperature Teff inthose equationsisincreasedby afactor(1/(1-
f/2))1/4 , or as much as 21/4 for a fully absorbing atmospheric layer. The parameter . is then reduced by that same
ratio (the increases in S and c cancel out, leaving a 1/Teff term). So the temperature curve of this radiatively
insulated planet is even more uniform than without the insulating layer. The average temperature can come within
a few percent of this higher Teff , or well above the limits for a planet with a transparent atmosphere.
A more realistic atmosphere would be characterized by more than one absorbing layer (or a thick layer with a
temperature differential and limited conductivity from bottom to top), which will further decrease outgoing thermal
radiation and increase surface temperatures. Details of absorption in the real atmosphere also depend on pressure;
nevertheless,thepresence of any absorption at allis whatqualitatively distinguishes agreenhouse-effectplanetfrom
one with a transparent atmosphere, and is what allows surface temperatures to climb above the effective radiative
limit.
V. CONCLUSION
Gerlich and Tscheuschner1 state, among more extravagant claims, that “Unfortunately, there is no source in the
literature, wherethegreenhouse effectisintroducedinharmony withthe scientific standards of theoreticalphysics.”
The above analysis I believe completely establishes, within perfectly simple and appropriate theoretical physics constructs, the main points. Namely that assuming “the atmosphere is transparent for visible light but opaque for
infrared radiation” leads to “a warming of the Earth’s surface” relative to firm limits established by basic physical
principles of energy conservation, for the case of an atmosphere transparent to both visible and infrared.
In particular, it has been shown that:
1. An average surface temperature for a planet is perfectly well defined with or without rotation, and with or
withoutinfrared absorbinggases
2. This average temperature is mathematically constrained to be less than the fourth root of the average fourth
power of the temperature, and can in some circumstances (a planet with no or very slow rotation, and low
surface thermal inertia) be much less
3.Foraplanetwith noinfrared absorbing orreflecting layerabovethesurface(and nosignificant fluxofinternal
energy), the fourth power of the surface temperature always eventually averages to a value determined by the
incoming stellar energy flux and relevant reflectivity and emissivity parameters.
4. The onlywaythefourthpower ofthe surfacetemperature can exceed thislimitis tobe coveredby an atmosphere
thatis atleastpartially opaquetoinfrared radiation. Thisisthe atmosphericgreenhouse effect.
5. The measured average temperature of Earth’s surfaceis 33 degreesC higher than the limit determined by items
(2)and(3). Therefore,Earthisproved tohave agreenhouse effect of atleast33K.
The specific contributions of individual gases such as CO2 to Earth’s greenhouse effect are covered well by the
standard treatments of the subject2,4,5 .
*
E-mail:apsmith@aps.org
1
“Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physics” by Gerhard Gerlich and Ralf D.
Tscheuschner, arXiv:0707.1161(2007).
2
See for example, An Introduction to Atmospheric Radiation, second edition by K.N.Liou(2002), section4.1..
3
For example, Thermal Physics, second edition by P.C.Riedi(1988), section10.2 onBlackBody radiation.
4
See Principles of Planetary Climate, by R. T. Pierrehumbert (retrieved February 2008 from
http://geosci.uchicago.edurtp1/ClimateBook/ClimateBook.html) -in particular section 8.3 on thermal inertia.
5
See The Discovery of Global Warming, by Spencer Weart -http://www.aip.org/history/climate/ -for a good discussion of
the development of more and more detailed climate models.