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New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary Temperature Model

Authors:
  • USFS Rocky Mountain Research Station, Fort Collins CO USA

Abstract and Figures

A recent study has revealed that the Earth’s natural atmospheric greenhouse effect is around 90 K or about 2.7 times stronger than assumed for the past 40 years. A thermal enhancement of such a magnitude cannot be explained with the observed amount of outgoing infrared long-wave radiation absorbed by the atmosphere (i.e. ≈ 158 W m-2), thus requiring a re-examination of the underlying Greenhouse theory. We present here a new investigation into the physical nature of the atmospheric thermal effect using a novel empirical approach toward predicting the Global Mean Annual near-surface equilibrium Temperature (GMAT) of rocky planets with diverse atmospheres. Our method utilizes Dimensional Analysis (DA) applied to a vetted set of observed data from six celestial bodies representing a broad range of physical environments in our Solar System, i.e. Venus, Earth, the Moon, Mars, Titan (a moon of Saturn), and Triton (a moon of Neptune). Twelve relationships (models) suggested by DA are explored via non-linear regression analyses that involve dimensionless products comprised of solar irradiance, greenhouse-gas partial pressure/density and total atmospheric pressure/density as forcing variables, and two temperature ratios as dependent variables. One non-linear regression model is found to statistically outperform the rest by a wide margin. Our analysis revealed that GMATs of rocky planets with tangible atmospheres and a negligible geothermal surface heating can accurately be predicted over a broad range of conditions using only two forcing variables: top-of-the-atmosphere solar irradiance and total surface atmospheric pressure. The hereto discovered interplanetary pressure-temperature relationship is shown to be statistically robust while describing a smooth physical continuum without climatic tipping points. This continuum fully explains the recently discovered 90 K thermal effect of Earth’s atmosphere. The new model displays characteristics of an emergent macro-level thermodynamic relationship heretofore unbeknown to science that has important theoretical implications. A key entailment from the model is that the atmospheric ‘greenhouse effect’ currently viewed as a radiative phenomenon is in fact an adiabatic (pressure-induced) thermal enhancement analogous to compression heating and independent of atmospheric composition. Consequently, the global down-welling long-wave ux presently assumed to drive Earth’s surface warming appears to be a product of the air temperature set by solar heating and atmospheric pressure. In other words, the so-called ‘greenhouse back radiation’ is globally a result of the atmospheric thermal effect rather than a cause for it. Our empirical model has also fundamental implications for the role of oceans, water vapour, and planetary albedo in global climate. Since produced by a rigorous attempt to describe planetary temperatures in the context of a cosmic continuum using an objective analysis of vetted observations from across the Solar System, these ndings call for a paradigm shift in our understanding of the atmospheric ‘greenhouse effect’ as a fundamental property of climate.
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Research Article
Environment Pollution and
Climate Change
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Nikolov and Zeller, Environ Pollut Climate Change 2017, 1:2
Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
Keywords: Greenhouse eect; Emergent model; Planetary
temperature; Atmospheric pressure; Greenhouse gas; Mars temperature
Introduction
In a recent study Volokin and ReLlez [1] demonstrated that the
strength of Earths atmospheric Greenhouse Eect (GE) is about 90 K
instead of 33 K as presently assumed by most researchers [2-7]. e new
estimate corrected a long-standing mathematical error in the application
of the Stefan–Boltzmann (SB) radiation law to a sphere pertaining to
Hölder’s inequality between integrals. Since the current greenhouse
theory strives to explain GE solely through a retention (trapping) of
outgoing long-wavelength (LW) radiation by atmospheric gases [2,5,7-
10], a thermal enhancement of 90 K creates a logical conundrum, since
satellite observations constrain the global atmospheric LW absorption
to 155–158 W m-2 [11-13]. Such a ux might only explain a surface
warming up to 35 K. Hence, more than 60% of Earth’s 90 K atmospheric
eect appears to remain inexplicable in the context of the current theory.
Furthermore, satellite- and surface-based radiation measurements have
shown [12-14] that the lower troposphere emits 42-44% more radiation
towards the surface (i.e. 341-346 W m-2) than the net shortwave ux
delivered to the Earth-atmosphere system by the Sun (i.e. 240 W m-2).
In other words, the lower troposphere contains signicantly more
kinetic energy than expected from solar heating alone, a conclusion also
supported by the new 90 K GE estimate. A similar but more extreme
situation is observed on Venus as well, where the atmospheric down-
welling LW radiation near the surface (>15,000 W m-2) exceeds the
total absorbed solar ux (65–150 W m-2) by a factor of 100 or more [6].
e radiative greenhouse theory cannot explain this apparent paradox
considering the fact that infrared-absorbing gases such as CO2, water
vapor and methane only re-radiate available LW emissions and do not
constitute signicant heat storage or a net source of additional energy to
the system. is raises a fundamental question about the origin of the
observed energy surplus in the lower troposphere of terrestrial planets
with respect to the solar input. e above inconsistencies between theory
and observations prompted us to take a new look at the mechanisms
controlling the atmospheric thermal eect.
We began our study with the premise that processes controlling
the Global Mean Annual near-surface Temperature (GMAT) of Earth
are also responsible for creating the observed pattern of planetary
temperatures across the Solar System. us, our working hypothesis was
that a general physical model should exist, which accurately describes
equilibrium GMATs of planets using a common set of drivers. If true,
such a model would also reveal the forcing behind the atmospheric
thermal eect.
Instead of examining existing mechanistic models such as 3-D
*Corresponding author: Ned Nikolov, Ksubz LLC, 9401 Shooy Lane, Wellington
CO 80549, USA, Tel: 970-980-3303, 970-206-0700; E-mail: ntconsulting@comcast.net
Received November 11, 2016; Accepted February 06, 2017; Published February
13, 2017
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the
Atmospheric Greenhouse Effect Deduced from an Empirical Planetary Temperature
Model. Environ Pollut Climate Change 1: 112.
Copyright: © 2017 Nikolov N, et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
Abstract
A recent study has revealed that the Earth’s natural atmospheric greenhouse effect is around 90 K or about 2.7 times stronger
than assumed for the past 40 years. A thermal enhancement of such a magnitude cannot be explained with the observed amount
of outgoing infrared long-wave radiation absorbed by the atmosphere (i.e. 158 W m-2), thus requiring a re-examination of the
underlying Greenhouse theory. We present here a new investigation into the physical nature of the atmospheric thermal effect using a
novel empirical approach toward predicting the Global Mean Annual near-surface equilibrium Temperature (GMAT) of rocky planets
with diverse atmospheres. Our method utilizes Dimensional Analysis (DA) applied to a vetted set of observed data from six celestial
bodies representing a broad range of physical environments in our Solar System, i.e. Venus, Earth, the Moon, Mars, Titan (a moon
of Saturn), and Triton (a moon of Neptune). Twelve relationships (models) suggested by DA are explored via non-linear regression
analyses that involve dimensionless products comprised of solar irradiance, greenhouse-gas partial pressure/density and total
atmospheric pressure/density as forcing variables, and two temperature ratios as dependent variables. One non-linear regression
model is found to statistically outperform the rest by a wide margin. Our analysis revealed that GMATs of rocky planets with tangible
atmospheres and a negligible geothermal surface heating can accurately be predicted over a broad range of conditions using
only two forcing variables: top-of-the-atmosphere solar irradiance and total surface atmospheric pressure. The hereto discovered
interplanetary pressure-temperature relationship is shown to be statistically robust while describing a smooth physical continuum
without climatic tipping points. This continuum fully explains the recently discovered 90 K thermal effect of Earth’s atmosphere. The
new model displays characteristics of an emergent macro-level thermodynamic relationship heretofore unbeknown to science that
has important theoretical implications. A key entailment from the model is that the atmospheric ‘greenhouse effect’ currently viewed
as a radiative phenomenon is in fact an adiabatic (pressure-induced) thermal enhancement analogous to compression heating
and independent of atmospheric composition. Consequently, the global down-welling long-wave ux presently assumed to drive
Earth’s surface warming appears to be a product of the air temperature set by solar heating and atmospheric pressure. In other
words, the so-called ‘greenhouse back radiation’ is globally a result of the atmospheric thermal effect rather than a cause for it. Our
empirical model has also fundamental implications for the role of oceans, water vapour, and planetary albedo in global climate. Since
produced by a rigorous attempt to describe planetary temperatures in the context of a cosmic continuum using an objective analysis
of vetted observations from across the Solar System, these ndings call for a paradigm shift in our understanding of the atmospheric
‘greenhouse effect’ as a fundamental property of climate.
New Insights on the Physical Nature of the Atmospheric Greenhouse
Effect Deduced from an Empirical Planetary Temperature Model
Ned Nikolov* and Karl Zeller
Ksubz LLC, 9401 Shooy Lane, Wellington CO 80549, USA
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
Page 2 of 22
Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
GCMs, we decided to try an empirical approach not constrained by a
particular physical theory. An important reason for this was the fact that
current process-oriented climate models rely on numerous theoretical
assumptions while utilizing planet-specic parameterizations of key
processes such as vertical convection and cloud nucleation in order
to simulate the surface thermal regime over a range of planetary
environments [15]. ese empirical parameterizations oentimes
depend on detailed observations that are not typically available for
planetary bodies other than Earth. Hence, our goal was to develop
a simple yet robust planetary temperature model of high predictive
power that does not require case-specic parameter adjustments while
successfully describing the observed range of planetary temperatures
across the Solar System.
Methods and Data
In our model development we employed a ‘top-down’ empirical
approach based on Dimensional Analysis (DA) of observed data
from our Solar System. We chose DA as an analytic tool because of
its ubiquitous past successes in solving complex problems of physics,
engineering, mathematical biology, and biophysics [16-21]. To our
knowledge DA has not previously been applied to constructing
predictive models of macro-level properties such as the average global
temperature of a planet; thus, the following overview of this technique
is warranted.
Dimensional analysis background
DA is a method for extracting physically meaningful relationships
from empirical data [22-24]. e goal of DA is to restructure a set of
original variables deemed critical to describing a physical phenomenon
into a smaller set of independent dimensionless products that may be
combined into a dimensionally homogeneous model with predictive
power. Dimensional homogeneity is a prerequisite for any robust
physical relationship such as natural laws. DA distinguishes between
measurement units and physical dimensions. For example, mass is a
physical dimension that can be measured in gram, pound, metric ton
etc.; time is another dimension measurable in seconds, hours, years,
etc. While the physical dimension of a variable does not change, the
units quantifying that variable may vary depending on the adopted
measurement system.
Many physical variables and constants can be described in terms of four
fundamental dimensions, i.e. mass [M], length [L], time [T], and absolute
temperature [Θ]. For example, an energy ux commonly measured in W
m-2 has a physical dimension [M T-3] since 1 W m-2 = 1 J s-1 m-2 = 1 (kg m2
s-2) s-1 m-2 = kg s-3. Pressure may be reported in units of Pascal, bar, atm.,
PSI or Torr, but its physical dimension is always [M L-1 T-2] because 1 Pa
= 1 N m-2 = 1 (kg m s-2) m-2 = 1 kg m-1 s-2. inking in terms of physical
dimensions rather than measurement units fosters a deeper understanding
of the underlying physical reality. For instance, a comparison between
the physical dimensions of energy ux and pressure reveals that a ux is
simply the product of pressure and the speed of moving particles [L T-1],
i.e. [M T-3] = [M L-1 T-2] [L T-1]. us, a radiative ux FR (W m-2) can be
expressed in terms of photon pressure Pph (Pa) and the speed of light c (m
s-1) as FR = c Pph. Since c is constant within a medium, varying the intensity
of electromagnetic radiation in a given medium eectively means altering
the pressure of photons. us, the solar radiation reaching Earths upper
atmosphere exerts a pressure (force) of sucient magnitude to perturb the
orbits of communication satellites over time [25,26].
e simplifying power of DA in model development stems from the
Buckingham Pi eorem [27], which states that a problem involving n
dimensioned xi variables, i.e.
( )
12 0
n
f x , x , , x
…=
can be reformulated into a simpler relationship of (n-m) dimensionless
πi products derived from xi, i.e.
ϕ(π1, π2, …. ,πn-m) = 0
where m is the number of fundamental dimensions comprising the
original variables. is theorem determines the number of non-
dimensional πi variables to be found in a set of products, but it does not
prescribe the number of sets that could be generated from the original
variables dening a particular problem. In other words, there might be,
and oentimes is more than one set of (n-m) dimensionless products to
analyze. DA provides an objective method for constructing the sets of
πi variables employing simultaneous equations solved via either matrix
inversion or substitution [22].
e second step of DA (aer the construction of dimensionless
products) is to search for a functional relationship between the πi
variables of each set using regression analysis. DA does not disclose
the best function capable of describing the empirical data. It is the
investigator’s responsibility to identify a suitable regression model
based on prior knowledge of the phenomenon and a general expertise
in the subject area. DA only guarantees that the nal model (whatever
its functional form) will be dimensionally homogeneous, hence it may
qualify as a physically meaningful relationship provided that it (a) is
not based on a simple polynomial t; (b) has a small standard error;
(c) displays high predictive skill over a broad range of input data; and
(d) is statistically robust. e regression coecients of the nal model
will also be dimensionless, and may reveal true constants of Nature by
virtue of being independent of the units utilized to measure the forcing
variables.
Selection of model variables
A planet’s GMAT depends on many factors. In this study, we focused
on drivers that are remotely measurable and/or theoretically estimable.
Based on the current state of knowledge we identied seven physical
variables of potential relevance to the global surface temperature: 1) top-
of-the-atmosphere (TOA) solar irradiance (S); 2) mean planetary surface
temperature in the absence of atmospheric greenhouse eect, hereto
called a reference temperature (Tr); 3) near-surface partial pressure
of atmospheric greenhouse gases (Pgh); 4) near-surface mass density
of atmospheric greenhouse gases (ρgh); 5) total surface atmospheric
pressure (P); 6) total surface atmospheric density (ρ); and 7) minimum
air pressure required for the existence of a liquid solvent at the surface,
hereto called a reference pressure (Pr). Table 1 lists the above variables
along with their SI units and physical dimensions. Note that, in order to
simplify the derivation of dimensionless products, pressure and density
are represented in Table 1 by the generic variables Px and ρx, respectively.
As explained below, the regression analysis following the construction
of
π
i
variables explicitly distinguished between models involving
partial pressure/density of greenhouse gases and those employing total
atmospheric pressure/density at the surface. e planetary Bond albedo
(αp) was omitted as a forcing variable in our DA despite its known eect
on the surface energy budget, because it is already dimensionless and
also partakes in the calculation of reference temperatures discussed
be low.
Appendix A details the procedure employed to construct the πi
variables. DA yielded two sets of πi products, each one consisting of two
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
Page 3 of 22
Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
dimensionless variables, i.e.
3
12
2
sx
rx
TP
;
T S
π= π=
ρ
and
12
sx
rr
TP
;
TP
π= π=
is implies an investigation of two types of dimensionally homogeneous
functions (relationships):
3
2
sx
rx
TP
T S

=
ρ

ƒ
(1)
and
( )
2
sx
rr
TP
f
TP

=


(2)
Note that π1 = Ts/Tr occurs as a dependent variable in both relationships,
since it contains the sought temperature Ts. Upon replacing the generic
pressure/density variables Px and ρx in functions (1) and (2) with
either partial pressure/density of greenhouse gases (Pgh and ρgh) or total
atmospheric pressure/density (P and ρ), one arrives at six prospective
regression models. Further, as explained below, we employed two
distinct kinds of reference temperature computed from dierent
formulas, i.e. an eective radiating equilibrium temperature (Te) and
a mean ‘no-atmosphere’ spherical surface temperature (Tna). is
doubled the πi instances in the regression analysis bringing the total
number of potential models for investigation to twelve.
Reference temperatures and reference pressure
A reference temperature (Tr) characterizes the average thermal
environment at the surface of a planetary body in the absence of
atmospheric greenhouse eect; hence, Tr is dierent for each body and
depends on solar irradiance and surface albedo. e purpose of Tr is
to provide a baseline for quantifying the thermal eect of planetary
atmospheres. Indeed, the Ts/Tr ratio produced by DA can physically be
interpreted as a Relative Atmospheric ermal Enhancement (RATE)
ideally expected to be equal to or greater than 1.0. Expressing the
thermal eect of a planetary atmosphere as a non-dimensional quotient
instead of an absolute temperature dierence (as done in the past)
allows for an unbiased comparison of the greenhouse eects of celestial
bodies orbiting at dierent distances from the Sun. is is because the
absolute strength of the greenhouse eect (measured in K) depends on
both solar insolation and atmospheric properties, while RATE being
a radiation-normalized quantity is expected to only be a function of a
planet’s atmospheric environment. To our knowledge, RATE has not
previously been employed to measure the thermal eect of planetary
atmospheres.
Two methods have been proposed thus far for estimating the
average surface temperature of a planetary body without the greenhouse
eect, both based on the SB radiation law. e rst and most popular
approach uses the planets global energy budget to calculate a single
radiating equilibrium temperature Te (also known as an eective
emission temperature) from the average absorbed solar ux [6,9,28],
i.e.
(3)
Here, S is the solar irradiance (W m-2) dened as the TOA
shortwave ux incident on a plane perpendicular to the incoming rays,
αp is the planetary Bond albedo (decimal fraction),
ε
is the planet’s
LW emissivity (typically 0.9 ≤ ε <1.0; in this study we assume ε = 0.98
based on lunar regolith measurements reported by Vasavada et al. [29],
and σ = 5.6704 × 10-8 W m-2 K-4 is the SB constant. e term S(1-αp )⁄4
represents a globally averaged shortwave ux absorbed by the planet-
atmosphere system. e rationale behind Eq. (3) is that the TOA energy
balance presumably denes a baseline temperature at a certain height
in the free atmosphere (around 5 km for Earth), which is related to the
planet’s mean surface temperature via the infrared optical depth of the
atmosphere [9,10]. Equation (3) was introduced to planetary science
in the early 1960s [30,31] and has been widely utilized ever since to
calculate the average surface temperatures of airless (or nearly airless)
bodies such as Mercury, Moon and Mars [32] as well as to quantify
the strength of the greenhouse eect of planetary atmospheres [2-
4,6,9,28]. However, Volokin and ReLlez [1] showed that, due to Hölder’s
inequality between integrals [33], Te is a non-physical temperature for
spheres and lacks a meaningful relationship to the planet’s Ts.
e second method attempts to estimate the average surface
temperature of a planet (Tna) in the complete absence of an atmosphere
using an explicit spatial integration of the SB law over a sphere. Instead
of calculating a single bulk temperature from the average absorbed
shortwave ux as done in Eq. (3), this alternative approach rst
computes the equilibrium temperature at every point on the surface of
an airless planet from the local absorbed shortwave ux using the SB
relation, and then spherically integrates the resulting temperature eld
to produce a global temperature mean. While algorithmically opposite
to Eq. (3), this method mimics well the procedure for calculating Earths
global temperature as an area-weighted average of surface observations.
Rubincam [34] proposed an analytic solution to the spherical
integration of the SB law (his Eq. 15) assuming no heat storage by the
regolith and zero thermal inertia of the ground. Volokin and ReLlez
[1] improved upon Rubincams formulation by deriving a closed-form
integral expression that explicitly accounts for the eect of subterranean
heat storage, cosmic microwave background radiation (CMBR) and
geothermal heating on the average global surface temperature of
airless bodies. e complete form of their analytic Spherical Airless-
Temperature (SAT) model reads:
Planetary Variable Symbol SI Units Physical Dimension
Global mean annual near-surface temperature (GMAT), the dependent variable TsK[Θ]
Stellar irradiance (average shortwave ux incident on a plane perpendicular to the stellar rays at the top of a planet’s
atmosphere) SW m-2 [M T-3]
Reference temperature (the planet’s mean surface temperature in the absence of an atmosphere or an atmospheric
greenhouse effect) TrK[Θ]
Average near-surface gas pressure representing either partial pressure of greenhouse gases or total atmospheric
pressure PxPa [M L-1 T-2]
Average near-surface gas density representing either greenhouse-gas density or total atmospheric density
x
ρ
kg m-3 [M L-3]
Reference pressure (the minimum atmospheric pressure required a liquid solvent to exists at the surface) PrPa [M L-1 T-2]
Table 1: Variables employed in the Dimensional Analysis aimed at deriving a general planetary temperature model. The variables are comprised of 4 fundamental physical
dimensions: mass [M], length [L], time [T] and absolute temperature [Θ].
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
Page 4 of 22
Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
Tna =
( ) ( )
( )
( ) ( )( )
( )
( )
( )( )
54 54
14
54 54
14
11
11
2
50 754 1
0 754 1
//
e e cg Cg
/
e e
//
e e c g C g
/
e e
S R R R R
S
. S RR RR
. S


−α + + +


+

−η −α εσ




−α + + +


− α εσ


η
η
η
(4a)
where αe is the eective shortwave albedo of the surface, ηe is the
eective ground heat storage coecient in a vacuum, Rc = σ 2.7254 =
3.13 × 10-6 W m-2 is the CMBR [35], and Rg is the spatially averaged
geothermal ux (W m-2) emanating from the subsurface. e heat
storage term ηe is dened as a fraction of the absorbed shortwave ux
conducted into the subsurface during daylight hour and subsequently
released as heat at night.
Since the eect of CMBR on Tna is negligible for S > 0.15 W m-2 [1]
and the geothermal contribution to surface temperatures is insignicant
for most planetary bodies, one can simplify Eq. (4a) by substituting Rc =
Rg = 0 is produces:
( ) ( )
( )
0.25
0.25 0.25
1
21 0.932
5
e
na e e
S
T
αηη
εσ


= −+


 (4b)
where 0.932 = 0.7540.25. e complete formula (4a) must only be used if
S ≤ 0.15 W m-2 and/or the magnitude of Rg is signicantly greater than
zero. For comparison, in the Solar System, the threshold S ≤ 0.15 W m-2
is encountered beyond 95 astronomical unis (AU) in the region of the
inner Oort cloud. Volokin and ReLlez [1] veried Equations (4a) and
(4b) against Moon temperature data provided by the NASA Diviner
Lunar Radiometer Experiment [29,36]. ese authors also showed that
accounting for the subterranean heat storage (ηe) markedly improves
the physical realism and accuracy of the SAT model compared to the
original formulation by Rubincam [34].
e conceptual dierence between Equations (3) and (4b) is that Τe
represents the equilibrium temperature of a blackbody disk orthogonally
illuminated by shortwave radiation with an intensity equal to the average
solar ux absorbed by a sphere having a Bond albedo αp, while Τna is the
area-weighted average temperature of a thermally heterogeneous airless
sphere [1,37]. In other words, for spherical objects, Τe is an abstract
mathematical temperature, while Tna is the average kinetic temperature
of an airless surface. Due to Hölders inequality between integrals, one
always nds Τe >> Τna when using equivalent values of stellar irradiance
and surface albedo in Equations (3) and (4b) [1].
To calculate the Tna temperatures for planetary bodies with tangible
atmospheres, we assumed that the airless equivalents of such objects
would be covered with a regolith of similar optical and thermo-physical
properties as the Moon surface. is is based on the premise that, in
the absence of a protective atmosphere, the open cosmic environment
would erode and pulverize exposed surfaces of rocky planets over time
in a similar manner [1]. Also, properties of the Moon surface are the
best studied ones among all airless bodies in the Solar System. Hence,
one could further simplify Eq. (4b) by combining the albedo, the heat
storage fraction and the emissivity parameter into a single constant
using applicable values for the Moon, i.e. αe = 0.132, ηe = 0.00971 and ε
= 0.98 [1,29]. is produces:
0.25
32.44
na
TS=
(4c)
Equation (4c) was employed to estimate the ‘no-atmosphere’ reference
temperatures of all planetary bodies participating in our analysis and
discussed below.
For a reference pressure, we used the gas-liquid-solid triple point of
water, i.e. Pr = 611.73 Pa [38] dening a baric threshold, below which water
can only exists in a solid/vapor phase and not in a liquid form. e results
of our analysis are not sensitive to the particular choice of a reference-
pressure value; hence, the selection of Pr is a matter of convention.
Regression analysis
Finding the best function to describe the observed variation of
GMAT among celestial bodies requires that the πi variables generated
by DA be subjected to regression analyses. As explained in Appendix A,
twelve pairs of πi variables hereto called Models were investigated. In
order to ease the curve tting and simplify the visualization of results,
we utilized natural logarithms of the constructed πi variables rather than
their absolute values, i.e. we modeled the relationship ln (π1) = f (ln(π2))
instead of π1 = f(π2). In doing so we focused on monotonic functions
of conservative shapes such as exponential, sigmoidal, hyperbolic,
and logarithmic, for their tting coecients might be interpretable in
physically meaningful terms. A key advantage of this type of functions
(provided the existence of a good t, of course) is that they also tend
to yield reliable results outside the data range used to determine their
coecients. We specically avoided non-monotonic functions such as
polynomials because of their ability to accurately t almost any dataset
given a suciently large number of regression coecients while at the
same time showing poor predictive skills beyond the calibration data
range. Due to their highly exible shape, polynomials can easily t
random noise in a dataset, an outcome we particularly tried to avoid.
e following four-parameter exponential-growth function was
found to best meet our criteria:
( ) ( )
exp exp y a bx c d x= +
(5)
where x = ln (π2) and y = ln (π1) are the independent and dependent
variable respectively while a, b, c and d are regression coecients. is
function has a rigid shape that can only describe specic exponential
patterns found in our data. Equation (5) was tted to each one of the
12 planetary data sets of logarithmic πi pairs suggested by DA using the
standard method of least squares. e skills of the resulting regression
models were evaluated via three statistical criteria: coecient of
determination (R2), adjusted R2, and standard error of the estimate (σest)
[39,40]. All calculations were performed with SigmaPlotTM 13 graphing
and analysis soware.
Planetary data
To ensure proper application of the DA methodology we compiled a
dataset of diverse planetary environments in the Solar System using the
best information available. Celestial bodies were selected for the analysis
based on three criteria: (a) presence of a solid surface; (b) availability
of reliable data on near-surface temperature, atmospheric composition,
and total air pressure/density preferably from direct observations; and
(c) representation of a broad range of physical environments dened
in terms of TOA solar irradiance and atmospheric properties. is
resulted in the selection of three planets: Venus, Earth, and Mars; and
three natural satellites: Moon of Earth, Titan of Saturn, and Triton of
Neptune.
Each celestial body was described by nine parameters shown in
Table 2 with data sources listed in Table 3. In an eort to minimize
the eect of unforced (internal) climate variability on the derivation
of our temperature model, we tried to assemble a dataset of means
representing an observational period of 30 years, i.e. from 1981 to 2010.
us, Voyager measurements of Titan from the early 1980s suggested
an average surface temperature of 94 ± 0.7 K [41]. Subsequent
observations by the Cassini mission between 2005 and 2010 indicated
a mean global temperature of 93.4 ± 0.6 K for that moon [42,43]. Since
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
Page 5 of 22
Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
Saturns orbital period equals 29.45 Earth years, we averaged the above
global temperature values to arrive at 93.7 ± 0.6 K as an estimate of
Titans 30-year GMAT. Similarly, data gathered in the late 1970s by the
Viking Landers on Mars were combined with more recent Curiosity-
Rover surface measurements and 1999-2005 remote observations by
the Mars Global Surveyor (MGS) spacecra to derive representative
estimates of GMAT and atmospheric surface pressure for the Red
Planet. Some parameter values reported in the literature did not meet
our criteria for global representativeness and/or physical plausibility
and were recalculated using available observations as described below.
e mean solar irradiances of all bodies were calculated as S = SE rau
-2
where rau is the body’s average distance (semi-major axis) to the Sun
(AU) and SE = 1,360.9 W m-2 is the Earths new lower irradiance at 1 AU
according to recent satellite observations reported by Kopp and Lean
[49]. Due to a design aw in earlier spectrometers, the solar irradiance
at Earth’s distance has been overestimated by ≈ 5 W m-2 prior to 2003
[49]. Consequently, our calculations yielded slightly lower irradiances
for bodies such as Venus and Mars compared to previously published
data. Our decision to recalculate S was based on the assumption that the
orbital distances of planets are known with much greater accuracy than
TOA solar irradiances. Hence, a correction made to Earth’s irradiance
requires adjusting the ‘solar constants’ of all other planets as well.
We found that quoted values for the mean global temperature and
surface atmospheric pressure of Mars were either improbable or too
uncertain to be useful for our analysis. us, studies published in the
last 15 years report Mars’ GMAT being anywhere between 200 K and
240 K with the most frequently quoted values in the range 210–220
K [6,32,76-81]. However, in-situ measurements by Viking Lander 1
suggest that the average surface air temperature at a low-elevation site
in the Martian subtropics does not exceed 207 K during the summer-
fall season (Appendix B). erefore, the Red Planet’s GMAT must be
lower than 207 K. e Viking records also indicate that average diurnal
temperatures above 210 K can only occur on Mars during summertime.
Hence, all such values must be signicantly higher than the actual mean
annual temperature at any Martian latitude. is is also supported by
results from a 3-D global circulation model of the Red Planet obtained
by Fenton et al. [82]. e surface atmospheric pressure on Mars varies
appreciably with season and location. Its global average value has
previously been reported between 600 Pa and 700 Pa [6,32,78,80,83,84],
a range that was too broad for the target precision of our study. Hence
our decision to calculate new annual global means of near-surface
temperature and air pressure for Mars via a thorough analysis of available
data from remote-sensing and in-situ observations. Appendix B details
our computational procedure with the results presented in Table 2. It is
noteworthy that our independent estimate of Mars’ GMAT (190.56 ±
0.7 K), while signicantly lower than values quoted in recent years, is in
perfect agreement with spherically integrated brightness temperatures
of the Red Planet derived from remote microwave measurements in the
late 1960s and early 1970s [85-87].
Moons GMAT was also not readily extractable from the published
literature. Although lunar temperatures have been measured for
more than 50 years both remotely and in situ [36] most studies focus
on observed temperature extremes across the lunar surface [56] and
rarely discuss the Moons average global temperature. Current GMAT
estimates for the Moon cluster around two narrow ranges: 250–255
K and 269–271 K [32]. A careful examination of the published data
reveals that the 250–255 K range is based on subterranean heat-ow
measurements conducted at depths between 80 and 140 cm at the
Apollo 15 and 17 landing sites located at 26oN; 3.6oE and 20oN; 30.6oE,
respectively [88]. Due to a strong temperature dependence of the lunar
regolith thermal conductivity in the topmost 1-2 cm soil, the Moons
average diurnal temperature increases steadily with depth. According
to Apollo measurements, the mean daily temperature at 35 cm
belowground is 40–45 K higher than that at the lunar surface [88]. e
diurnal temperature uctuations completely vanish below a depth of 80
cm. At 100 cm depth, the temperature of the lunar regolith ranged from
250.7 K to 252.5 K at the Apollo 15 site and between 254.5 K and 255.5 K
at the Apollo 17 site [88]. Hence, reported Moon average temperatures
in the range 250-255 K do not describe surface conditions. Moreover,
since measured in the lunar subtropics, such temperatures do not likely
even represent Moons global thermal environment at these depths. On
the other hand, frequently quoted Moon global temperatures of ~270 K
have actually been calculated from Eq. (3) and are not based on surface
measurements. However, as demonstrated by Volokin and ReLlez [1],
Parameter Venus Earth Moon Mars Titan Triton
Average distance to the Sun, rau (AU) 0.7233 1.0 1.0 1.5237 9.582 30.07
Average TOA solar irradiance, S (W m-2)2,601.3 1,360.9 1,360.9 586.2 14.8 1.5
Bond albedo, αp (decimal fraction) 0.900 0.294 0.136 0.235 0.265 0.650
Average absorbed shortwave radiation, Sa = S(1-αp)/4 (W m-2)65.0 240.2 294.0 112.1 2.72 0.13
Global average surface atmospheric pressure, P (Pa) 9,300,000.0 ±
100,000 98,550.0 ± 6.5 2.96 × 10-10 ±
10-10 685.4 ± 14.2 146,700.0 ± 100 4.0 ± 1.2
Global average surface atmospheric density, ρ (kg m-3) 65.868 ± 0.44 1.193 ± 0.002 2.81 × 10-15 ±
9.4 × 10-15
0.019 ± 3.2 ×
10-4 5.161 ± 0.03 3.45 × 10-4 ± 9.2
× 10-5
Chemical composition of the lower atmosphere (% of volume)
96.5 CO2
3.48 N2
0.02 SO2
77.89 N2
20.89 O2
0.932 Ar
0.248 H2O
0.040 CO2
26.7 4He
26.7 20Ne
23.3 H2
20.0 40Ar
3.3 22Ne
95.32 CO2
2.70 N2
1.60 Ar
0.13 O2
0.08 CO
0.021 H2O
95.1 N2
4.9 CH4
99.91 N2
0.060 CO
0.024 CH4
Molar mass of the lower atmosphere, M (kg mol-1)0.0434 0.0289 0.0156 0.0434 0.0274 0.0280
GMAT, Ts (K) 737.0 ± 3.0 287.4 ± 0.5 197.35 ± 0.9 190.56 ± 0.7 93.7 ± 0.6 39.0 ± 1.0
Table 2: Planetary data set used in the Dimensional Analysis compiled from sources listed in Table 3. The estimation of Mars’ GMAT and the average surface atmospheric
pressure are discussed in Appendix B. See text for details about the computational methods employed for some parameters.
Planetary Body Information Sources
Venus [32,44-48]
Earth [12,13,32,49-55]
Moon [1,29,32,48,56-59]
Mars [32,48,60-63], Appendix B
Titan [32,41-43,64-72]
Triton [48,73-75]
Table 3: Literature sources of the planetary data presented in Table 2.
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
Eq. (3) overestimates the mean global surface temperature of spheres
by about 37%. In this study, we employed the spherical estimate of
Moons GMAT (197.35 K) obtained by Volokin and ReLlez [1] using
output from a NASA thermo-physical model validated against Diviner
observations [29].
Surprisingly, many publications report incorrect values even
for Earth’s mean global temperature. Studies of terrestrial climate
typically focus on temperature anomalies and if Earth’s GMAT is
ever mentioned, it is oen loosely quoted as 15 C (~288 K) [2-4,6].
However, observations archived in the HadCRUT4 dataset of the
UK Met Oce’s Hadley Centre [50,89] and in the Global Historical
Climatology Network [51,52,90,91] indicate that, between 1981 and
2010, Earth’s mean annual surface air temperature was 287.4 K (14.3
C) ± 0.5 K. Some recent studies acknowledge this more accurate lower
value of Earth’s absolute global temperature [92]. For Earth’s mean
surface atmospheric pressure we adopted the estimate by Trenberth et
al. [53] (98.55 kPa), which takes into account the average elevation of
continental landmasses above sea level; hence, it is slightly lower than
the typical sea-level pressure of ≈ 101.3 kPa.
e average near-surface atmospheric densities (ρ, kg m-3) of
planetary bodies were calculated from reported means of total
atmospheric pressure (P), molar mass (M, kg mol-1) and temperature
(Ts) using the Ideal Gas Law, i.e.
( )
6
s
PM
RT
ρ
= (6)
where R = 8.31446 J mol-1 K-1 is the universal gas constant. is
calculation was intended to make atmospheric densities physically
consistent with independent data on pressure and temperature utilized
in our study. e resulting ρ values were similar to previously published
data for individual bodies. Standard errors of the air-density estimates
were calculated from reported errors of P and Τs for each body using
Eq. (6).
Data in Table 2 were harnessed to compute several intermediate
variables and all dimensionless πi products necessary for the regression
analyses. e results from these computations are shown in Table 4.
Greenhouse gases in planetary atmospheres represented by the major
constituents carbon dioxide (CO2), methane (CH4) and water vapor
(H2O) were collectively quantied via three bulk parameters: average
molar mass (Mgh, kg mol-1), combined partial pressure (Pgh, Pa) and
combined partial density (ρgh, kg m-3). ese parameters were estimated
from reported volumetric concentrations of individual greenhouse
gases (Cx, %) and data on total atmospheric pressure and density in
Table 2 using the formulas:
( )
CO2 CH 4 H2O
0.044 0.016 0.018 /
gh gh
M C C CC= ++
(7)
( )
0.01
gh gh
PP C=
(8)
( )( )
0.01 /
gh gh gh
CMM
ρρ
=
(9)
where Cgh = CCO2 + CCH4 + CH2O is the total volumetric concentration
of major greenhouse gases (%). e reference temperatures Τe and Τna
were calculated from Equations (3) and (4c), respectively.
Results
Function (5) was tted to each one of the 12 sets of logarithmic πi
pairs generated by Equations (1) and (2) and shown in Table 4. Figures
1 and 2 display the resulting curves of individual regression models
with planetary data plotted in the background for reference. Table 5 lists
the statistical scores of each non-linear regression. Model 12 depicted
in Figure 2f had the highest R2 = 0.9999 and the lowest standard error
σest = 0.0078 among all regressions. Model 1 (Figure 1a) provided the
second best t with R2 = 0.9844 and σest = 0.1529. Notably, Model 1
shows almost a 20-time larger standard error on the logarithmic scale
than Model 12. Figure 3 illustrates the dierence in predictive skills
between the two top-performing Models 1 and 12 upon conversion
of vertical axes to a linear scale. Taking an antilogarithm weakens
the relationship of Model 1 to the point of becoming immaterial and
highlights the superiority of Model 12. e statistical results shown in
Table 5 indicate that the explanatory power and descriptive accuracy of
Model 12 surpass those of all other models by a wide margin.
Since Titan and Earth nearly overlap on the logarithmic scale of Figure
2f, we decided to experiment with an alternative regression for Model 12,
Intermediate Variable or Dimensionless Product Venus Earth Moon Mars Titan Triton
Average molar mass of greenhouse gases, Mgh (kg mol-1)
(Eq. 7) 0.0440 0.0216 0.0 0.0440 0.0160 0.0160
Near-surface partial pressure of greenhouse gases, Pgh (Pa)
(Eq. 8)
8,974,500.0 ±
96,500 283.8 ± 0.02 0.0 667.7 ± 13.8 7,188.3 ± 4.9 9.6 × 10-4 ± 2.9
× 10-4
Near-surface density of greenhouse gases, ρgh (kg m-3) (Eq. 9) 64.441 ± 0.429 2.57 × 10-3 ± 4.3
× 10-6 0.0 0.018 ± 3.1 ×
10-4
0.148 ± 8.4 ×
10-4
4.74 × 10-8 ± 1.3
× 10-8
Radiating equilibrium temperature, Te (K) (Eq. 3) 185.0 256.4 269.7 211.9 83.6 39.2
Average airless spherical temperature, Tna (K) (Eq. 4c) 231.7 197.0 197.0 159.6 63.6 35.9
Ts/ Te3.985 ± 0.016 1.121 ± 0.002 0.732 ± 0.003 0.899 ± 0.003 1.120 ± 0.008 0.994 ± 0.026
Ts/Tna 3.181 ± 0.013 1.459 ± 0.002 1.002 ± 0.004 1.194 ± 0.004 1.473 ± 0.011 1.086 ± 0.028
ln(Ts/Te) 1.3825 ± 0.0041 0.1141 ± 0.0017 -0.3123 ± 0.0046 -0.1063 ± 0.0037 0.1136 ± 0.0075 -5.2×10-3 ±
0.0256
ln(Ts/Tna) 1.1573 ± 0.0041 0.3775 ± 0.0017 1.59×10-3 ±
0.0046 0.1772 ± 0.0037 0.3870 ± 0.0075 0.0828 ± 0.0256
ln[Pgh
3/(ρgh S2)] 28.1364 8.4784 Undened 10.7520 23.1644 -4.7981
ln[P3/(ρgh S2)] 28.2433 26.0283 +∞ 10.8304 32.2122 20.2065
ln[Pgh
3/(ρ S2)] 28.1145 2.3370 Undened 10.7396 19.6102 -13.6926
ln[Pgh/Pr] 9.5936 -0.7679 Undened 0.0876 2.4639 -13.3649
ln[P3/(ρ S2)] 28.2214 19.8869 -46.7497 10.8180 28.6580 11.3120
ln(P/Pr)9.6292 ± 0.0108 5.0820 ±
6.6×10-5
-28.3570 ±
0.3516 0.1137 ± 0.0207 5.4799 ±
6.8×10-4
-5.0300 ±
0.3095
Table 4: Intermediate variables and dimensionless products required for the regression analyses and calculated from data in Table 2. Equations used to compute
intermediate variables are shown in parentheses. The reference pressure is set to the barometric triple point of water, i.e. Pr = 611.73 Pa.
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
which excludes Titan from the input dataset. is new curve had R2 =
1.0 and σest = 0.0009. Although the two regression equations yield similar
results over most of the relevant pressure range, we chose the one without
Titan as nal for Model 12 based on the assumption that Earths GMAT
is likely known with a much greater accuracy than Titan’s mean annual
temperature. Taking an antilogarithm of the nal regression equation,
which excludes Titan, yielded the following expression for Model 12:
0.150263 1.04193
5
na
+ 1. exp 0.17 84205 31 0 21 1
s
rr
TP P
TP P

 
=
 

 

×
(10a)
e regression coecients in Eq. (10a) are intentionally shown in
full precision to allow an accurate calculation of RATE (i.e. the Ts/
Tna ratios) provided the strong non-linearity of the relationship and
to facilitate a successful replication of our results by other researchers.
Figure 4 depicts Eq. (10a) as a dependence of RATE on the average
surface air pressure. Superimposed on this graph are the six planetary
bodies from Table 4 along with their uncertainty ranges.
Equation (10a) implies that GMATs of rocky planets can be
calculated as a product of two quantities: the planet’s average surface
temperature in the absence of an atmosphere (Tna, K) and a non-
dimensional factor (Ea ≥ 1.0) quantifying the relative thermal eect of
the atmosphere, i.e.
( )
10b
s na a
T TE=
(10b)
where Τna is obtained from the SAT model (Eq. 4a) and Ea is a function
of total pressure (P) given by:
( ) ( )
0.150263 1.04193
5
exp 0.174205 exp 1.83121 10 11
a
rr
PP
EP PP
 
 
 
= ×
 
 
 
 
(11)
Note that, as P approaches 0 in Eq. (11), Ea approaches the physically
realistic limit of 1.0. Other physical aspects of this equation are
discussed below.
For bodies with tangible atmospheres (such as Venus, Earth,
Figure 1: The relative atmospheric thermal enhancement (Ts/Tr) as a function of various dimensionless forcing variables generated by DA using data on solar
irradiance, near-surface partial pressure/density of greenhouse gases, and total atmospheric pressure/density from Table 4. Panels a through f depict six regression
models suggested by DA with the underlying celestial bodies plotted in the background for reference. Each pair of horizontal graphs represents different reference
temperatures (Tr) dened as either Tr = Te (left) or Tr = Tna (right).
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
Mars, Titan and Triton), one must calculate Tna using αe = 0.132 and
ηe = 0.00971, which assumes a Moon-like airless reference surface in
accordance with our pre-analysis premise. For bodies with tenuous
atmospheres (such as Mercury, the Moon, Calisto and Europa), Tna
should be calculated from Eq. (4a) (or Eq. 4b respectively if S > 0.15
W m-2 and/or Rg ≈ 0 W m-2) using the body’s observed values of Bond
albedo αe and ground heat storage fraction ηe. In the context of this
model, a tangible atmosphere is dened as one that has signicantly
modied the optical and thermo-physical properties of a planets
surface compared to an airless environment and/or noticeably
impacted the overall planetary albedo by enabling the formation of
clouds and haze. A tenuous atmosphere, on the other hand, is one that
has not had a measurable inuence on the surface albedo and regolith
thermo-physical properties and is completely transparent to shortwave
radiation. e need for such delineation of atmospheric masses when
calculating Tna arises from the fact that Eq. (10a) accurately describes
RATEs of planetary bodies with tangible atmospheres over a wide
range of conditions without explicitly accounting for the observed large
dierences in albedos (i.e. from 0.235 to 0.90) while assuming constant
values of αe and ηe for the airless equivalent of these bodies. One possible
explanation for this counterintuitive empirical result is that atmospheric
pressure alters the planetary albedo and heat storage properties of the
surface in a way that transforms these parameters from independent
controllers of the global temperature in airless bodies to intrinsic
byproducts of the climate system itself in worlds with appreciable
atmospheres. In other words, once atmospheric pressure rises above a
certain level, the eects of albedo and ground heat storage on GMAT
become implicitly accounted for by Eq. (11). Although this hypothesis
requires a further investigation beyond the scope of the present study,
one nds an initial support for it in the observation that, according to
data in Table 2, GMATs of bodies with tangible atmospheres do not
show a physically meaningful relationship with the amounts of absorbed
shortwave radiation determined by albedos. Our discovery for the
need to utilize dierent albedos and heat storage coecients between
airless worlds and worlds with tangible atmospheres is not unique as a
methodological approach. In many areas of science and engineering,
it is sometime necessary to use disparate model parameterizations to
successfully describe dierent aspects of the same phenomenon. An
example is the distinction made in uid mechanics between laminar
and turbulent ow, where the non-dimensional Reynolds number is
employed to separate the two regimes that are subjected to dierent
mathematical treatments.
Figure 2: The same as in Figure 1 but for six additional regression models (panels a through f).
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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We do not currently have sucient data to precisely dene the limit
between tangible and tenuous atmospheres in terms of total pressure for
the purpose of this model. However, considering that an atmospheric
pressure of 1.0 Pa on Pluto causes the formation of layered haze [93],
we surmise that this limit likely lies signicantly below 1.0 Pa. In this
study, we use 0.01 Pa as a tentative threshold value. us, in the context
of Eq. (10b), we recommend computing Tna from Eq. (4c) if P > 10-2 Pa,
and from Eq. (4a) (or Eq. 4b, respectively) using observed values of αe
and ηe if P ≤ 10-2 Pa. Equation (4a) should also be employed in cases,
where a signicant geothermal ux exists such as on the Galilean moons
of Jupiter due to tidal heating, and/or if S ≤ 0.15 W m-2. Hence, the
30-year mean global equilibrium surface temperature of rocky planets
depends in general on ve factors: TOA stellar irradiance (S), a reference
airless surface albedo (αe), a reference airless ground heat storage fraction
No. Functional Model Coefcient of Determination (R2) Adjusted R2Standard Error σest
1
3
2
e
gh
s
gh
P
Tf
TS
ρ

=


0.9844 0.9375 0.1529
2
3
2
na
gh
s
gh
P
Tf
TS
ρ

=


0.9562 0.8249 0.1773
3
3
2
e
s
gh
TP
f
TS
ρ

=


0.1372 -2.4511 1.1360
4
3
2
na
s
gh
TP
f
TS
ρ

=


0.2450 -2.0200 0.7365
5
3
2
e
gh
s
P
Tf
TS
ρ

=


0.9835 0.9339 0.1572
6
3
2
na
gh
s
P
Tf
TS
ρ

=


0.9467 0.7866 0.1957
7
e
gh
s
r
P
Tf
TP

=


0.9818 0.9274 0.1648
8
na
gh
s
r
P
Tf
TP

=

0.9649 0.8598 0.1587
9
3
2
e
s
TP
f
TS
ρ

=


0.4488 -0.3780 0.7060
10
3
2
na
s
TP
f
TS
ρ

=

0.6256 0.0639 0.4049
11
e
s
r
TP
f
TP

=

0.9396 0.8489 0.2338
12
na
s
r
TP
f
TP

=


0.9999 0.9997 0.0078
Table 5: Performance statistics of the twelve regression models suggested by DA. Statistical scores refer to the model logarithmic forms shown in Figures 1 and 2.
Figure 3: Comparison of the two best-performing regression models according to statistical scores listed in Table 5. Vertical axes use linear scales to better illustrate
the difference in skills between the models.
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
(ηe), the average geothermal ux reaching the surface (Rg), and the total
surface atmospheric pressure (P). For planets with tangible atmospheres
(P > 10-2 Pa) and a negligible geothermal heating of the surface (Rg ≈ 0),
the equilibrium GMAT becomes only a function of two factors: S and
P, i.e. Τs = 32.44 S0.25Eα(P). e nal model (Eq. 10b) can also be cast
in terms of Ts as a function of a planets distance to the Sun (rau, AU) by
replacing S in Equations (4a), (4b) or (4c) with 1360.9 rau
-2.
Environmental scope and numerical accuracy of the new
model
Figure 5 portrays the residuals between modeled and observed
absolute planetary temperatures. For celestial bodies participating in
the regression analysis (i.e. Venus, Earth, Moon, Mars and Triton), the
maximum model error does not exceed 0.17 K and is well within the
uncertainty of observations. e error for Titan, an independent data
point, is 1.45 K or 1.5% of that moon’s current best-known GMAT (93.7
K). Equation (10b) produces 95.18 K for Titan at Saturns semi-major
axis (9.582 AU) corresponding to a solar irradiance S = 14.8 W m-2. is
estimate is virtually identical to the 95 K average surface temperature
reported for that moon by the NASA JPL Voyager Mission website
[94]. e Voyager spacecra 1 and 2 reached Saturn and its moons in
November 1980 and August 1981, respectively, when the gas giant was
at a distance between 9.52 AU and 9.60 AU from the Sun corresponding
approximately to Saturns semi-major axis [95].
Data acquired by Voyager 1 suggested an average surface
temperature of 94 ± 0.7 K for Titan, while Voyager 2 indicated a
temperature close to 95 K [41]. Measurements obtained between 2005
and 2010 by the Cassini-Huygens mission revealed Ts ≈ 93.4 ± 0.6 K
[42,43]. Using Saturn’s perihelion (9.023 AU) and aphelion (10.05 AU)
one can compute Titans TOA solar irradiance at the closest and furthest
approach to the Sun, i.e. 16.7 W m-2 and 13.47 W m-2, respectively.
Inserting these values into Eq. (10b) produces the expected upper and
lower limit of Titan’s mean global surface temperature according to
our model, i.e. 92.9 K ≤ Ts ≤ 98.1 K. Notably this range encompasses
all current observation-based estimates of Titan’s GMAT. Since both
Voyager and Cassini mission covered shorter periods than a single
Titan season (Saturns orbital period is 29.45 Earth years), the available
measurements may not well represent that moon’s annual thermal
cycle. In addition, due to a thermal inertia, Titan’s average surface
temperature likely lags variations in the TOA solar irradiance caused
by Saturns orbital eccentricity. us, the observed 1.45 K discrepancy
between our independent model prediction and Titan’s current
best-known GMAT seems to be within the range of plausible global
temperature uctuations on that moon. Hence, further observations are
needed to more precisely constrain Titans long-term GMAT.
Measurements conducted by the Voyager spacecra in 1989
indicated a global mean temperature of 38 ± 1.0 K and an average
atmospheric pressure of 1.4 Pa at the surface of Triton [73]. Even
though Eq. (10a) is based on slightly dierent data for Triton (i.e. Ts =
39 ±1.0 K and P = 4.0 Pa) obtained by more recent stellar occultation
measurements [73], employing the Voyager-reported pressure in Eq.
(10b) produces Ts = 38.5 K for Triton’s GMAT, a value well within the
uncertainty of the 1989 temperature measurements.
e above comparisons indicate that Eq. (10b) rather accurately
describes the observed variation of the mean surface temperature across
a wide range of planetary environments in terms of solar irradiance
(from 1.5 W m-2 to 2,602 W m-2), total atmospheric pressure (from
near vacuum to 9,300 kPa) and greenhouse-gas concentrations (from
0.0% to over 96% per volume). While true that Eq. (10a) is based on
data from only 6 celestial objects, one should keep in mind that these
constitute virtually all bodies in the Solar System meeting our criteria
for availability and quality of measured data. Although function (5)
has 4 free parameters estimated from just 5-6 data points, there are no
signs of model overtting in this case because (a) Eq. (5) represents
a monotonic function of a rigid shape that can only describe well
certain exponential pattern as evident from Figures 1 and 2 and the
statistical scores in Table 5; (b) a simple scatter plot of ln (P/Pr) vs. ln(Ts/
Tna) visibly reveals the presence of an exponential relationship free of
data noise; and (c) no polynomial can t the data points in Figure 2f
as accurately as Eq. (5) while also producing a physically meaningful
response curve similar to known pressure-temperature relationships in
other systems. ese facts indicate that Eq. (5) is not too complicated
to cause an over-tting but just right for describing the data at hand.
e fact that only one of the investigated twelve non-linear
regressions yielded a tight relationship suggests that Model 12 describes
Figure 4: The relative atmospheric thermal enhancement (Ts/Tna ratio) as a
function of the average surface air pressure according to Eq. (10a) derived from
data representing a broad range of planetary environments in the solar system.
Saturn’s moon Titan has been excluded from the regression analysis leading
to Eq. (10a). Error bars of some bodies are not clearly visible due to their small
size relative to the scale of the axes. See Table 2 for the actual error estimates.
Figure 5: Absolute differences between modeled average global temperatures
by Eq. (10b) and observed GMATs (Table 2) for the studied celestial bodies.
Saturn’s moon Titan represents an independent data point, since it was excluded
from the regression analysis leading to Eq. (10a).
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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a macro-level thermodynamic property of planetary atmospheres
heretofore unbeknown to science. A function of such predictive power
spanning the entire breadth of the Solar System cannot be just a result
of chance. Indeed, complex natural systems consisting of myriad
interacting agents have been known to sometime exhibit emergent
responses at higher levels of hierarchical organization that are amenable
to accurate modeling using top-down statistical approaches [96].
Equation (10a) also displays several other characteristics discussed
below that lend further support to the above notion.
Model robustness
Model robustness denes the degree to which a statistical
relationship would hold when recalculated using a dierent dataset. To
test the robustness of Eq. (10a) we performed an alternative regression
analysis, which excluded Earth and Titan from the input data and
only utilized logarithmic pairs of Ts/Tna and P/Pr for Venus, the Moon,
Mars and Triton from Table 4. e goal was to evaluate how well the
resulting new regression equation would predict the observed mean
surface temperatures of Earth and Titan. Since these two bodies occupy
a highly non-linear region in Model 12 (Figure 2f), eliminating them
from the regression analysis would leave a key portion of the curve
poorly dened. As in all previous cases, function (5) was tted to the
incomplete dataset (omitting Earth and Titan), which yielded the
following expression:
( )
0.150275 3.32375
15
na
exp 0.174222 5.25043 10
12a
s
rr
T
PP
TP P

 

= +×
 

 

(12a)
Substituting the reference temperature Tna in Eq. (12a) with its
equivalent from Eq. (4c) and solving for Ts produces
( )
0.150275 3.32375
0.25 15
32.44 exp 0.174222 exp 5.25043 10 12b
s
rr
PP
TS PP
 
 
 
= ×
 
 
 
 
(12b)
It is evident that the regression coecients in the rst exponent term of
Eq. (12a) are nearly identical to those in Eq. (10a). is term dominates
the Ts-P relationship over the pressure range 0-400 kPa accounting
for more than 97.5% of the predicted temperature magnitudes. e
regression coecients of the second exponent dier somewhat between
the two formulas causing a divergence of calculated RATE values
over the pressure interval 400–9,100 kPa. e models converge again
between 9,000 kPa and 9,300 kPa. Figure 6 illustrates the similarity of
responses between Equations (10a) and (12a) over the pressure range
0–300 kPa with Earth and Titan plotted in the foreground for reference.
Equation (12b) reproduces the observed global surface temperature
of Earth with an error of 0.4% (-1.0 K) and that of Titan with an error
of 1.0% (+0.9 K). For Titan, the error of the new Eq. (12b) is even
slightly smaller than that of the original model (Eq. 10b). e ability
of Model 12 to predict Earths GMAT with an accuracy of 99.6% using
a relationship inferred from disparate environments such as those
found on Venus, Moon, Mars and Triton indicates that (a) this model
is statistically robust, and (b) Earths temperature is a part of a cosmic
thermodynamic continuum well described by Eq. (10b). e apparent
smoothness of this continuum for bodies with tangible atmospheres
(illustrated in Figure 4) suggests that planetary climates are well-
buered and have no ‘tipping points’ in reality, i.e. states enabling
rapid and irreversible changes in the global equilibrium temperature
as a result of destabilizing positive feedbacks assumed to operate within
climate systems. is robustness test also serves as a cross-validation
suggesting that the new model has a universal nature and it is not a
product of overtting.
e above characteristics of Eq. (10a) including dimensional
homogeneity, high predictive accuracy, broad environmental scope of
validity and statistical robustness indicate that it represents an emergent
macro-physical model of theoretical signicance deserving further
investigation. is conclusive result is also supported by the physical
meaningfulness of the response curve described by Eq. (10a).
Discussion
Given the high statistical scores of the new model discussed above,
it is important to address its physical signicance, potential limitations,
and broad implications for the current climate theory.
Similarity of the new model to Poisson’s formula and the SB
radiation law
e functional response of Eq. (10a) portrayed in Figure 4 closely
resembles the shape of the dry adiabatic temperature curve in Figure
7a described by the Poisson formula and derived from the First Law of
ermodynamics and the Ideal Gas Law [4], i.e.
( )
/
13
p
Rc
oo
Tp
Tp

=

(13)
Here, To and po are reference values for temperature and pressure
typically measured at the surface, while T and p are corresponding scalars
in the free atmosphere, and cp is the molar heat capacity of air (J mol-1
K-1). For the Earths atmosphere, R/cp = 0.286. Equation (13) essentially
describes the direct eect of pressure p on the gas temperature (T) in
the absence of any heat exchange with the surrounding environment.
Equation (10a) is structurally similar to Eq. (13) in a sense that
both expressions relate a temperature ratio to a pressure ratio, or more
precisely, a relative thermal enhancement to a ratio of physical forces.
However, while the Poisson formula typically produces 0 ≤ T/To ≤ 1.0,
Eq. (10a) always yields Ts/Tna ≥ 1.0. e key dierence between the two
models stems from the fact that Eq. (13) describes vertical temperature
changes in a free and dry atmosphere induced by a gravity-controlled
pressure gradient, while Eq. (10a) predicts the equilibrium response of a
planet’s global surface air temperature to variations in total atmospheric
Figure 6: Demonstration of the robustness of Model 12. The solid black curve
depicts Eq. (10a) based on data from 5 celestial bodies (i.e. Venus, Earth, Moon,
Mars and Triton). The dashed grey curve portrays Eq. (12a) derived from data of
only 4 bodies (i.e. Venus, Moon, Mars and Triton) while excluding Earth and Titan
from the regression analysis. The alternative Eq. (12b) predicts the observed
GMATs of Earth and Titan with accuracy greater than 99% indicating that Model
12 is statistically robust.
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
pressure. In essence, Eq. (10b) could be viewed as a predictor of the
reference temperature To in the Poisson formula. us, while qualitatively
similar, Equations (10a) and (13) are quantitatively rather dierent. Both
functions describe eects of pressure on temperature but in the context of
disparate physical systems. erefore, estimates obtained from Eq. (10a)
should not be confused with results inferred from the Poisson formula.
For example, Eq. (10b) cannot be expected to predict the temperature
lapse rate and/or vertical temperature proles within a planetary
atmosphere as could be using Eq. (13). Furthermore, Eq. (10a) represents
a top-down empirical model that implicitly accounts for a plethora of
thermodynamic and radiative processes and feedbacks operating in real
climate systems, while the Poisson formula (derived from the Ideal Gas
Law) only describes pressure-induced temperature changes in a simple
mixture of dry gases without any implicit or explicit consideration of
planetary-scale mechanisms such as latent heat transport and cloud
radiative forcing.
Equation (10a) also shows a remarkable similarity to the SB law
relating the equilibrium skin temperature of an isothermal blackbody
(Tb, K) to the electromagnetic radiative ux (I, W m-2) absorbed/
emitted by the body’s surface, i.e. Tb = (I ⁄ σ)0.25. Dividing each side of
this fundamental relationship by the irreducible temperature of deep
Space Tc = 2.725 K and its causative CMBR Rc = 3.13 × 10-6 W m-2
respectively, yields Tb⁄Tc = (IRc )0.25. Further, expressing the radiative
uxes I and Rc on the right-hand side as products of photon pressure
and the speed of light (c, m s-1) in a vacuum, i.e. I = cPph and Rc = cPc,
leads to the following alternative form of the SB law:
( )
0.25
14
ph
b
cc
P
T
TP

=

(14)
where Pc = 1.043 × 10-14 Pa is the photon pressure of CMBR. Clearly, Eq.
(10a) is analogous to Eq. (14), while the latter is structurally identical to
the Poisson formula (13). Figure 7b depicts Eq. (14) as a dependence of
the Tb/Tc ratio on photon pressure Pph.
It is evident from Figures 4 and 7 that formulas (10a), (13) and (14)
describe qualitatively very similar responses in quantitatively vastly
dierent systems. e presence of such similar relations in otherwise
disparate physical systems can fundamentally be explained by the fact
that pressure as a force per unit area represents a key component of
the internal kinetic energy (dened as a product of gas volume and
pressure), while temperature is merely a physical manifestation of this
energy. Adding a force such as gas pressure to a physical system inevitably
boosts the internal kinetic energy and raises its temperature, a process
known in thermodynamics as compression heating. e direct eect
of pressure on a system’s temperature is thermodynamically described
by adiabatic processes. e pressure-induced thermal enhancement
at a planetary level portrayed in Figure 4 and accurately quantied by
Eq. (10a or 11) is analogous to a compression heating, but not fully
identical to an adiabatic process. e latter is usually characterized by
a limited duration and oentimes only applies to nite-size parcels of
air moving vertically through the atmosphere. Equation (11), on the
other hand, describes a surface thermal eect that is global in scope and
permanent in nature as long as an atmospheric mass is present within
the planet’s gravitational eld. Hence, the planetary RATE (Ts/Tna ratio)
could be understood as a net result of countless simultaneous adiabatic
processes continuously operating in the free atmosphere. Figures 4 and
7 also suggest that the pressure control of temperature is a universal
thermodynamic principle applicable to systems ranging in complexity
from a simple isothermal blackbody absorbing a homogeneous ux of
electromagnetic radiation to diverse planetary atmospheres governed
by complex non-linear process interactions and cloud-radiative
feedbacks. To our knowledge, this cross-scale similarity among various
pressure-temperature relationships has not previously been identied
and could provide a valuable new perspective on the working of
planetary climates.
Nevertheless, important dierences exist between Eq. (10a) and
these other simpler pressure-temperature relations. us, while the
Poisson formula and the SB radiation law can mathematically be
derived from ‘rst principles’ and experimentally tested in a laboratory,
Eq. (10a) could neither be analytically deduced from known physical
laws nor accurately simulated in a small-scale experiment. is is
because Eq. (10a) describes an emergent macro-level property of
planetary atmospheres representing the net result of myriad process
interactions within real climate systems that are not readily computable
using mechanistic (bottom-up) approaches adopted in climate models
or fully reproducible in a laboratory setting.
Potential limitations of the planetary temperature model
Equation (10b) describes long-term (30-year) equilibrium GMATs
of planetary bodies and does not predict inter-annual global temperature
variations caused by intrinsic uctuations of cloud albedo and/or ocean
heat uptake. us, the observed 0.82 K rise of Earths global temperature
since 1880 is not captured by our model, as this warming was likely
Figure 7: Known pressure-temperature kinetic relations: (a) Dry adiabatic response of the air/surface temperature ratio to pressure changes in a free dry atmosphere
according to Poisson’s formula (Eq. 13) with a reference pressure set to po = 100 kPa; (b) The SB radiation law expressed as a response of a blackbody temperature
ratio to variations in photon pressure (Eq. 14). Note the qualitative striking similarity of shapes between these curves and the one portrayed in Figure 4 depicting the
new planetary temperature model (Eq. 10a).
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
not the result of an increased atmospheric pressure. Recent analyses of
observed dimming and brightening periods worldwide [97-99] suggest
that the warming over the past 130 years might have been caused by a
decrease in global cloud cover and a subsequent increased absorption of
solar radiation by the surface. Similarly, the mega shi of Earths climate
from a ‘hothouse’ to an ‘icehouse’ evident in the sedimentary archives
over the past 51 My cannot be explained by Eq. (10b) unless caused by
a large loss of atmospheric mass and a corresponding signicant drop
in surface air pressure since the early Eocene. Pleistocene uctuations
of global temperature in the order of 3.0–8.0 K during the last 2 My
revealed by multiple proxies [100] are also not predictable by Eq. (10b)
if due to factors other than changes in total atmospheric pressure and/
or TOA solar irradiance.
e current prevailing view mostly based on theoretical
considerations and results from climate models is that the Pleistocene
glacial-interglacial cycles have been caused by a combination of three
forcing agents: Milankovitch orbital variations, changes in atmospheric
concentrations of greenhouse gases, and a hypothesized positive ice-
albedo feedback [101,102]. However, recent studies have shown that
orbital forcing and the ice-albedo feedback cannot explain key features
of the glacial-interglacial oscillations such as the observed magnitudes
of global temperature changes, the skewness of temperature response
(i.e. slow glaciations followed by rapid meltdowns), and the mid-
Pleistocene transition from a 41 Ky to 100 Ky cycle length [103-105]. e
only signicant forcing remaining in the present paleo-climatological
toolbox to explicate the Pleistocene cycles are variations in greenhouse-
gas concentrations. Hence, it is dicult to explain, from a standpoint
of the current climate theory, the high accuracy of Eq. (11) describing
the relative thermal eect of diverse planetary atmospheres without any
consideration of greenhouse gases. If presumed forcing agents such as
greenhouse-gas concentrations and the planetary albedo were indeed
responsible for the observed past temperature dynamics on Earth, why
did these agents not show up as predictors of contemporary planetary
temperatures in our analysis as well? Could it be because these agents
have not really been driving Earths climate on geological time scales?
We address the potential role of greenhouse gases in more details below.
Since the relationship portrayed in Figure 4 is undoubtedly real, our
model results point toward the need to reexamine some fundamental
climate processes thought to be well understood for decades. For
example, we are currently testing a hypothesis that Pleistocene glacial
cycles might have been caused by variations in Earth’s total atmospheric
mass and surface air pressure. Preliminary results based on the ability
of an extended version of our planetary model (simulating meridional
temperature gradients) to predict the observed polar amplication
during the Last Glacial Maximum indicate that such a hypothesis is not
unreasonable. However, conclusive ndings from this research will be
discussed elsewhere.
According to the present understanding, Earths atmospheric
pressure has remained nearly invariant during the Cenozoic era (last
65.5 My). However, this notion is primarily based on theoretical
analyses [106], since there are currently no known geo-chemical proxies
permitting a reliable reconstruction of past pressure changes in a
manner similar to that provided by various temperature proxies such as
isotopic oxygen 18, alkenones and TEX86 in sediments, and Ar-N isotope
ratios and deuterium concentrations in ice. e lack of independent
pressure proxies makes the assumption of a constant atmospheric mass
throughout the Cenozoic a priori and thus questionable. Although
this topic is beyond the scope of our present study, allowing for the
possibility that atmospheric pressure on Earth might have varied
signicantly over the past 65.5 My could open exciting new research
venues in Earth sciences in general and paleoclimatology in particular.
Role of greenhouse gasses from a perspective of the new
model
Our analysis revealed a poor relationship between GMAT and the
amount of greenhouse gases in planetary atmospheres across a broad
range of environments in the Solar System (Figures 1-3 and Table 5).
is is a surprising result from the standpoint of the current Greenhouse
theory, which assumes that an atmosphere warms the surface of a planet
(or moon) via trapping of radiant heat by certain gases controlling the
atmospheric infrared optical depth [4,9,10]. e atmospheric opacity
to LW radiation depends on air density and gas absorptivity, which in
turn are functions of total pressure, temperature, and greenhouse-gas
concentrations [9]. Pressure also controls the broadening of infrared
absorption lines in individual gases. erefore, the higher the pressure,
the larger the infrared optical depth of an atmosphere, and the stronger
the expected greenhouse eect would be. According to the current
climate theory, pressure only indirectly aects global surface temperature
through the atmospheric infrared opacity and its presumed constraint on
the planet’s LW emission to Space [9,107].
ere are four plausible explanations for the apparent lack of a
close relationship between GMAT and atmospheric greenhouse gasses
in our results: 1) e amounts of greenhouse gases considered in our
analysis only refer to near-surface atmospheric compositions and
do not describe the infrared optical depth of the entire atmospheric
column; 2) e analysis lumped all greenhouse gases together and did
not take into account dierences in the infrared spectral absorptivity of
individual gasses; 3) e eect of atmospheric pressure on broadening
the infrared gas absorption lines might be stronger in reality than
simulated by current radiative-transfer models, so that total pressure
overrides the eect of a varying atmospheric composition across a wide
range of planetary environments; and 4) Pressure as a force per unit area
directly impacts the internal kinetic energy and temperature of a system
in accordance with thermodynamic principles inferred from the Gas
Law; hence, air pressure might be the actual physical causative factor
controlling a planets surface temperature rather than the atmospheric
infrared optical depth, which merely correlates with temperature due to
its co-dependence on pressure.
Based on evidence discussed earlier, we argue that option #4 is
the most likely reason for the poor predictive skill of greenhouse
gases with respect to planetary GMATs revealed in our study (Figures
1-3). By denition, the infrared optical depth of an atmosphere is a
dimensionless quantity that carries no units of force or energy [3,4,9].
erefore, it is dicult to fathom from a fundamental physics standpoint
of view, how this non-dimensional parameter could increase the kinetic
energy (and temperature) of the lower troposphere in the presence of
free convection provided that the latter dominates the heat transport in
gaseous systems. Pressure, on the other hand, has a dimension of force
per unit area and as such is intimately related to the internal kinetic
energy of an atmosphere E (J) dened as the product of gas pressure (P,
Pa) and gas volume (V, m3), i.e. E (J) = PV. Hence, the direct eect of
pressure on a system’s internal energy and temperature follows straight
from fundamental parameter denitions in classical thermodynamics.
Generally speaking, kinetic energy cannot exist without a pressure
force. Even electromagnetic radiation has pressure.
In climate models, the eect of infrared optical depth on surface
temperature is simulated by mathematically decoupling radiative
transfer from convective heat exchange. Specically, the LW
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
Page 14 of 22
Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
radiative transfer is calculated in these models without simultaneous
consideration of sensible- and latent heat uxes in the solution matrix.
Radiative transfer modules compute the so-called heating rates (K/
day) strictly as a function of atmospheric infrared opacity, which
under constant-pressure conditions solely depends on greenhouse-
gas concentrations. ese heating rates are subsequently added to the
thermodynamic portion of climate models and distributed throughout
the atmosphere. In this manner, the surface warming becomes a
function of an increasing atmospheric infrared opacity. is approach to
modeling of radiative-convective energy transport rests on the principle
of superposition, which is only applicable to linear systems, where the
overall solution can be obtained as a sum of the solutions to individual
system components. However, the integral heat transport within a
free atmosphere is inherently nonlinear with respect to temperature.
is is because, in the energy balance equation, radiant heat transfer
is contingent upon power gradients of absolute temperatures, while
convective cooling/heating depends on linear temperature dierences
in the case of sensible heat ux and on simple vapor pressure gradients
in the case of latent heat ux [4]. e latent heat transport is in turn
a function of a solvent’s saturation vapor pressure, which increases
exponentially with temperature [3]. us, the superposition principle
cannot be employed in energy budget calculations. e articial
decoupling between radiative and convective heat-transfer processes
adopted in climate models leads to mathematically and physically
incorrect solutions with respect to surface temperature. e LW
radiative transfer in a real climate system is intimately intertwined
with turbulent convection/advection as both transport mechanisms
occur simultaneously. Since convection (and especially the moist one)
is orders of magnitude more ecient in transferring energy than LW
radiation [3,4], and because heat preferentially travels along the path
of least resistance, a properly coupled radiative-convective algorithm
of energy exchange will produce quantitatively and qualitatively
dierent temperature solutions in response to a changing atmospheric
composition than the ones obtained by current climate models.
Specically, a correctly coupled convective-radiative system will render
the surface temperature insensitive to variations in the atmospheric
infrared optical depth, a result indirectly supported by our analysis as
well. is topic requires further investigation beyond the scope of the
present study.
e direct eect of atmospheric pressure on the global surface
temperature has received virtually no attention in climate science thus
far. However, the results from our empirical data analysis suggest that it
deserves a serious consideration in the future.
eoretical implications of the new interplanetary
relationship
e hereto discovered pressure-temperature relationship quantied
by Eq. (10a) and depicted in Figure 4 has broad theoretical implications
that can be summarized as follows:
Physical nature of the atmospheric ‘greenhouse eect’: According
to Eq. (10b), the heating mechanism of planetary atmospheres is
analogous to a gravity-controlled adiabatic compression acting upon
the entire surface. is means that the atmosphere does not function
as an insulator reducing the rate of planet’s infrared cooling to space as
presently assumed [9,10], but instead adiabatically boosts the kinetic
energy of the lower troposphere beyond the level of solar input through
gas compression. Hence, the physical nature of the atmospheric
‘greenhouse eect’ is a pressure-induced thermal enhancement
(PTE) independent of atmospheric composition. is mechanism
is fundamentally dierent from the hypothesized ‘trapping’ of LW
radiation by atmospheric trace gases rst proposed in the 19th century
and presently forming the core of the Greenhouse climate theory.
However, a radiant-heat trapping by freely convective gases has never
been demonstrated experimentally. We should point out that the hereto
deduced adiabatic (pressure-controlled) nature of the atmospheric
thermal eect rests on an objective analysis of vetted planetary
observations from across the Solar System and is backed by proven
thermodynamic principles, while the ‘trapping’ of LW radiation by an
unconstrained atmosphere surmised by Fourier, Tyndall and Arrhenius
in the 1800s was based on a theoretical conjecture. e latter has later
been coded into algorithms that describe the surface temperature as a
function of atmospheric infrared optical depth (instead of pressure) by
articially decoupling radiative transfer from convective heat exchange.
Note also that the Ideal Gas Law (PV = nRT) forming the basis of
atmospheric physics is indierent to the gas chemical composition.
Eect of pressure on temperature: Atmospheric pressure
provides in and of itself only a relative thermal enhancement (RATE)
to the surface quantied by Eq. (11). e absolute thermal eect of an
atmosphere depends on both pressure and the TOA solar irradiance.
For example, at a total air pressure of 98.55 kPa, Earth’s RATE is 1.459,
which keeps our planet 90.4 K warmer in its present orbit than it would
be in the absence of an atmosphere. Hence, our model fully explains
the new ~90 K estimate of Earths atmospheric thermal eect derived
by Volokin and ReLlez [1] using a dierent line of reasoning. If one
moves Earth to the orbit of Titan (located at ~9.6 AU from the Sun)
without changing the overall pressure, our planet’s RATE will remain
the same, but the absolute thermal eect of the atmosphere would drop
to about 29.2 K due to a vastly reduced solar ux. In other words, the
absolute eect of pressure on a systems temperature depends on the
background energy level of the environment. is implies that the
absolute temperature of a gas may not follow variations of pressure
if the gas energy absorption changes in opposite direction to that of
pressure. For instance, the temperature of Earth’s stratosphere increases
with altitude above the tropopause despite a falling air pressure, because
the absorption of UV radiation by ozone steeply increases with height,
thus osetting the eect of a dropping pressure. If the UV absorption
were constant throughout the stratosphere, the air temperature would
decrease with altitude.
Atmospheric back radiation and surface temperature: Since
(according to Eq. 10b) the equilibrium GMAT of a planet is mainly
determined by the TOA solar irradiance and surface atmospheric
pressure, the down-welling LW radiation appears to be globally a product
of the air temperature rather than a driver of the surface warming. In
other words, on a planetary scale, the so-called back radiation is a
consequence of the atmospheric thermal eect rather than a cause for
it. is explains the broad variation in the size of the observed down-
welling LW ux among celestial bodies irrespective of the amount of
absorbed solar radiation. erefore, a change in this thermal ux brought
about by a shi in atmospheric LW emissivity cannot be expected to
impact the global surface temperature. Any variation in the global
infrared back radiation caused by a change in atmospheric composition
would be compensated for by a corresponding shi in the intensity of
the vertical convective heat transport. Such a balance between changes
in atmospheric infrared heating and the upward convective cooling at
the surface is required by the First Law of ermodynamics. However,
current climate models do not simulate this compensatory eect of
sensible and latent heat uxes due to an improper decoupling between
radiative transfer and turbulent convection in the computation of total
energy exchange.
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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Role of planetary albedos: e fact that Eq. (10b) accurately
describes planetary GMATs without explicitly accounting for the
observed broad range of albedos, i.e. from 0.136 to 0.9 (Table 2),
indicates that the shortwave reectivity of planetary atmospheres is
mostly an intrinsic property (a byproduct) of the climate system itself
rather than an independent driver of climate as currently believed. In
other words, it is the internal energy of the atmosphere maintained by
solar irradiance and air pressure that controls the bulk of the albedo.
An indirect support for this unorthodox conclusion is provided by
the observation that the amounts of absorbed shortwave radiation
determined by albedos show no physically meaningful relationship
with planetary GMATs. For example, data in Table 2 indicate that
Venus absorbs 3.7 times less solar energy per unit area than Earth, yet
its surface is about 450 K hotter than that of Earth; the Moon receives
on average 54 W m-2 more net solar radiation than Earth, but it is
about 90 K cooler on average than our planet. e hereto proposed
passive nature of planetary albedos does not imply that the global
cloud cover could not be inuenced by an external forcing such as solar
wind, galactic cosmic rays, and/or gravitational elds of other celestial
objects. Empirical evidence strongly suggests that it can [108-113], but
the magnitude of such inuences is expected to be small compared to
the total albedo due to the presence of stabilizing negative feedbacks
within the system. We also anticipate that the sensitivity of GMATs to
an albedo change will greatly vary among planetary bodies. Viewing
the atmospheric reectivity as a byproduct of the available internal
energy rather than a driver of climate can also help explain the observed
remarkable stability of Earth’s albedo [54,114].
Climate stability: Our semi-empirical model (Equations 4a, 10b
and 11) suggests that, as long as the mean annual TOA solar ux and
the total atmospheric mass of a planet are stationary, the equilibrium
GMAT will remain stable. Inter-annual and decadal variations of global
temperature forced by uctuations of cloud cover, for example, are
expected to be small compared to the magnitude of the background
atmospheric warming because of strong negative feedbacks limiting
the albedo changes. is implies a relatively stable climate for a planet
such as Earth absent signicant shis in the total atmospheric mass
and the planet’s orbital distance to the Sun. Hence, planetary climates
appear to be free of tipping points, i.e. functional states fostering
rapid and irreversible changes in the global temperature as a result of
hypothesized positive feedbacks thought to operate within the system.
In other words, our results suggest that the Earth’s climate is well
buered against sudden changes.
Eect of oceans and water vapor on global temperature: e new
model shows that the Earths global equilibrium temperature is a part
of a cosmic thermodynamic continuum controlled by atmospheric
pressure and total solar irradiance. Since our planet is the only one
among studied celestial bodies harboring a large quantity of liquid
water on the surface, Eq. (10b) implies that the oceans play virtually no
role in determining Earths GMAT. is nding may sound inexplicable
from the standpoint of the radiative Greenhouse theory, but it follows
logically from the new paradigm of a pressure-induced atmospheric
warming. e presence of liquid water on the surface of a planet requires
an air pressure greater than 612 Pa and an ambient temperature above
273.2 K. ese conditions are provided by the planet’s size and gravity,
its distance to the Sun, and the mass of the atmosphere. Hence, the
water oceans on Earth seem to be a thermodynamic consequence of
particular physical conditions set by cosmic arrangements rather than
an active controller of the global climate. Similarly, the hydrocarbon
lakes on the surface of Titan [115,116] are the result of a high
atmospheric pressure and an extremely cold environment found on that
moon. us, our analysis did not reveal evidence for the existence of a
feedback between planetary GMAT and a precipitable liquid solvent on
the surface as predicted by the current climate theory. Consequently,
the hypothesized runaway greenhouse, which requires a net positive
feedback between global surface temperature and the atmospheric LW
opacity controlled by water vapor [117], appears to be a model artifact
rather than an actual physical possibility. Indeed, as illustrated in Figure
4, the hot temperature of Venus oen cited as a product of a ‘runaway
greenhouse’ scenario [117,118] ts perfectly within the pressure-
dependent climate continuum described by Equations (10b) and (11).
Model Application and Validation
Encouraged by the high predictive skill and broad scope of validity
of Model 12 (Figure 2f) we decided to apply Eq. (10b) to four celestial
bodies spanning the breadth of the Solar System, i.e. Mercury, Europa,
Callisto and Pluto, which global surface temperatures are not currently
known with certainty. Each body is the target of either ongoing or
planned robotic exploration missions scheduled to provide surface
thermal data among other observations, thus oering an opportunity
to validate our planetary temperature model against independent
measurements.
e MESSENGER spacecra launched in 2004 completed the rst
comprehensive mapping of Mercury in March 2013 (http://messenger.
jhuapl.edu/). Among other things, the spacecra also took infrared
measurements of the planet’s surface using a special spectrometer
[119] that should soon become available. e New Horizons spacecra
launched in January 2006 [120] reached Pluto in July of 2015 and
performed a thermal scan of the dwarf planet during a yby. e
complete dataset from this yby were received on Earth in October of
2016 and are currently being analyzed. A proposed joint Europa-Jupiter
System Mission by NASA and the European Space Agency is planned to
study the Jovian moons aer year 2020. It envisions exploring Europas
physical and thermal environments both remotely via a NASA Orbiter
and in situ by a Europa Lander [121].
All four celestial bodies have somewhat eccentric orbits around the
Sun. However, while Mercury’s orbital period is only 88 Earth days,
Europa and Callisto circumnavigate the Sun once every 11.9 Earth
years while Pluto takes 248 Earth years. e atmospheric pressure on
Pluto is believed to vary between 1.0 and 4.0 Pa over the course of its
orbital period as a function of insolation-driven sublimation of nitrogen
and methane ices on the surface [122]. Each bodys temperature was
evaluated at three orbital distances from the Sun: aphelion, perihelion,
and the semi-major axis. Since Mercury, Europa and Callisto harbor
tenuous atmospheres (P << 10-2 Pa), the reference temperature Tna in
Eq. (10b) must be calculated from Eq. (4a), which requires knowledge
of the actual values of αe, ηe, and Rg. We assumed that Mercury had Rg =
0.0 W m-2, αe = 0.068 [123] and Moon-like thermo-physical properties
of the regolith (ηe = 0.00971). Input data for Europa and Callisto were
obtained from Spencer et al. [124] and Moore et al. [125], respectively.
Specically, in order to calculate ηe and Rg for these moons we utilized
equatorial temperature data provided by Spencer et al. [124] in their
Figure 1, and by Moore et al. [125] in their Fig. 17.7 along with a
theoretical formula for computing the average nighttime surface
temperature T at the equator based on the SB law, i.e.
( ) ( )
0.25
1 15
0.98
eg
SR
T
αη
σ
−+

=

(15)
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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where S(1-α)ηe is the absorbed solar ux (W m-2) stored as heat into
the subsurface. e geothermal heat ux on Europa is poorly known.
However, based on thermal observations of Io reported by Veeder et al.
[126], we assumed Rg = 2.0 W m-2 for Europa. Using S = 50.3 W m-2, an
observed nighttime equatorial temperature T = 90.9 K and an observed
average night-side albedo α = 0.58 [124], we solved Eq. (15) for the
surface heat storage fraction to obtain ηe = 0.085 for Europa. A similar
computational procedure was employed for Callisto using α = 0.11 and
equatorial surface temperature data from Fig. 17.7 in Moore et al. [125].
is produced Rg = 0.5 W m-2 and ηe = 0.057. Using these values in
Eq. (15) correctly reproduced Callistos nighttime equatorial surface
temperature of86.0 K. e much higher ηe estimates for Europa and
Callisto compared to ηe = 0.00971 for the Moon can be explained with
the large water-ice content on the surface of these Galilean moons.
Europa is almost completely covered by a thick layer of water ice, which
has a much higher thermal conductivity than the dry regolith. Also,
sunlight penetrates deeper into ice than it does into powdered regolith.
All this enables a much larger fraction of the absorbed solar radiation to
be stored into the subsurface as heat and later released at night boosting
the nighttime surface temperatures of these moons. Volokin and ReLlez
[1] showed that GMAT of airless bodies is highly sensitive to ηe.
Tabl e 6 lists the average global surface temperatures of the four
celestial bodies predicted by Eq. (10b) along with the employed input
data. According to our model, Mercury is about 117 K cooler on average
than NASAs current estimate of 440 K [32], which is based on Eq. (3)
and does not represent a spherically averaged surface temperature [1].
Our prediction of Europas GMAT, 99.4 K, agrees well with the100
K estimate reported for this moon by Sotin et al. [127]. Our estimate
of Plutos average surface temperature at perihelion (38.6 K) is similar
to the mean temperature computed for that dwarf planet by Olkin et
al. [124] using a mechanistic model of nitrogen ice volatilization at
the surface. Stern et al. [128] and Gladstone et al. [93] reported initial
results from yby observations of Pluto taken by the Radio Experiment
(REX) instrument aboard the New Horizons spacecra in July 2015,
when the dwarf planet was approximately at 32.9 AU from the Sun.
Using the observed surface pressure of 1.05 ± 0.1 Pa (10.5 ± 1 μbar)
[93] our model predicts an average global temperature of 36.7 K for
Pluto. Stern et al. [128] reported a near-surface temperature of38
K. However, this value was calculated from pre-yby global brightness
measurements rather than derived via spherical integration of spatially
resolved surface temperatures (Stern, personal communication). Since
global brightness temperatures tend to be higher than spherically
averaged kinetic surface temperatures [1], our model prediction may
well be within the uncertainty of Plutos true global temperature. We
will know more about this in 2017 when spatially resolved thermal
measurements obtained by New Horizons become available.
One should use caution when comparing results from Eq. (10b)
to remotely sensedaverage temperatures commonly quoted for
celestial bodies with tenuous atmospheres such as the moons of Jupiter
and Neptune. Studies oentimes report the so-calledbrightness
temperatures’ retrieved at specic wavelengths that have not been
subjected to a proper spherical integration. As pointed out by Volokin
and ReLlez [1], due to Hölder’s inequality between integrals, calculated
brightness temperatures of spherical objects can be signicantly higher
than actual mean kinetic temperatures of the surface. Since Eq. (10b)
yields spherically averaged temperatures, its predictions for airless
bodies are expected to be lower than the disk-integrated brightness
temperatures typically quoted in the literature.
Conclusion
For 190 years the atmosphere has been thought to warm Earth
by absorbing a portion of the outgoing LW infrared radiation and
reemitting it back toward the surface, thus augmenting the incident
solar ux. is conceptualized continuous absorption and downward
reemission of thermal radiation enabled by certain trace gases known
to be transparent to solar rays while opaque to electromagnetic
long-wavelengths has been likened to the trapping of heat by glass
greenhouses, hence the term ‘atmospheric greenhouse eect’. Of course,
we now know that real greenhouses preserve warmth not by trapping
infrared radiation but by physically obstructing the convective heat
exchange between a greenhouse interior and the exterior environment.
Nevertheless, the term ‘greenhouse eect’ stuck in science.
e hypothesis that a freely convective atmosphere could retain
(trap) radiant heat due its opacity has remained undisputed since its
introduction in the early 1800s even though it was based on a theoretical
conjecture that has never been proven experimentally. It is important to
note in this regard that the well-documented enhanced absorption of
thermal radiation by certain gases does not imply an ability of such gases
to trap heat in an open atmospheric environment. is is because, in
gaseous systems, heat is primarily transferred (dissipated) by convection
(i.e. through uid motion) rather than radiative exchange. If gases of
high LW absorptivity/emissivity such as CO2, methane and water vapor
were indeed capable of trapping radiant heat, they could be used as
insulators. However, practical experience has taught us that thermal
radiation losses can only be reduced by using materials of very low LW
Surface Atmospheric
Pressure (Pa)
αe (fraction)
ηe (fraction)
Rg (W m-2)
Predicted Average Global
Surface Temperature at Specic Orbital Distances from the Sun
Aphelion Semi-major Axis Perihelion
Mercury 5 × 10-10
αe = 0.068
ηe = 0.00971
Rg = 0.0
296.8 K
(0.459 AU)
323.3 K
(0.387 AU)
359.5 K
(0.313 AU)
Europa 10-7
αe = 0.62
ηe = 0.085
Rg = 2.0
98.1 K
(5.455 AU)
99.4 K
(5.203 AU)
100.7 K
(4.951 AU)
Callisto 7.5 × 10-7
αe = 0.11
ηe = 0.057
Rg = 0.5
101.2 K
(5.455 AU)
103.2 K
(5.203 AU)
105.4 K
(4.951 AU)
Pluto 1.05
αe = 0.132
ηe = 0.00971
Rg = 0.0
30.0 K
(49.310 AU)
33.5 K
(39.482 AU)
38.6 K
(29.667 AU)
Table 6: Average global surface temperatures predicted by Eq. (10b) for Mercury, Europa, Calisto and Pluto. Input data on orbital distances (AU) and total atmospheric
pressure (Pa) were obtained from the NASA Solar System Exploration [48] website, the NASA Planetary Factsheet [32] and Gladstone et al. [93]. Solar irradiances required
by Eq. (10b) were calculated from reported orbital distances as explained in the text. Values of αe, ηe and Rg for Europa and Callisto were estimated from observed data by
Spencer et al. [124] and Moore et al. [125] respectively (see text for details).
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
absorptivity/emissivity and correspondingly high thermal reectivity
such as aluminum foil. ese materials are known among engineers at
NASA and in the construction industry as radiant barriers [129]. It is
also known that high-emissivity materials promote radiative cooling.
Yet, all climate models proposed since 1800s are built on the premise
that the atmosphere warms Earth by limiting radiant heat losses of the
surface through the action of infrared absorbing gases alo.
If a trapping of radiant heat occurred in Earth’s atmosphere, the
same mechanism should also be expected to operate in the atmospheres
of other planetary bodies. us, the Greenhouse concept should be able
to mathematically describe the observed variation of average planetary
surface temperatures across the Solar System as a continuous function
of the atmospheric infrared optical depth and solar insolation. However,
to our knowledge, such a continuous description (model) does not
exist. Furthermore, measured magnitudes of the global down-welling
LW ux on planets with thick atmospheres such as Earth and Venus
indicate that the lower troposphere of these bodies contains internal
kinetic energy far exceeding the solar input [6,12,14]. is fact cannot
be explained via re-radiation of absorbed outgoing thermal emissions
by gases known to supply no additional energy to the system. e desire
to explicate the sizable energy surplus evident in the tropospheres of
some terrestrial planets provided the main impetus for this research.
We combined high-quality planetary data from the last three
decades with the classical method of dimensional analysis to search for
an empirical model that might accurately and meaningfully describe
the observed variation of global surface temperatures throughout the
Solar System while also providing a new perspective on the nature of the
atmospheric thermal eect. Our analysis revealed that the equilibrium
global surface temperatures of rocky planets with tangible atmospheres
and a negligible geothermal surface heating can reliably be estimated
across a wide range of atmospheric compositions and radiative regimes
using only two forcing variables: TOA solar irradiance and total surface
atmospheric pressure (Eq. 10b with Tna computed from Eq. 4c).
Furthermore, the relative atmospheric thermal enhancement (RATE)
dened as a ratio of the planet’s actual global surface temperature to
the temperature it would have had in the absence of atmosphere is fully
explicable by the surface air pressure alone (Eq. 10a and Figure 4). At
the same time, greenhouse-gas concentrations and/or partial pressures
did not show any meaningful relationship to surface temperatures
across a broad span of planetary environments considered in our study
(see Figures 1 and 2 and Table 5).
Based on statistical criteria including numerical accuracy,
robustness, dimensional homogeneity and a broad environmental
scope of validity, the new relationship (Figure 4) quantied by Eq. (10a)
appears to describe an emergent macro-level thermodynamic property
of planetary atmospheres heretofore unbeknown to science. e
physical signicance of this empirical model is further supported by its
striking qualitative resemblance to the dry adiabatic temperature curve
described by the Poisson formula (Eq. 13) and to the photon-pressure
form of the SB radiation law (Eq. 14). Similar to these well-known
kinetic relations, Eq. (10a) also predicts the direct eect of pressure on
temperature albeit in the context of a dierent macro-physical system.
To our knowledge, this is the rst model accurately describing the
average surface temperatures of planetary bodies throughout the Solar
System in the context of a thermodynamic continuum using a common
set of drivers.
e planetary temperature model consisting of Equations (4a),
(10b), and (11) has several fundamental theoretical implications, i.e.
e ‘greenhouse eect’ is not a radiative phenomenon driven
by the atmospheric infrared optical depth as presently believed,
but a pressure-induced thermal enhancement analogous to
adiabatic heating and independent of atmospheric composition;
e down-welling LW radiation is not a global driver of surface
warming as hypothesized for over 100 years but a product of
the near-surface air temperature controlled by solar heating
and atmospheric pressure;
e albedo of planetary bodies with tangible atmospheres is not
an independent driver of climate but an intrinsic property (a
byproduct) of the climate system itself. is does not mean that
the cloud albedo cannot be inuenced by external forcing such
as solar wind or galactic cosmic rays. However, the magnitude
of such inuences is expected to be small due to the stabilizing
eect of negative feedbacks operating within the system. is
understanding explains the observed remarkable stability of
planetary albedos;
e equilibrium surface temperature of a planet is bound to
remain stable (i.e. within ± 1 K) as long as the atmospheric
mass and the TOA mean solar irradiance are stationary. Hence,
Earth’s climate system is well buered against sudden changes
and has no tipping points;
e proposed net positive feedback between surface
temperature and the atmospheric infrared opacity controlled
by water vapor appears to be a model artifact resulting from
a mathematical decoupling of the radiative-convective heat
transfer rather than a physical reality.
e hereto reported ndings point toward the need for a paradigm
shi in our understanding of key macro-scale atmospheric properties and
processes. e implications of the discovered planetary thermodynamic
relationship (Figure 4, Eq. 10a) are fundamental in nature and require
careful consideration by future research. We ask the scientic community
to keep an open mind and to view the results presented herein as a possible
foundation of a new theoretical framework for future exploration of
climates on Earth and other worlds.
Appendices
Appendix A. Construction of the Dimensionless π Variables
Table 1 lists 6 generic variables (Ts, Tr, S, Px, Pr and ρx) composed of
4 fundamental dimensions: mass [M], length [L], time [T], and absolute
temperature [Θ]. According to the Buckingham Pi theorem [27], this
implies the existence of two dimensionless πi products per set. To
derive the πi variables we employed the following objective approach.
First, we hypothesized that a planets GMAT (Ts) is a function of all 5
independent variables listed in Table 1, i.e.
( )
rs xr x
T T , S , P , P , = ρ
ƒ
(A.1)
is unknown function is described to a rst approximation as a simple
product of the driving variables raised to various powers, i.e.
abc d e
s r xr x
T T SPP
ρ
(A.2)
where a, b, c, d and e are rational numbers. In order to determine the
power coecients, Eq. (A.2) is cast in terms of physical dimensions of
the participating variables, i.e.
[ ] [ ]
3 12 12 3
M T M L T M L T M L
− −− −−

Θ≈Θ

b c de
a (A.3)
Satisfying the requirement for dimensional homogeneity of Eq.
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
(A.2) implies that the sum of powers of each fundamental dimension
must be equal on both sides of Eq. (A.3). is allows us to write four
simultaneous equations (one per fundamental dimension) containing
ve unknowns, i.e.
[ ]
[ ]
[ ]
[ ]
1 :
0 :
3 0 :
3 2 2 0 :
a
bcd e M
cd e L
bcd T
= Θ
++ +=
−− − =
−− − =
(A.4)
System (A.4) is underdetermined and has the following solution: a
= 1, b = 2e, and c = -(3e + d). Note that, in the DA methodology,
one oentimes arrives at underdetermined systems of equations,
simply because the number of independent variables usually exceeds
the number of fundamental physical dimensions comprising such
variables. However, this has no adverse eect on the derivation of the
sought dimensionless πi products.
Substituting the above roots in Eq. (A.2) reduces the original ve
unknowns to two: d and e, i.e.
( )
3
12
ed
e de
s r x rx
T TS P P
ρ
−+
(A.5a)
ese solution powers may now be assigned arbitrary values, although
integers such as 0, 1 and -1 are preferable, for they oer the simplest
solution leading to the construction of proper πi variables. Setting d = 0
and e = -1 reduces Eq. (A.5a) to
1 23 1
s r xx
T TS P
ρ
−−
(A.5b)
providing the rst pair of dimensionless products:
3
12
2
;
π= π=
sx
rx
TP
TS
ρ
(A.6)
e second pair of πi variables emerges upon setting d = -1 and e = 0 in
Eq. (A.5a), i.e.
12
; π= π=
sx
rr
TP
TP
(A.7)
us, the original function (A.1) consisting of six dimensioned
variables has been reduced to a relationship between two dimensionless
quantities, i.e. π1 = f (π2). is relationship must further be investigated
through regression analysis.
Appendix B. Estimation of Mars’ GMAT and Surface
Atmospheric Pressure
Although Mars is the third most studied planetary body in the
Solar System aer Earth and the Moon, there is currently no consensus
among researchers regarding its mean global surface temperature (TM).
TM values reported over the past 15 years span a range of 40 K. Examples
of disparate GMATs quoted for the Red Planet include 200 K [79], 202
K [82,130], 210 K [32], 214 K [80], 215 K [6,81], 218 K [77], 220 K [76],
227 K [131] and 240 K [78]. e most frequently cited temperatures fall
between 210 K and 220 K. However, a close examination of the available
thermal observations reveals a high improbability for any of the above
estimates to represent Mars’ true GMAT.
Figure B.1 depicts hourly temperature series measured at 1.5 m
aboveground by Viking Landers 1 and 2 (VL1 and VL2 respectively) in
the late 1970s [60]. e VL1 record covers about half of a Martian year,
while the VL2 series extends to nearly 1.6 years. e VL1 temperature
series captures a summer-fall season on a site located at about 1,500 m
below Datum elevation in the subtropics of Mars’ Northern Hemisphere
(22.5o N). e arithmetic average of the series is 207.3 K (Fig. B.1a).
Since the record lacks data from the cooler winter-spring season, this
value is likely higher than the actual mean annual temperature at that
location. Furthermore, observations by the Hubble telescope from the
mid-1990s indicated that the Red Planet may have cooled somewhat
since the time of the Viking mission [132,133]. Because of a thin
atmosphere and the absence of signicant cloud cover and perceptible
water, temperature uctuations near the surface of Mars are tightly
coupled to diurnal, seasonal and latitudinal variations in incident solar
radiation. is causes sites located at the same latitude and equivalent
altitudes to have similar annual temperature means irrespective of
their longitudes [134]. Hence, one could reliably estimate a latitudinal
temperature average on Mars using point observations from any
elevation by applying an appropriate lapse-rate correction for the
average terrain elevation of said latitude.
At 22.5o absolute latitude, the average elevation between Northern
and Southern Hemisphere on Mars is close to Datum level, i.e. about
1,500 m above the VL1 site. Adjusting the observed 207.3 K temperature
average at VL1 to Datum elevation using a typical near-surface Martian
lapse rate of -4.3 K km-1 [78] produces ~201 K for the average summer-
fall temperature at that latitude. Since the mean surface temperature
Figure B.1: Near-surface hourly temperatures measured on Mars by (a) Viking Lander 1 at Chryse Planitia (22.48° N, 49.97° W, Elevation: -1,500 m); and (b) Viking
Lander 2 at Utopia Planitia (47.97° N, 225.74° W, Elevation: -3,000 m) (Kemppinen et al. [60]; data downloaded from: http://www-k12.atmos.washington.edu/k12/
resources/mars_data-information/data.html). Black dashed lines mark the arithmetic average (Tmean) of each series. Grey dashed lines highlight the range of most
frequently reported GMAT values for Mars, i.e. 210–240 K. The average diurnal temperature can only exceed 210 K during the summer; hence, all Martian latitudes
outside the Equator must have mean annual temperatures signicantly lower than 210 K.
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
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Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
of a sphere is typically lower than its subtropical temperature average,
we can safely conclude based on Figure B.1a that Mars’ GMAT is likely
below 201 K. e mean temperature at the VL2 site located at ~48o N
latitude and 3,000 m below Datum elevation is 191.1 K (Fig. B.1b). e
average terrain elevation between Northern and Southern Hemisphere
at 48o absolute latitude is about -1,500 m. Upon adjusting the VL2
annual temperature mean to -1,500 m altitude using a lapse rate of
-4.3 K km-1 we obtain 184.6 K. Since a planet’s GMAT numerically falls
between the mean temperature of the Equator and that of 42o absolute
latitude, the above calculations suggest that Mars’ GMAT is likely
between 184 K and 201 K.
A close examination of the Viking record also reveals that average
diurnal temperatures above 210 K only occur on Mars during the
summer season and, therefore, cannot possibly represent an annual
mean for any Martian latitude outside the Equator. On the other hand,
frequently reported values of Mars’ GMAT in excess of 210 K appear to
be based on the theoretical expectation that a planet’s average surface
temperature should exceed the corresponding eective radiating
temperature produced by Eq. (3) [6,78], which is Te ≈ 212 K for Mars.
is presumption is rooted in the a priori assumption that Te represents
a planet’s average surface temperature in the absence of atmospheric
greenhouse eect. However, Volokin and ReLlez [1] have shown
that, due to Hölder’s inequality between integrals, the mean physical
temperature of a spherical body with a tenuous atmosphere is always
lower than its eective radiating temperature computed from the
globally integrated absorbed solar ux. In other words, Eq. (3) yields
non-physical temperatures for spheres. Indeed, based on results from
a 3-D climate model Haberle [130] concluded that Mars’ mean global
surface temperature is at least 8 K cooler than the planet’s eective
radiating temperature. erefore, Mars’ GMAT must be inferred from
actual measurements rather than from theoretical calculations.
In order to obtain a reliable estimate of Mars’ GMAT, we calculated
the mean annual temperatures at several Martian latitudes employing
near-surface time series measured in-situ by Viking Landers and the
Curiosity Rover, and remotely by the Mars Global Surveyor (MGS)
spacecra. e Radio Science Team (RST) at Stanford University
utilized radio occultation of MGS refraction data to retrieve seasonal
time-series of near-surface atmospheric temperature and pressure on
Mars [61,62,135]. We utilized MGS-RST data obtained between 1999
and 2005. Calculated mean temperatures from in-situ measurements
were adjusted to corresponding average terrain elevations of target
latitudes using a lapse rate of -4.3 K km-1 [78]. Figure B.2 portrays
the estimated Mean Annual near-surface Temperatures (MAT) at ve
absolute Martian latitudes (gray dots) along with their standard errors
(vertical bars). e equatorial MAT was calculated from Curiosity Rover
observations; temperatures at absolute latitudes 0.392 rad (22.48o) and
0.837 rad (47.97o) were derived from VL measurements, while these
at latitudes 1.117 rad (64o) and 1.396 rad (80o) were estimated from
MGS-RST data. e black curve represents a third-order polynomial
tted through the latitudinal temperature averages and described by the
polynomial:
( ) ( )
23
202.888 0.781801 22.3673 3.16594 B.1TL L L L=−−
with L being the absolute latitude (rad). MAT values predicted by
Eq. (B.1) for Mars’ Equatorial and Polar Regions agree well with
independent near-surface temperatures remotely measured by the
Mars Climate Sounder (MCS), a platform deployed aer MGS in
2006 [136]. Shirley et al. [136] showed that, although separated in
time by 2-5 years, MCS temperature proles match quite well those
retrieved by MGS-RST especially in the lower portion of the Martian
atmosphere. Figures 2 and 3 of Shirley et al. [136] depict nighttime
winter temperature proles over the Mars’ northern and southern Polar
Regions, respectively at about 75o absolute latitude. e average winter
surface temperature between the two Hemispheres for this latitude
is about 148.5 K. is compares favorably with 156.4 K produced by
Eq. (B.1) for 75o (1.309 rad) latitude considering that MAT values are
expected to be higher than winter temperature averages. Figures 4 and
5 of Shirley et al. [136] portray average temperature proles retrieved
by MGS-RST and MCS over lowlands (165o – 180o E) and highlands
(240o - 270o E) of the Mars’ equatorial region (8o N - 8o S), respectively.
For highlands (≈5 km above Datum), the near-surface temperature
appears to be around 200 K, while for lowlands (≈2.5 km below Datum)
it is ≈211 K. Since most of Mars’ equatorial region lies above Datum, it
is likely that Mars’ equatorial MAT would be lower than 205.5 K and
close to our independent estimate of ≈203 K based on Curiosity Rover
measurements.
Mars’ GMAT (TM) was calculated via integration of polynomial
(B.1) using the formula:
( ) ( )
/2
0
cos B.2
M
T T L L dL
π
=
(B.2)
where 0 ≤ cosL ≤ 1 is a polar-coordinate area-weighting factor.
e result is TM = 190.56 ± 0.7 K (Figure B.2). is estimate, while
signicantly lower than GMAT values quoted in recent publications,
agrees quite well with spherically integrated brightness temperatures
of Mars retrieved from remote microwave observations during the
late 1960s and early 1970s [85-87]. us, according to Hobbs et al.
[85] and Klein [86], the Martian mean global temperature (inferred
from measurements at wavelengths between 1 and 21 cm) is 190 –
193 K. Our TM estimate is also consistent with the new mean surface
temperature of the Moon (197.35 K) derived by Volokin and ReLlez
[1] using output from a validated NASA thermo-physical model [29].
Since Mars receives 57% less solar ittadiance than the Moon and has
a thin atmosphere that only delivers a weak greenhouse eect [9], it
makes a physical sense that the Red Planet would be on average cooler
than our Moon (i.e. TM < 197.3K). Moreover, if the average temperature
Figure B.2: Mean annual surface air temperatures at ve Martian absolute
latitudes (gray dots) estimated from data provided by Viking Landers, Curiosity
Rover, and the Mars Global Surveyor Radio Science Team. Each dot represents
a mean annual temperature corresponding to the average terrain elevation
between Northern and Southern Hemisphere for particular latitude. The black
curve depicts a third-order polynomial (Eq. B.1) tted through the latitudinal
temperature means using a non-linear regression. Mars’ GMAT, TM = 190.56
K (marked by a horizontal gray dashed line) was calculated via integration of
polynomial (B.1) using formula (B.2).
Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary
Temperature Model. Environ Pollut Climate Change 1: 112.
Page 20 of 22
Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal
of the lunar equator (Moon’s warmest latitude) is 213 K as revealed by
NASA Diviner observations [1,29], it is unlikely that Mars’ mean global
temperature would be equal to or higher than 213 K as assumed by
many studies [6,76-78,80,131]
Published values of Mars’ average surface atmospheric pressures
range from 600 Pa to 700 Pa [6,32,78,80,83,84]. Since this interval was
too broad for the target precision of our study, we employed MGS-RST
data retrieved from multiple latitudes and seasons between 1999 and
2005 to calculate a new mean surface air pressure for the Red Planet.
Our analysis produced P = 685.4 ± 14.2 Pa, an estimate within the
range of previously reported values.
Funding Sources
This research did not receive any specic grant from funding agencies in the
public, commercial, or not-for-prot sectors.
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