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Response to R. I. Holmes on
Total Solar Irradiance and Temperature at 1 Bar
Robert Wentworth1*
1Ph.D. Applied Physics, retired
Abstract
This work responds to Holmes (2019) in Earth Sciences 8.6, DOI: 10.11648/j.earth.20190806.15, which suggests that the
temperatures of terrestrial planets with thick atmospheres depend primarily on total solar irradiance and pressure, based on
examination of Venus, Earth, and Titan at an altitude where the pressure is 1 bar. A re-evaluation of the data invalidates the
hypothesis in the case of Titan. Calculating a temperature for Earth based on Titan data yields a temperature for Earth that is too
low by 25-30 K. The proposed numerical relationship is confirmed to exist between data for Venus and Earth, within a narrow
range of pressures. However, there is no indication that this single-data-point correlation is associated with causation. To the
contrary, the hypothesis is non-physical, in that it postulates that solar energy reflected away from a planet influences planetary
temperature just as much as solar energy that is absorbed by the planet. Holmes had suggested that it would be difficult for
explanations in terms of albedo and the greenhouse effect to account for the numerical coincidence, given that the albedo and
greenhouse gas characteristics of the two planets are very different. However, an analysis in terms of the albedo and greenhouse
effect of each planet was found to be fully consistent with the empirical relationship between the temperatures of Venus and
Earth at 1 atm. A separate analysis was offered to show that percent greenhouse gases is a meaningless metric for predicting the
relative size of the greenhouse effect on different planets; an improved metric was offered. No statistical or physical basis was
found for believing the relationship identified by Holmes is general or reflects a causal relationship. No justification was found to
interpret the single-data-point relationship between the temperatures of Venus and Earth as anything other than a coincidence.
1 Introduction
In 2019, Robert Ian Holmes published “On the Appar-
ent Relationship Between Total Solar Irradiance and
the Atmospheric Temperature at 1 Bar on Three Terres-
trialtype Bodies” [1]. The paper argues that the tem-
perature of terrestrial planets with thick atmospheres
at a pressure of 1 bar is proportional to the fourth
root of total solar irradiance. This work evaluates that
hypothesis and related claims.
2 Claimed Relationship
2.1 Hypothesis
Holmes [1] suggests that, for terrestrial planets with
thick atmospheres (of which there are only three in our
solar system), the ratio of atmospheric temperatures
measured at a pressure of 1 atm
1
is given by the fourth
root of the ratio of the total solar irradiance,
TSI
, of
each planet:
*Corresponding author: correspondence33@climatepuzzles.org
Published: January 2, 2024 (Revision 0)
1
Holmes uses 1 bar and 1 atm interchangeably, which is not unrea-
sonable in this context. Note that 1 atm = 1.013 bar.
T1(1 atm)
T2(1 atm)≈4
sTSI1
TSI2
=sR2
R1
(1)
where
R
denotes the planet’s distance from the Sun.
This is equivalent to asserting that for these planets
TE(1 atm)≈T2(1 atm)
4
√rTSI2
. (2)
where
TE
is the temperature for Earth; and
rTSI
is the
total solar irradiance relative to that of Earth,
rTSI2=
TSI2/TSIE
. (Alternatively, another planet could, be
used as the reference planet.)
2.2 Planetary Data
Holmes [1] estimates the temperature of Venus at 1 atm
to be 340 K. This estimate seems reasonable given Fig-
ure 1 and other available data [3, Fig. 21, Fig. 15][4].2
Venus orbits with a mean distance from the Sun of
0.723 AU, while Earth’s mean distance is 1.00 AU [5,
6].
Titan has a surface pressure of 1.47 bar [3]. Tabulated
data from Cassini radio occultation measurements in-
dicate that, at a pressure of 1 bar, Titan’s atmosphere
2Venus has a pressure of 1 atm at an altitude just under 50 km [4].
Response to Holmes on TSI and Temperature at 1 Bar
has a consistent temperature of about 85.5 K and an
average altitude of 7.2 km [7, Tables 2-3]. Graphed
data from the Huygens atmospheric probe suggest a
possible temperature at 1 bar as high as 87 K [8, Fig. 3].
These values are at the low end of the 85-90 K range
estimated by Holmes [1].
3
However, I see nothing in
the available data to justify an estimate in the higher
portion of that range.
The estimate of Titan’s atmospheric temperature is
based on Cassini-Huygens data obtained in 2005-2006
[3, 7, 8]. Saturn’s orbit has a period of 29.4 years and an
eccentricity of 6 percent, so this timing matters. Saturn
and Titan were at their closest to the Sun (perihelion) on
July 11, 2003 and at their most distant point (aphelion)
in 2018 [10]. At the time when temperature data was
Figure 1: Venus temperature profile and winds [2, Fig. 6].
3
Holmes [1] reports estimating 85-90 K based on surface temper-
atures varying from 90-94 K between the poles and the equator,
and accounting for the thermal gradient described by Robinson
and Catling [9]. However, my reading of Robinson and Catling
[9] indicates a temperature lapse with altitude larger than what
Holmes inferred. The sources I have relied upon offer direct infor-
mation concerning the temperature at a pressure of 1 bar, without
requiring inferences.
taken,
4
Titan was closer to the Sun than usual, at a
distance of 9.11 AU or less.
2.3 Checking the Hypothesis
The planetary data is tabulated in Table 1. The results
of evaluating equation (2) appear in the final column.
•
As suggested by Holmes, there is excellent agree-
ment between Venus and Earth.
•
However, extrapolating the temperature of Titan
to Earth via equation (2) computes a temperature
for Earth in the range 258-263 K, which is too low
by 25-30 K.
The error associated with Titan is much too large to allow
equations (1) and (2) to be considered useful for estimating
planetary temperature.
This large error for Titan was obscured in Holmes
[1] in a number of ways:
•
Holmes estimated that Titan’s temperature at 1
atm could be as high as 90 K, based on a ques-
tionable inference, uninformed by the specific data
indicating a lower value.
•
Titan’s closer-than-average proximity to the Sun
during the measurement period was not accounted
for.
•
Holmes calculated a temperature for Titan based
on the temperature of Earth, rather than doing
the calculation in the other direction, as has been
done here. That lead to readers trying to compare
temperatures in an unfamiliar range (the range of
temperatures relevant for Titan) rather than mak-
ing comparisons in a familiar range (the range of
temperatures relevant for Earth). When the tem-
peratures are placed in a more familiar context, it
becomes clearer that the size of the error is unac-
ceptably large.
Our solar system contains four terrestrial bodies with
significant atmospheres (Venus, Earth, Mars, Titan).
Holmes did not try to explain the temperature of Mars
due to its low atmospheric pressure, even though the
temperature of Mars has previously been explained
using standard methods of planetary temperature anal-
ysis [11]. We now see that Titan must be eliminated
as well, as not satisfying Holmes’s hypothesis to any
useful degree.
Thus, we find that Holmes has identified a “relation-
ship” between the temperatures of only two planets,
Venus and Earth.
4The final observation date was May 20, 2006 [7].
2ClimatePuzzles.org
Response to Holmes on TSI and Temperature at 1 Bar
Table 1: Numerical check of Holmes’s hypothesis.
R(AU) Relative TSI 4
√rTSI T(1 atm)T(1 atm)
4
√rTSI
Venus 0.723 1.913 1.176 340 K 289 K
Earth 1.00 1 1 288 K 288 K
Titan (observations) 9.11 0.0120 0.331 85.5-87 K 258-263 K
Titan (mean distance)∗9.54 0.0110 0.324 85.5-87 K 271-276 K
∗Calculations using Titan’s mean distance from the Sun are provided for comparison only.
Figure 2: Temperature profiles of Venus and Earth [12, Fig. 2].
2.4 Limited range of pressures
It may be seen from Figure 2 that the temperature pro-
files of Venus and Earth parallel one another over only
a rather limited range of pressures. While the relation-
ship proposed by Holmes holds at around 1 atm, the
match would have been poor if the temperatures had
been compared at a pressure corresponding to high in
Earth’s troposphere.
So, Holmes has identified a relationship between
temperatures that applies to only two planets, and only
over a limited range of pressures.
3 Discussion
3.1 Origin of Hypothesis
In introducing his hypothesis, Holmes writes:
“It is known from the Stefan-Boltzmann law
that the radiating temperature of an isolated
planetary body in space, (one which possesses
no atmosphere), varies with the fourth-root of
the power incident upon it. And given that
previous works have detailed that a principle
factor in determining atmospheric tempera-
tures on planetary bodies with thick atmo-
spheres is atmospheric pressure, logic dictates
that these may be combined - initially at a
standard for pressure - for example, 1 bar.”
What Holmes describes is an approach which is un-
conventional, illogical, and ineffective, when it comes
to addressing scientific and engineering questions in
areas where there is a wealth of pre-existing knowledge
to build upon.
The principles for understanding and predicting the
temperatures of things have been well-understood, and
used routinely, for over a century.5
Understanding those principles allows one to deduce
what temperatures will be. Yet, instead of doing that,
Holmes pastes together mismatched ideas which he
understands poorly—and then guesses what the answer
might be.
Holmes clearly doesn’t understand why the fourth-
root of power shows up in some analyses, or he would
know that it makes no sense to apply it to
TSI
without
accounting for other factors, such as albedo.
The people Holmes cites as saying that pressure is
involved are (a) himself; (b) Robinson and Catling [9]
who say pressure is involved in affecting the infrared
radiative properties of atmospheres—properties which
Holmes argues are not involved in determining plane-
tary temperature; and (c) Nikolov and Zeller [13], who
use a similar flawed approach of guessing answers (that
are unjustified and provably false [14]) rather than de-
ducing answers. So, there isn’t much, if any, justification
for Holmes including pressure in the way that he does.
Holmes offers no basis for taking his hypothesis
seriously. Predicting temperatures is an activity in
which guesswork is unnecessary and inappropriate.
5
Steady-state and transient temperature analysis are basic engineer-
ing skills, not cutting-edge science.
ClimatePuzzles.org 3
Response to Holmes on TSI and Temperature at 1 Bar
3.2 Implications for Standard Theories
Holmes [1] writes:
“Whenever a hypothesis is hypothesis is used
to explain the Earth’s temperature. . . it must
explain. . . how the atmospheric temperature
at 1 bar, in all three terrestrial-type bodies
which possess thick atmospheres seem to be
related to the quaternary root of relative dif-
ferences in total irradiation at equal pressure,
regardless of the very different albedos and
greenhouse gas percentages of these bodies."
6
and:
“Logic perhaps dictates that the widely dif-
fering albedos and ‘greenhouse’ gas content
must mean something for planetary tempera-
tures. Yet the temperatures at 1 bar, calculated
from other planets, using relative TSI alone,
are surprisingly close to the measured tem-
peratures; it is insufficient to simply say they
are wrong. Why and how they are wrong will
need to be comprehensively explained.”
Holmes is unjustified in imagining that his obser-
vation poses any problems for standard theories of
planetary temperature.
It is a common failing of analyses claiming to fal-
sify the greenhouse effect that they rely on assumptions
about what the theory would predict, rather than ap-
plying the theory quantitatively to see what it actually
says about a given situation.
In this case, Holmes seemingly suspects that it would
be difficult for standard approaches for planetary tem-
perature analysis to explain how the observed rela-
tionship could hold for Venus and Earth, given that
the albedos and greenhouse effect values for the two
planets are so different. Yet, in the absence of any
calculations, Holmes is simply offering uninformed
speculation that there would be a problem.
In Appendix C, I have applied standard principles
of planetary temperature physics to investigate the
observed relationship between temperatures on Venus
and Mars. It turns out that the observed relationship
between the two planets is entirely consistent with an
explanation in terms of the albedo and greenhouse effect:
6
Homes seems to imagine that the greenhouse gas percentage is a
relevant metric for estimating the size of the greenhouse effect on
a planet. However, it is not. The greenhouse gas percentage bears
no relationship to the size of the greenhouse effect on different
planets, as is explained in Appendix B.
•
The normalized greenhouse effect on Venus at a
pressure of 1 atm is 0.80, much smaller than the
0.99 normalized greenhouse effect at the surface.
7
Taking this into account, one may verify that the
claimed relationship between the temperatures of
Venus and Earth matches what would be expected
to be observed, based on standard physics.
•
Albedo cools a planet, while the greenhouse ef-
fect warms it. The large albedo and greenhouse
effect (at 1 atm) on Venus have a net effect that
matches the net effect of Earth’s smaller albedo
and greenhouse effect.
So, Holmes’s observation doesn’t even hint in the
direction of there being any problem with standard
approaches to analyzing planetary temperature.
The analysis does not, however, suggest that the re-
lationship between temperatures on Venus and Earth
is more than a numerical coincidence, or that the re-
lationship would be expected to hold between other
planets.
3.3 Physical Laws
Holmes [1] writes:
“This result. . . is more confirmation that the
main determinants of atmospheric tempera-
tures in the regions of terrestrial planetary at-
mospheres which are >0.1 bar, is overwhelm-
ingly the result of two factors; solar insolation
and atmospheric pressure.”
Holmes forgets the maxim that “correlation is not cau-
sation.” All Holmes has demonstrated is a correla-
tion—with a single confirming data point! The only
other available data point (Titan) falsifies the hypothe-
sis. There is no statistical significance whatsoever to a
correlation involving only a single data point.
Noticing such a correlation does not in any way
establish that planetary temperatures are caused by only
solar insolation and atmospheric pressure. Holmes
seems to be engaged in wishful thinking, regarding the
significance of his observation.
Holmes also writes:
“Whenever a hypothesis is used to explain
the Earth’s temperature, it must also take into
account the universality of the physical laws
of nature."
7
This means that on Venus the amount of thermal radiation which
escapes to space is 80% less than the amount of thermal radiation
emitted upwards from a pressure of 1 atm, and 99% less than the
amount of thermal radiation emitted by the surface.
4ClimatePuzzles.org
Response to Holmes on TSI and Temperature at 1 Bar
This statement is rather ironic, insofar as Holmes seems
determined to ignore every established law of physics
(other than the Ideal Gas Law), in favor of speculating
about a possible new physical law that is incompatible
with the previously known laws.
In particular, Holmes seemingly hopes to dethrone
albedo and the greenhouse effect as having any role
in determining planetary temperature. Yet, we know
that these must play a role in planetary temperature
because of some bedrock laws of physics:
1.
Energy cannot be created or destroyed. If more en-
ergy enters a region than leaves, then the balance
must be reflected by an accumulation of energy
within that region.
2.
Temperature is a measure of the thermal energy
content of matter.
Albedo is defined to be the fraction of the solar energy
incident on a planet that is reflected away. Reflected
sunlight is not retained by the planet or its atmosphere.
(Holmes asks us to believe that
TSI
is what deter-
mines planetary temperature, so that the portion of
sunlight that was reflected away from a planet some-
how magically contributes just as much to planetary
temperature as does the amount of sunlight that was
absorbed. How does that honor the laws of physics?)
The greenhouse effect is simply a measure of the rate
at which energy leaves a planet, for a given surface
temperature.
So, by its definition, albedo plays a role in deter-
mining the rate at which energy arrives. And, by its
definition, the greenhouse effect plays a role in deter-
mining the rate at which energy leaves.
The difference between the rate of energy arriving
and the rate of energy leaving determines the rate of
energy accumulating, in accordance with principle #1
above. And, the rate of energy accumulating drives
the rate of temperature change, in accordance with
principle #2 above.8
Thus, Holmes’s effort to reject the relevance of albedo
and the greenhouse effect constitutes a serious failure
to “take into account the universality of the physical
laws of nature.”
8
There are nuances to the argument, but when those nuances are
addressed this analysis becomes entirely rigorous [15].
4 Conclusions
Holmes writes:
“This seems to point to the main determi-
nants of planetary atmospheric temperatures
of terrestrial-type bodies which possess thick
atmospheres, being atmospheric pressure and
TSI, not albedo and greenhouse gas content.
If this relationship proves to be a real feature
of planetary atmospheric physics, it will have
far-reaching effects for how albedo and ‘green-
house’ gas content are treated when calculat-
ing atmospheric temperatures in the future.”
Holmes gets “way ahead of himself” in imaging that
the coincidental relationship he identified, between
two planets within a narrow pressure range, could
potentially be a “real feature of planetary atmospheric
physics.” It would have been more realistic to have
hoped to have found a crude “rule of thumb.”’
Yet, the evidence is far too weak to establish
Holmes’s hypothesis as more than a coincidence that
happened between two planets, but which doesn’t gen-
eralize to a third planet.
Holmes identified a point where two curves of tem-
perature versus altitude happen to intersect—after an
adjustment with no physical basis is performed.
9
The
only marginally interesting thing about the observation
is that the intersection of curves happens to occur at a
pressure that is meaningful to humans. However, such
a coincidence is not scientifically meaningful.
Extraordinary claims require extraordinary evidence.
The hypothesis Holmes offers is supported by a sin-
gle data point, is falsified by Titan data, and is un-
moored from physics.
The single data point falls very far short of constitut-
ing meaningful evidence of anything—not even a viable
“rule of thumb.”
It does not remotely constitute extraordinary evi-
dence of new physics.
9
Adjusting to account for differences in the amount of sunlight
absorbed by different planets would have a physical basis; adjusting
to account for differences in incident sunlight and ignoring the fact
that much of that light is reflected away has no basis in physics.
ClimatePuzzles.org 5
Response to Holmes on TSI and Temperature at 1 Bar
A Appendix: Ideal Gas Law
Holmes [1] makes a number of erroneous arguments
involving the Ideal Gas Law (IGL).
Holmes defines ‘anomalous’:
“‘Anomalous’ meaning an effect outside of the
contributions from their three basic properties
of density, pressure and molar mass. . . ”
and asserts:
“For a ‘greenhouse effect’ caused by ‘green-
house gases’ to occur in a convecting atmo-
sphere (one of >10kPa), a large anomalous
change must happen in the density, the pressure
or both. No anomalous changes of this mag-
nitude have been detected in any planetary
atmospheres.”
Holmes does not explain why he believes that the
greenhouse effect would be associated with ‘anoma-
lous changes’, i.e., changes that cause the IGL to be
violated. However, that assertion is baseless. As is
illustrated in Table 2, the greenhouse effect has been
observed and quantified in the atmospheres of the four
terrestrial planets with atmospheres. Yet, in each of
these atmospheres, the gas law (the IGL or a more
precise Real Gas Law) is fully honored. So, contrary
to Holmes’s assertion, there is no conflict between the
greenhouse effect and the IGL.
The fundamental error in Holmes’s thinking seems
to relate to ignoring the role of energy in determining the
temperature and density of a gas.
For example, Holmes writes:
“The molar mass version of the ideal gas law
accurately determines - and allows - an atmo-
spheric temperature to be determined based
only on a gas constant and three gas prop-
erties; namely pressure, density and molar
mass. No reference to the radiative properties
of a gas are needed or included.”
The three parameters temperature, density, and pres-
sure, encode the state of a gas, after that state has been
established. These parameters contain no information
about how the gas got to that state. Nor do they forbid
changes to the state, as will occur if thermal energy is
added or taken away.
The radiative properties of gases alter energy flows,
potentially impacting the energy content of the gas.
The gas law does not care about the radiative properties
of gases because, by itself, the IGL offers no information
about what, if any processes might alter the energy content
of the gas, thereby altering the temperature and density. All
the IGL tells us is that, when energy content changes,
the state variables will change in a way that maintains
the relationship between temperature, density, and
pressure.
Continuing, Holmes writes:
“the same concentrations of gases cannot pro-
vide different temperatures at different times.
The formula
T=P M/Rρ
which is derived
from the ideal gas law, forbids it.”
Yet, that statement is clearly false. The only thing that
the concentrations determine in Holmes’s equation is
the molar mass,
M
. The temperature,
T
, and density,
ρ
, are both free to change without causing the IGL to
be violated. And, both
T
and
ρ
will change, if more
thermal energy is added to the air. Setting the concen-
tration of gases in a gas mixture does not forever freeze
the temperature and density of the gas mixture, so that
these must remain unchanged for all time. Holmes’s
assertion is a non-sequitur.
Homes also writes:
The reason for the terminal conflict is because
it is stated in all IPCC reports that there exists
a time delay to reach ‘equilibration’. . . if there
was a sudden doubling in the atmospheric
greenhouse gas CO2, the greenhouse gas ef-
fect from this would operate slowly, causing
an eventual 3c of warming over centuries
to millennia. . . the temperature must rise sig-
nificantly over time, with the same prevail-
ing atmospheric gas concentrations, and there
would be no rapid equilibration, as the ideal
gas law and it’s derivative, the molar mass
version, demand.”
Holmes seems confused, perhaps because there are
different processes of ‘equilibration’ that occur on dif-
ferent time scales:
•
As thermal energy is added to air, the temperature
and density adjust almost instantaneously to re-
flect the new energy content and the new state of
the gas. Likely this is the ‘rapid equilibration’ that
Holmes refers to.
•
Changing the radiative properties of the atmo-
sphere alters energy flows so that more energy
is arriving to Earth than is leaving. Most of this
energy ends up in the oceans. As energy is added
to the oceans, their temperatures slowly increase
6ClimatePuzzles.org
Response to Holmes on TSI and Temperature at 1 Bar
over centuries to millennia. (Surely, Holmes is not
demanding that the oceans should jump to their
final temperatures instantly?) As the oceans warm,
they cause the energy content of the atmosphere
to increase, so that the mean temperature of the
atmosphere rises as well.
So, there are two different types of equilibration hap-
pening, one with a short time scale and one with a long
time scale. There is no contradiction involved.
For more on Holmes and the IGL, see Wentworth
[16].
Holmes’s arguments involving the IGL are uniformly
mistaken.
B Appendix: Comparing
Greenhouse Gas Amounts
Holmes [1] writes:
“The data shows that the ‘greenhouse gas’ con-
centration varies widely from the low 2.7%
and 2.5% for Titan and Earth respectively, to
the very high 96.5% for Venus; the implica-
tion must be that there cannot be any special
warming effect from the so-called ‘greenhouse’
gases."
Holmes uses the percentage of greenhouse gases as the
basis of his reasoning about the greenhouse effect on
different planets. Yet, his is a largely meaningless metric
to focus on when comparing different planets. It bears
no relationship to the size of the greenhouse effect on
each planet.
Use of this metric in this way involves misunder-
standing the role of non-absorbing atmospheric con-
stituents. Comparing the amount of greenhouse gases
to the total amount of gas implicitly assumes that trans-
parent gases in an atmosphere reduces the impact of
whatever greenhouse gases are present. Yet, this is not
the case.
To the contrary, non-absorbing gases boost the impact
of each greenhouse gas molecule, by raising the overall
atmospheric pressure, which broadens absorption:
•
The ability of an atmosphere to absorb radiation as
it travels upward is proportional the the number of
greenhouse gas molecules that must be traversed.
•
How much radiation is absorbed by greenhouse
gases on a per-molecule basis increases as pres-
sure increases, through the mechanism of collision-
induced spectral line broadening [17, Section 4.4].
Figure 3: Absorption by the same amount of atmospheric CO
2
(1 mole/m
2
) in different pressure regimes, in the spectral region of
peak absorption. Low pressure and temperature cause absorption
to be high only within very narrow spectral lines. As pressure and
temperature increase, absorption lines broaden and overlap, leading
to greater overall absorption of thermal radiation.
This phenomenon is illustrated in Figure 3, which
shows how absorption compares at different al-
titudes on Earth and near the surface of Mars.
On Venus, high pressures lead to CO
2
exhibiting
continuum absorption, in which radiation is ab-
sorbed very broadly [18]. Having non-absorbing
molecules present in an atmosphere enhances the ab-
sorbing power of whatever greenhouse gas molecules are
present, by increasing pressure, thereby broadening
the range of wavelengths that can be absorbed.
Collision-broadening is related to the rate at which
a molecule experiences collisions. This rate is pro-
portional to
ρ·√T
or
P/√T
where
ρ
is density,
P
is
pressure, and Tis temperature.10
What is most relevant to planetary temperature is the
value of the greenhouse effect itself. A theoretical value
of the greenhouse effect can be calculated using the
radiative transfer equation [19], or an observed value
can be determined using measurements of radiative
fluxes.
In the absence of doing such a calculation, focusing
on the percentage of greenhouse gases is not an appro-
priate proxy for the size of the greenhouse effect on
different planets. Considering the percentage of green-
house gases is only helpful when analyzing a single
planet for which those percentages are changing.
If one must have a crude proxy for the likely size of
the greenhouse effect of a planet, one might consider
10√Tappears because it is associated with molecular velocity.
ClimatePuzzles.org 7
Response to Holmes on TSI and Temperature at 1 Bar
using the following rough proxy for the amount of
absorbing power in the atmosphere, AP:
AP =GMA ·Ps/√Tsa (3)
where
Ps
and
Tsa
are surface pressure and near-surface
air temperature; and
GMA
, or greenhouse-gas moles
per unit area, is the number of moles of greenhouse
gas in a fixed-area vertical column through the atmo-
sphere.
GMA
indicates the density of greenhouse gas
molecules that must be traversed to reach space, while
Ps/√Tsa
estimates the amount of pressure broadening.
Note that
GMA
is only a crude metric for the amount
of greenhouse gases, since different greenhouse gases
have different effects, on a per-molecule basis. Sim-
ilarly,
AP
is only a crude metric for estimating how
how much capacity the atmosphere has for absorbing
thermal radiation. Note that the greenhouse effect is
determined by both absorption and emission. Even
so,
AP
is a much better indicator of how large the
greenhouse effect is likely to be than is the “percent
greenhouse gases.”
The total number of moles of gas in a column is
given by
Ps/(g′·Ma)
assuming pressure is measured
in pascals; where
g′
is the gravitational constant for that
planet; and
Ma
is the mean molar mass of atmospheric
gases. So, if
fGHG
is the molar fraction of greenhouse
gases (percentage/100), then the greenhouse-gas moles
per unit area is:
GMA =fGHG ·Ps
g′·Ma. (4)
The actual greenhouse effect may be calculated in
terms of any of these metrics [20, 21, 11]:
G=SLR −OLR (5)
˜
g=G/SLR =1−OLR/SLR (6)
∆Tg= (SLR/σ)1/4 −(OLR/σ)1/4 (7)
where
SLR
is the radiative flux of longwave thermal
radiation emitted by the surface,
OLR
is the flux of
longwave radiation reaching space, and
σ
is the Stefan-
Boltzmann constant.
SLR
is essentially given by
SLR =σT4
s
, where
Ts
is
the surface skin temperature,
11
though there are some
nuances [15]. For a planet whose temperature is stable,
the rate of outgoing energy,
OLR
, must be very nearly
equal to the rate of incoming energy. If there is no
significant internal heating source (as there is for the
gas giants, but not for terrestrial planets), then the rate
11
Skin temperature, in this context, refers to the temperature of a thin
layer of matter that interacts with the air above. Typically,
Ts>Tsa
.
of incoming energy is just the rate of absorbed solar
radiation,
ASR = (
1
−a)MSI = (
1
−a)TSI/
4. Thus,
OLR = (1−a)TSI/4.
These formulas allow the greenhouse effect on each
planet to be calculated from observed data.12
A larger value of the absorption proxy metric,
AP
, is
generally likely to correspond to a larger normalized
greenhouse effect,
˜
g
, although the relationship is not at
all linear.
The value of the absorption proxy,
AP
, and of the
greenhouse effect metrics are tabulated in Table 2.
13
As may be seen from this table, the proxy
AP
correctly
sorts planets according to which has the largest nor-
malized greenhouse effect, ˜
g.
The fraction of the atmosphere that is greenhouse gases,
fGHG
, has no relationship whatsoever to the size of the ob-
served greenhouse effect, when comparing different plan-
ets.
C Appendix: Analysis Using
Radiative Concepts
C.1 Formulas
Let us analyze what physics tells us about the hypoth-
esis expressed by equations (1)-(2).
Formulas for planetary temperature typically de-
pend on the mean solar irradiance,
MSI
, measured on
a conceptual spherical surface above a planet’s atmo-
sphere. The intensity of the Sun is reported in terms of
the total solar irradiance,
TSI
, which is measured on a
flat plane perpendicular to the Sun’s rays. A geometri-
cal argument allows one to convert between these two
12
The analysis here is broadly correct. However, to obtain accurate
values, it is important to pay attention to details. In the case of
Earth,
OLR
is about 1 W/m
2
less than
(
1
−a)TSI/
4 because there
is currently an energy imbalance and planetary warming; and
the mentioned nuances (related to emissivity and temperature
variations [22]) mean that, for Earth, it is simplest to use a reported
value for
SLR
rather than trying to infer it from temperature. For
Mars it is essential to consider the effects of temperature variations
[11]. A two-part analysis is needed in the case of Titan, because
Titan has both a greenhouse effect and an anti-greenhouse effect
[23]; the above formulas would combine these two opposing effects
into one nominal greenhouse effect, underestimating the size of
the actual greenhouse effect.
13
Table 2 is based on data from [5, 6, 24, 25, 26, 27, 11, 23]. Earth’s
atmosphere overall has 0.4% water vapor by volume [28, 29].
8ClimatePuzzles.org
Response to Holmes on TSI and Temperature at 1 Bar
Table 2: Greenhouse effect and absorption proxy for different planets.
GHG Amount Absorption Proxy Greenhouse Effect
PsTsa g′MafGHG GMA Ps/T
1
2
sa AP ˜
g G ∆Tg
Venus 92 737 8.87 43.5 95.6 23000 3.4 77000 0.99 16600 510
Titan 1.47 94 1.35 27.5 5.0 200 0.15 30 0.63 2.8 21
Earth 1.01 288 9.82 29.0 0.44 1.6 0.060 0.094 0.40 158 34
Mars 0.0064 201 3.73 43.5 95.1 3.7 0.00045 0.0017 0.09 11 5
bar K m/s2kg/kmol % kmol/m2bar/K 1
2kmol bar/m2K1
2W/m2K
quantities, leading to the conclusion
MSI =TSI/
4.
1415
It follows that, comparing planets 1 and 2:
TSI1
TSI2
=MSI1
MSI2
=R2
R12
(8)
where
R1
and
R2
are the radiuses of each planet’s orbit
around the Sun.
For a given planet, the air temperature at different
pressures, P, may be written16
T(P) = 4
rMSI
σ·K(P)(9)
where
MSI
is mean solar irradiance above the atmo-
sphere, σis the Stefan-Boltzmann constant, and
K(P) = (1−a)·w(P)4
1−˜
g(P). (10)
Here,
a
is the planetary albedo;
˜
g(P)
is the normal-
ized greenhouse effect between a particular pressure
level and space (i.e., the fraction by which upwelling
14
The total power intercepted by the planet is
πr2·TSI
where
r
is the
radius of the planet; dividing this total power over the full surface
area of the planet, 4
πr2
, leads to
MSI =TSI/
4. This result must
be true, given the way the quantities
TSI
and
MSI
are defined. If
one takes into account Earth being slightly flattened relative to a
perfect sphere, the relationship becomes MSI =TSI/4.0034 [30].
15
Some climate skeptics seem strangely resistant to this mathematical
reality. Using an average over the full surface area of the planet
does not mean that climate scientists don’t recognize that there are
differences between day and night. It simply means that they want
to define a flux of incoming energy which can be compared to the
flux of outgoing energy; and energy leaves to space from the entire
surface of the planet. They are being careful to ensure that their
calculations are consistent with the principle of conservation of
energy. The math is entirely rigorous.
16
This assumes energy balance, so that the outgoing longwave radi-
ation flux
OLR
is equal to the absorbed energy flux
(
1
−a)MSI
.
Given that, equation (9) turns out to be a tautology, i.e., a equa-
tion that mathematically is always true. Verifying the equation is
simply a matter of taking into account the definitions of all the
quantities involved. There is no actual physics required, beyond
the assumption that the rates of incoming and outgoing energy are
equal.
longwave radiation at the the given pressure will be
reduced before space is reached); and
w(P)
is the ra-
tio between the air temperature at that pressure and
the effective temperature associated with upwelling
radiation at that pressure.17
So, when the temperature of different planets at the
same pressure, Pm, is compared, we find
T1(Pm)
T2(Pm)=4
sMSI1
MSI2·K1(Pm)
K2(Pm)(11)
Because
MSI1/MSI2=TSI1/TSI2
, this means that
Holmes’ hypothesis, equation (1), is equivalent to the
claim that
4
sK1(Pm)
K2(Pm)≈1 (12)
when Pm=1 atm.
The preceding equations (9)-(11) are are valid for
any planet in equilibrium and heated only by sunlight.
Equation (10) may be simplified when
P
is a pres-
sure for which the atmosphere is opaque to thermal
radiation. In this case, the atmosphere radiates as a
black-body; so the actual and effective temperatures
are equal, and
w(P) =
1.
1819
In this situation, equation
(10) becomes
K(P) = 1−a
1−˜
g(P). (13)
17
In other words, If
U(P)
is the flux of upwelling longwave radiation
at pressure
P
, so that
OLR =U(
0
)
, then
˜
g(P) =
1
−U(
0
)/U(P)
and w(P) = T(P)/[U(P)/σ]1/4 .
18
Aside from mentions of conservation of energy and energy balance,
this is actually the first point in this analysis where I say anything
that isn’t trivially and automatically true mathematically, based on
the definitions of the quantities involved. However, this statement
is trivially true based on the physics involved.
19
In parts of the atmosphere that are partially transparent to thermal
radiation,
w(P)
will typically be somewhat less than 1, except when
there is a thermal inversion with temperatures increasing at higher
altitudes, as occurs in the upper atmosphere.
ClimatePuzzles.org 9
Response to Holmes on TSI and Temperature at 1 Bar
Table 3: Numerical check of hypothesis using radiative analysis.
Albedo Greenhouse Effect K(1 atm) = K(1 atm)1/4 =
aSurface ˜
g(Ps)˜
g(1 atm) (1−a)/(1−˜
g(1 atm)) 4
p(1−a)/(1−˜
g(1 atm))
Venus 0.77 0.99 0.80 1.17 1.04
Earth 0.29 0.40 0.40 1.17 1.04
Figure 4: Taking the fourth root of
K2/K1
produces a result that
is much closer to one than was the initial value.
The Venusian atmosphere is opaque to thermal ra-
diation in those portions of the atmosphere where the
temperature exceeds about 260 K, as may be inferred
by examining the spectrum of thermal radiation emit-
ted to space [18, Fig. 3c]. Since
TV(
1
atm) =
340 K
for Venus, this means that the Venusian atmosphere is
effectively opaque to thermal radiation at this altitude.
Thus, equation (13) applies to Venus at a pressure of 1
atm.
C.2 Discussion and Application
So, to understand what physics has to say about the
hypothesis, i.e., equation (12), we need to examine the
behavior of [K2(P)K2(P)]1/4.
The first thing to realize is that taking the forth
root of a quantity always brings that result much
closer to one. This is illustrated in Figure 4. Thus,
if
K2(P)/K1(P)
is even vaguely close to 1, then taking
its fourth root will produce a result much closer to 1.
Next, we consider the behavior of
˜
g(P)
.
20
The
pressure-dependent greenhouse effect,
˜
g(P)
, decreases
with decreasing pressure, until reaching zero at zero
pressure. The greenhouse effect parameter is largest
at the surface,
21
where it has the value
˜
g(Ps)
, with
Ps
20I’ll be describing the behavior of ˜
g(P)based on an understanding
of the underlying physics.
21
That
˜
g(P)
is largest at the surface does not mean that the influence
of the greenhouse effect is localized to the surface. The warming
impact of the greenhouse effect is a systemic effect, which involves
the entire vertical atmospheric profile. The parameter
˜
g(P)
repre-
sents the influence of the greenhouse effect over the altitude range
from pressure
P
out to space. Naive reasoning about this is likely
to be incorrect.
being surface pressure.
The surface pressure on Venus is about 92 bar. When
considering an altitude where the pressure is 1 atm, the
greenhouse effect parameter at that altittude,
˜
g(
1
atm)
is expected to be substantially smaller than the green-
house effect parameter at the surface, ˜
g(Ps).
The thermal opacity of the atmosphere at 1 atm al-
lows us to deduce that the upward longwave radiation
flux at this level is
UV(
1
atm) = σ·(
340 K
)4=
758
W/m
2
. Venus has a Bond albedo of
aV=
0.77 and
TSIV=
2601W/m
2
[5]. So, we find
OLRV=ASRV=
(1−aV)MSIV=150 W/m2and ˜
gV(1 atm) = 0.80.
This value of
˜
gV(
1
atm) =
0.80 is a large reduction
relative to the surface value
˜
gV(Ps) =
0.99. However,
this is to be expected since most of the atmosphere of
Venus is below this altitude.
Plugging these values into equation (9) yields
TV(1 atm) = 340 K, as expected.
Earth’s albedo is
aE=
0.29; and its normalized green-
house effect is ˜
g=0.40 [27].
These values are tabulated in Table 3, along with
calculated values for
K(
1
atm)
and
4
pK(1 atm)
. As can
be seen, the value of
4
pK(1 atm)
is identical for Venus
and Earth. Thus, equation (12), which corresponds to
the relationship proposed by Holmes, is satisfied for
Venus and Earth, according to a radiative analysis of
these two planets.
Thus we find that the relationship regarding the rela-
tionship between the temperatures of Venus and Earth at a
pressure of 1 atm is consistent with an analysis based on
the albedo and the greenhouse effect of each planet, despite
the the albedo and greenhouse effect both being much
larger on Venus.
10 ClimatePuzzles.org
Response to Holmes on TSI and Temperature at 1 Bar
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