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- Corresponding Author:
- Ned Nikolov

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Wellington CO 80549, USA

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**E-mail:**ntconsulting@comcast.net

**Received date:** November 11, 2016; **Accepted date:** February 06, 2017; **Published date:** 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.s

**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.

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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 flux 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 findings call for a paradigm shift in our understanding of the atmospheric ‘greenhouse effect’ as a fundamental property of climate.

Greenhouse effect; Emergent model; Planetary temperature; Atmospheric pressure; Greenhouse gas; Mars temperature

In a recent study Volokin et al. [1] demonstrated that the strength of
Earth’s atmospheric Greenhouse Effect (GE) is about 90 K instead of 33
K as presently assumed by most researchers e.g. [2-7]. The 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 flux might only explain a surface
warming up to 35 K. Hence, more than 60% of Earth’s 90 K atmospheric effect 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 flux
delivered to the Earth-atmosphere system by the Sun (i.e., 240 W m^{-2}).
In other words, the lower troposphere contains significantly 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 downwelling
LW radiation near the surface (>15,000 W m^{-2}) exceeds the
total absorbed solar flux (65–150 W m^{-2}) by a factor of 100 or more [6].
The radiative greenhouse theory cannot explain this apparent paradox
considering the fact that infrared-absorbing gases such as CO_{2}, water vapor and methane only re-radiate available LW emissions and do not
constitute significant heat storage or a net source of additional energy to
the system. This 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. The above inconsistencies between theory
and observations prompted us to take a new look at the mechanisms
controlling the atmospheric thermal effect.

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. Thus, our working hypothesis was that a general physical model should exist, which accurately describes GMATs of planets using a common set of drivers. If so, then such a model would also reveal the forcing behind the atmospheric thermal effect.

Instead of examining existing mechanistic models such as 3-D 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-specific 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]. These empirical parameterizations oftentimes 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-specific parameter adjustments while successfully describing the observed range of planetary temperatures across the Solar System.

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]. The 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 (s), hour (h), 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 flux 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}. Thinking 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 flux and
pressure reveals that a flux 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}]. Thus,
a radiative flux 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=cPph. Since c is constant
within a medium, varying the intensity of electromagnetic radiation in a
given medium effectively means altering the pressure of photons. Thus,
the solar radiation reaching Earth’s upper atmosphere exerts a pressure
(force) of sufficient magnitude to perturb the orbits of communication
satellites over time [25,26].

The simplifying power of DA in model development stems from the Buckingham Pi Theorem [27], which states that a problem involving n dimensioned xi variables, i.e.,

can be reformulated into a simpler relationship of (n-m)
dimensionless π_{i} products derived from xi, i.e.,

where m is the number of fundamental dimensions comprising
the original variables. This theorem determines the number of nondimensional
π_{i} 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 defining a particular problem. In other words, there might be,
and oftentimes is more than one set of (n-m) dimensionless products to
analyze. DA provides an objective method for constructing the sets of
π_{i} variables employing simultaneous equations solved via either matrix
inversion or substitution [22].

The second step of DA (after the construction of dimensionless
products) is to search for a functional relationship between the π_{i} 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 final 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 fit; (b) has a small standard error;
(c) displays high predictive skill over a broad range of input data; and
(d) is statistically robust. The regression coefficients of the final 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 identified seven physical
variables of potential relevance to the global surface temperature: 1) topof-
the-atmosphere (TOA) solar irradiance (S); 2) mean planetary surface
temperature in the absence of atmospheric greenhouse effect, hereto
called a reference temperature (Tr); 3) near-surface partial pressure
of atmospheric greenhouse gases (P_{gh}); 4) near-surface mass density
of atmospheric greenhouse gases (ρ_{gh}); 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 P_{x} and ρ_{x}, 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. The planetary Bond albedo
(α_{p}) was omitted as a forcing variable in our DA despite its known effect
on the surface energy budget, because it is already dimensionless and
also partakes in the calculation of reference temperatures discussed
below.

Planetary Variable | Symbol | SI Units | Physical Dimension |
---|---|---|---|

Global mean annual near-surface temperature (GMAT), the dependent variable | T_{s} |
K | [Θ] |

Stellar irradiance (average shortwave flux incident on a plane perpendicular to the stellar rays at the top of a planet’s atmosphere) | S | W m^{-2} |
[M T^{-3}] |

Reference temperature (the planet’s mean surface temperature in the absence of an atmosphere or an atmospheric greenhouse effect) | T_{r} |
K | [Θ] |

Average near-surface gas pressure representing either partial pressure of greenhouse gases or total atmospheric pressure | P_{x} |
Pa | [M L^{-1} T^{-2}] |

Average near-surface gas density representing either greenhouse-gas density or total atmospheric density | P_{x} |
kg m^{-3} |
[M L^{-3}] |

Reference pressure (the minimum atmospheric pressure required a liquid solvent to exists at the surface) | P_{r} |
Pa | [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].

Appendix A details the procedure employed to construct the π_{i} variables. DA yielded two sets of π_{i} products, each one consisting of two dimensionless variables, i.e.,

and

This implies an investigation of two types of dimensionally homogeneous functions (relationships):

and

(2)

Note that π_{1}=T_{s}/T_{r} occurs as a dependent variable in both
relationships, since it contains the sought temperature T_{s}. Upon
replacing the generic pressure/density variables P_{x} and ρ_{x} in functions
(1) and (2) with either partial pressure/density of greenhouse gases (P_{gh} and ρ_{gh}) or total atmospheric pressure/density (P and ρ), one arrives
at six prospective regression models. Further, as explained further, we
employed two distinct kinds of reference temperature computed from
different formulas, i.e., an effective radiating equilibrium temperature
(Te) and a mean ‘no-atmosphere’ spherical surface temperature (T_{na})
(**Table 1**). This doubled the π_{i} 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 effect; hence, Tr is different for each body and
depends on solar irradiance and surface albedo. The purpose of Tr is
to provide a baseline for quantifying the thermal effect of planetary
atmospheres. Indeed, the T_{s}/T_{r} ratio produced by DA can physically be
interpreted as a Relative Atmospheric Thermal Enhancement (RATE)
ideally expected to be equal to or greater than 1.0. Expressing the
thermal effect of a planetary atmosphere as a non-dimensional quotient
instead of an absolute temperature difference (as done in the past)
allows for an unbiased comparison of the greenhouse effects of celestial
bodies orbiting at different distances from the Sun. This is because the
absolute strength of the greenhouse effect (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 effect of planetary
atmospheres.

Two methods have been proposed thus far for estimating the average surface temperature of a planetary body without the greenhouse effect, both based on the SB radiation law. The first and most popular approach uses the planet’s global energy budget to calculate a single radiating equilibrium temperature Te (also known as an effective emission temperature) from the average absorbed solar flux [6,9,28], i.e.,

(3)

Here, S is the solar irradiance (W m^{-2}) defined as the TOA
shortwave flux incident on a plane perpendicular to the incoming rays,
α_{p} 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. The term S(1-α_{p} )⁄4
represents a globally averaged shortwave flux absorbed by the planetatmosphere
system. The rationale behind Eq. (3) is that the TOA energy
balance presumably defines 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 effect of planetary atmospheres [2-4,6,9,28].
However, Volokin et al. [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 T_{s}.

The second method attempts to estimate the average surface
temperature of a planet (T_{na}) 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 flux as done in Eq. (3), this alternative approach first
computes the equilibrium temperature at every point on the surface of
an airless planet from the local absorbed shortwave flux using the SB
relation, and then spherically integrates the resulting temperature field
to produce a global temperature mean. While algorithmically opposite
to Eq. (3), this method mimics well the procedure for calculating Earth’s
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 et al. [1] improved upon Rubincam’s formulation by deriving a closed-form integral expression that explicitly accounts for the effect of subterranean heat storage, cosmic microwave background radiation (CMBR) and geothermal heating on the average global surface temperature of airless bodies. The complete form of their analytic Spherical Airless- Temperature (SAT) model reads:

(4a)

where αe is the effective shortwave albedo of the surface, η_{e} is the
effective ground heat storage coefficient in a vacuum, R_{c}=σ 2.725^{4}=3.13
× 10-6 W m^{-2} is the CMBR [35], and Rg is the spatially averaged
geothermal flux (W m^{-2}) emanating from the subsurface. The heat
storage term η_{e} is defined as a fraction of the absorbed shortwave flux
conducted into the subsurface during daylight hour and subsequently
released as heat at night.

Since the effect of CMBR on T_{na} is negligible for S>0.15 W m^{-2} [1]
and the geothermal contribution to surface temperatures is insignificant
for most planetary bodies, one can simplify Eq. (4a) by substituting
R_{c}=R_{g}=0 This produces:

(4b)

where 0.932=0.754^{0.25}. The complete formula (4a) must only be used
if S ≤ 0.15 W m^{-2} and/or the magnitude of Rg is significantly 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 et al. [1] verified Equations (4a) and
(4b) against Moon temperature data provided by the NASA Diviner
Lunar Radiometer Experiment [29,36]. These authors also showed that
accounting for the subterranean heat storage (η_{e}) markedly improves
the physical realism and accuracy of the SAT model compared to the
original formulation by Rubincam [34].

The conceptual difference between Equations (3) and (4b) is that Τ_{e} represents the equilibrium temperature of a blackbody disk orthogonally
illuminated by shortwave radiation with an intensity equal to the average
solar flux absorbed by a sphere having a Bond albedo α_{p}, while Τna is the
area-weighted average temperature of a thermally heterogeneous airless
sphere [1,37]. In other words, for spherical objects, Τ_{e} is an abstract
mathematical temperature, while T_{na} is the average kinetic temperature
of an airless surface. Due to Hölder’s inequality between integrals, one
always finds Τ_{e}>>Τna when using equivalent values of stellar irradiance
and surface albedo in Equations (3) and (4b) [1].

To calculate the T_{na} 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. This 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, η_{e}=0.00971 and
ε=0.98 [1,29]. This produces:

(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., P_{r}=611.73P_{a} [38] defining a baric threshold, below which water
can only exists in a solid/vapor phase and not in a liquid form. The results
of our analysis are not sensitive to the particular choice of a referencepressure
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 π_{i} variables generated
by DA be subjected to regression analyses. As explained in Appendix A,
twelve pairs of π_{i} variables hereto called Models were investigated. In
order to ease the curve fitting and simplify the visualization of results,
we utilized natural logarithms of the constructed π_{i} variables rather than
their absolute values, i.e., we modeled the relationship In (π_{1})=f (In(π_{2}))
nstead of π_{1}=f(π_{2}) In doing so we focused on monotonic functions of
conservative shapes such as exponential, sigmoidal, hyperbolic, and
logarithmic, for their fitting coefficients might be interpretable in
physically meaningful terms. A key advantage of this type of functions
(provided the existence of a good fit, of course) is that they also tend
to yield reliable results outside the data range used to determine their
coefficients. We specifically avoided non-monotonic functions such as
polynomials because of their ability to accurately fit almost any dataset
given a sufficiently large number of regression coefficients while at the
same time showing poor predictive skills beyond the calibration data
range. Due to their highly flexible shape, polynomials can easily fit
random noise in a dataset, an outcome we particularly tried to avoid.

The following four-parameter exponential-growth function was found to best meet our criteria:

(5)

where x=In π_{2} (and y=In π_{1}) are the independent and dependent
variable respectively while a,b,c and d are regression coefficients. This
function has a rigid shape that can only describe specific exponential
patters found in our data. Equation (5) was fitted to each one of the 12
planetary data sets of logarithmic π_{i} pairs suggested by DA using the
standard method of least squares. The skills of the resulting regression
models were evaluated via three statistical criteria: coefficient of
determination (R^{2}), adjusted R^{2}, and standard error of the estimate
(σ_{est}) [39,40]. All calculations were performed with the SigmaPlot^{™} 13
graphing and analysis software.

**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 defined in terms of TOA solar irradiance and atmospheric properties. This 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 effort to minimize
the effect 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. Thus, 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
Saturn’s 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
Titan’s 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) spacecraft to derive representative
estimates of GMAT and atmospheric surface pressure for the Red Planet
(**Table 2**). 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
(**Table 3**).

Parameter | Venus | Earth | Moon | Mars | Titan | Triton |
---|---|---|---|---|---|---|

Average distance to the Sun, (AU) | 0.7233 | 1.0 | 1.0 | 1.5237 | 9.582 | 30.07 |

Average TOA solar irradiance, (W m-2) | 2,601.3 | 1,360.9 | 1,360.9 | 586.2 | 14.8 | 1.5 |

Bond albedo, (decimal fraction) | 0.900 | 0.294 | 0.136 | 0.235 | 0.265 | 0.650 |

Average absorbed shortwave radiation, (W m-2) | 65.0 | 240.2 | 294.0 | 112.1 | 2.72 | 0.13 |

Global average surface atmospheric pressure, (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, (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 CO_{2}3.48 N _{2
} 0.02 SO2 |
77.89 N_{2} 20.89 O _{2
} 0.932 Ar 0.248 H _{2}O0.040 CO _{2} |
26.7 4He 26.7 20Ne 23.3 H _{2}20.0 40Ar 3.3 22Ne |
95.32 CO_{2} 2.70 N _{2} 1.60 Ar 0.13 O _{2}0.08 CO 0.021 H _{2}O |
95.1 N_{2
} 4.9 CH_{4} |
99.91 N_{2}0.060 CO 0.024 CH _{4} |

Molar mass of the lower atmosphere, (kg mol^{-1}) |
0.0434 | 0.0289 | 0.0156 | 0.0434 | 0.0274 | 0.0280 |

GMAT, (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 Section 2.5 for details about the computational methods employed for some parameters.

Planetary Body | Information Sources |
---|---|

Venus |
[32-48] |

Earth |
[12,13,32,49-55] |

Moon |
[1,29,32,48,56-59] |

Mars |
[32,48,60-63] |

Titan |
[32,41-43,64-72] |

Triton |
[48,73-75] |

**Table 3:** Literature sources of the planetary data presented in Table 2.

The mean solar irradiances of all bodies were calculated as S=S_{E} r_{au}^{-2} where r_{au} is the body’s average distance (semi-major axis) to the Sun
(AU) and S_{E}=1,360.9 W m^{-2} is the Earth’s new lower irradiance at 1 AU
according to recent satellite observations reported by Kopp and Lean
[49]. Due to a design flaw 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 to our analysis. Thus, 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 summerfall
season (Appendix B). Therefore, the Red Planet’s GMAT must be
lower than 207 K. The Viking records also indicate that average diurnal
temperatures above 210 K can only occur on Mars during summertime.
Hence, all such values must be significantly higher than the actual mean annual temperature at any Martian latitude. This is also supported by
results from a 3-D global circulation model of the Red Planet obtained
by Fenton et al. [82]. The 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 significantly 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].

Moon’s 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 Moon’s 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-flow measurements conducted at depths between 80 and 140 cm at the Apollo 15 and 17 landing sites located at 26oN; 3.6° E and 20° N; 30.6° E, respectively [88]. Due to a strong temperature dependence of the lunar regolith thermal conductivity in the topmost 1-2 cm soil, the Moon’s 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]. The diurnal temperature fluctuations 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 Moon’s global thermal environment at these depths. On the other hand, frequently quoted Moon global temperatures of ~270 K are actually calculated from Eq. (3) and not based on surface measurements. However, as demonstrated by Volokin et al. [1], Eq. (3) overestimates the mean global surface temperature of spheres by about 37%. In this study, we employed the spherical estimate of Moon’s GMAT (197.35 K) obtained by Volokin et al. [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 often loosely quoted as 15 C (~288 K) [2-4,6]. However, observations archived in the HadCRUT4 dataset of the UK Met Office’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.

The average near-surface atmospheric densities (p, kg m^{-3})
of planetary bodies were calculated from reported means of total
atmospheric pressure (P), molar mass (M, kg moL^{-1}) and temperature
(T_{s}) using the Ideal Gas Law, i.e.,

(6)

where R=8.31446 J moL^{-1} K^{-1} is the universal gas constant. This
calculation was intended to make atmospheric densities physically
consistent with independent data on pressure and temperature utilized
in our study. The resulting p values were similar to previously published
data for individual bodies. Standard errors of the air-density estimates
were calculated from reported errors of and Τ_{s} for each body using Eq.
(6).

Data in **Table 2** were harnessed to compute several intermediate
variables and all dimensionless π_{i} products necessary for the regression
analyses. The results from these computations are shown in **Table 4**.
Greenhouse gases in planetary atmospheres represented by the major
constituents carbon dioxide (CO_{2}), methane (CH_{4}) and water vapor (H_{2}O) were collectively quantified via three bulk parameters: average
molar mass (Mgh, kg moL^{-1}), combined partial pressure (P_{gh}, Pa) and
combined partial density (ρ_{gh}, kg m^{-3}). These 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 (**Table 4**):

Intermediate Variable or Dimensionless Product | Venus | Earth | Moon | Mars | Titan | Triton |
---|---|---|---|---|---|---|

Average molar mass of greenhouse gases, (kg mol-1) (Eq. 7) | 0.0440 | 0.0216 | 0.0 | 0.0440 | 0.0160 | 0.0160 |

Near-surface partial pressure of greenhouse gases, (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, (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, (K) (Eq. 3) | 185.0 | 256.4 | 269.7 | 211.9 | 83.6 | 39.2 |

Average airless spherical temperature, (K) (Eq. 4c) | 231.7 | 197.0 | 197.0 | 159.6 | 63.6 | 35.9 |

T_{s}/ T_{e} |
3.985 ± 0.016 | 1.121 ± 0.002 | 0.732 ± 0.003 | 0.899 ± 0.003 | 1.120 ± 0.008 | 0.994 ± 0.026 |

T_{s}/ T_{e} |
3.181 ± 0.013 | 1.459 ± 0.002 | 1.002 ± 0.004 | 1.194 ± 0.004 | 1.473 ± 0.011 | 1.086 ± 0.028 |

In(T_{s}/T_{na} |
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 |

In(P_{gh} 3/P_{gh} S^{2} |
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 |

In[P3/P_{gh} S^{2}] |
28.1364 | 8.4784 | Undefined | 10.7520 | 23.1644 | -4.7981 |

In[P_{gh} 3/p S^{2}] |
28.2433 | 26.0283 | +∞ | 10.8304 | 32.2122 | 20.2065 |

In[P_{gh} 3/p S^{2}] |
28.1145 | 2.3370 | Undefined | 10.7396 | 19.6102 | -13.6926 |

In[P_{gh}/pr] |
9.5936 | -0.7679 | Undefined | 0.0876 | 2.4639 | -13.3649 |

In[P3/p S^{2}] |
28.2214 | 19.8869 | -46.7497 | 10.8180 | 28.6580 | 11.3120 |

In[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., P_{r}=611.73 Pa.

(7)

(8)

(9)

where Cgh=CCO_{2}+CCH_{4}+CH_{2}O is the total volumetric concentration
of major greenhouse gases (%). The reference temperatures Τ_{e} and Τna
were calculated from Equations (3) and (4c), respectively.

Function (5) was fitted to each one of the 12 sets of logarithmic π_{i} 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 R^{2}=0.9999 and the lowest standard error
σ_{est}=0.0078 among all regressions. Model 1 (**Figure 1a**) provided the
second best fit with R^{2}=0.9844 and σ_{est}=0.1529 Notably, Model 1 shows
almost a 20-time larger standard error on the logarithmic scale than
Model 12. **Figure 3** illustrates the difference 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. The statistical results shown in **Table 5** indicate
that the explanatory power and descriptive accuracy of Model 12
surpass these of all other models by a wide margin.

**Figure 1:** The relative atmospheric thermal enhancement (T_{s}/T_{r}) 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 (T_{r}) defined as either T_{r}/T_{e} (left) or Tr/Tna (right).

No. | Functional Model | Coefficient of Determination (R2) | Adjusted R2 | Standard Error |
---|---|---|---|---|

1 | 0.9844 | 0.9375 | 0.1529 | |

2 | 0.9562 | 0.8249 | 0.1773 | |

3 | 0.1372 | -2.4511 | 1.1360 | |

4 | 0.2450 | -2.0200 | 0.7365 | |

5 | 0.9835 | 0.9339 | 0.1572 | |

6 | 0.9467 | 0.7866 | 0.1957 | |

7 | 0.9818 | 0.9274 | 0.1648 | |

8 | 0.9649 | 0.8598 | 0.1587 | |

9 | 0.4488 | -0.3780 | 0.7060 | |

10 | 0.6256 | 0.0639 | 0.4049 | |

11 | 0.9396 | 0.8489 | 0.2338 | |

12 | 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.

Since Titan and Earth nearly overlap on the logarithmic scale of **Figure 2f**, we decided to experiment with an alternative regression for
Model 12, which excludes Titan from the input dataset. This new curve
had R2=1.0 and σ_{est}=0.0009. Although the two regression equations yield similar results over most of the relevant pressure range, we chose
the one without Titan as final for Model 12 based on the assumption
that Earth’s GMAT is likely known with a much greater accuracy than
Titan’s mean annual temperature. Taking an antilogarithm of the
final regression equation, which excluded Titan, yields the following
expression for Model 12:

(10a)

The regression coefficients in Eq. (10a) are intentionally shown in
full precision to allow an accurate calculation of RATE (i.e., the T_{s}/T_{na} ratios) provided the strong non-linearity of (**Figures 1-3** and **Table 5**)
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.

**Figure 4:** The relative atmospheric thermal enhancement (T_{s}/T_{na} 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.

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 (T_{na}, K) and a nondimensional
factor (Ea ≥ 1.0) quantifying the relative thermal effect of
the atmosphere, i.e.,

(10b)

where Τna is obtained from the SAT model (Eq. 4a) and Ea is a function of total pressure (P) given by:

(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, Mars,
Titan and Triton), one must calculate T_{na} using αe=0.132 and η_{e}=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), T_{na} should be calculated from
Eq. (4a) (or Eq. 4b respectively if S>0.15 W m^{-2} and/or R_{g} ≈ 0 W m^{-2})
using the body’s observed values of Bond albedo αe and ground heat
storage fraction η_{e}. In the context of this model, a tangible atmosphere
is defined as one that has significantly modified the optical and
thermo-physical properties of a planet’s 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 influence on
the surface albedo and regolith thermos-physical properties and is
completely transparent to shortwave radiation. The need for such
delineation of atmospheric masses when calculating T_{na} 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 differences in albedos (i.e.,
from 0.235 to 0.90) while assuming constant values of αe and η_{e} 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 effects of
albedo and ground heat storage on GMAT become implicitly accounted
for by Eq. (11). Although this hypothesis requires an investigation
beyond the scope of the present study, one finds 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 different
albedos and heat storage coefficients 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 different
aspects of the same phenomenon. An example is the distinction made
in fluid mechanics between laminar and turbulent flow, where the nondimensional
Reynold’s number is employed to separate the two regimes
that are subjected to different mathematical treatments.

We do not currently have sufficient data to precisely define 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 significantly below 1.0 Pa. In this
study, we use 0.01 Pa as a tentative threshold value. Thus, in the context
of Eq. (10b), we recommend computing T_{na} from Eq. (4c) if P>10-2Pa,
and from Eq. (4a) (or Eq. 4b, respectively) using observed values of αe
and η_{e} if P ≤ 10-2Pa. Equation (4a) should also be employed in cases, where a significant geothermal flux Rg>>0 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 five factors: TOA stellar irradiance (S),
a reference airless surface albedo (αe), a reference airless ground heat
storage fraction (η_{e}), the average geothermal flux reaching the surface
(R_{g}), and the total surface atmospheric pressure (P). For planets with tangible atmospheres (P>10-2Pa) and a negligible geothermal heating of
the surface (R_{g} ≈ 0), the equilibrium GMAT becomes only a function
of two factors: S and P, i.e., Τ_{s}=32.44S0.25Eα (P). The final model (Eq.
10b) can also be cast in terms of T_{s} as a function of a planet’s distance
to the Sun (r_{au}, AU) by replacing S in Equations (4a), (4b) or (4c) with
1360.9 r_{au}^{-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. The 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 Saturn’s semi-major axis (9.582 AU) corresponding to a solar irradiance S=14.8 W m^{-2}. This
estimate is virtually identical to the 95 K average surface temperature
reported for that moon by the NASA JPL Voyager Mission website
[94]. The Voyager spacecraft 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 Saturn’s 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 T_{s} ≈ 93.4 ± 0.6 K
[42,43]. Using Saturn’s perihelion (9.023 AU) and aphelion (10.05 AU)
one can compute Titan’s 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 ≤ T_{s} ≤ 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 (Saturn’s 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 Saturn’s orbital eccentricity. Thus, 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 fluctuations on that moon. Hence, further observations are
needed to more precisely constrain Titan’s long-term GMAT.

Measurements conducted by the Voyager spacecraft 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 different data for Triton (i.e., T_{s} =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 T_{s}=38.5 K for Triton’s GMAT, a value well within the
uncertainty of the 1989 temperature measurements (**Figure 5**).

The 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 overfitting 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 statistical scores
in **Table 5**; (b) a simple scatter plot of In (P/P_{r}) vs. In(T_{s}/T_{na})visibly
reveals the presence of an exponential relationship free of data noise;
and (c) no polynomial can fit 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.
These facts indicate that Eq. (5) is not too complicated to cause an over
fitting but just right for describing the data at hand.

The fact that only one of the investigated twelve non-linear regressions yielded a tight relationship suggests that Model 12 describes 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 defines the degree to which a statistical
relationship would hold when recalculated using a different 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 T_{s}/T_{na} and P/P_{r} for Venus, the Moon,
Mars and Triton from **Table 4**. The 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 defined. As in all previous cases, function (5) was fitted to the
incomplete dataset (omitting Earth and Titan), which yielded the
following expression:

(12a)

Substituting the reference temperature T_{na} in Eq. (12a) with its
equivalent from Eq. (4c) and solving for T_{s} produces

(12b)

It is evident that the regression coefficients in the first exponent
term of Eq. (12a) are nearly identical to those in Eq. (10a). This
term dominates the T_{s}-Prelationship over the pressure range 0-400
kPa accounting for more than 97.5% of the predicted temperature
magnitudes. The regression coefficients of the second exponent differ
somewhat between the two formulas causing a divergence of calculated
RATE values over the pressure interval 400 –9,100 kPa. The 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 (**Figure 6**).

**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.

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). The ability
of Model 12 to predict Earth’s 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) Earth’s temperature is a part of a cosmic
thermodynamic continuum well described by Eq. (10b). The apparent
smoothness of this continuum for bodies with tangible atmospheres
(illustrated in **Figure 4**) suggests that planetary climates are wellbuffered
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. This robustness test also serves as a cross-validation
suggesting that the new model has a universal nature and is not a
product of over fitting.

The 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 potential theoretical significance deserving further investigation. This conclusive result is also supported by the physical meaningfulness of the response curve described by Eq. (10a).

Given the high statistical scores of the new model (Eq. 10b) discussed above, it is important to address its physical significance, potential limitations, and broad implications for the current climate theory.

**Similarity of the new model to Poisson’s formula and the SB
radiation law**

The 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
Thermodynamics and the Ideal Gas Law [4], i.e.,

**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).

(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 Earth’s atmosphere, R/cp=0.286. Equation (13)
essentially describes the direct effect of a pressure p on 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 ≤ 1.0Eq.
(10a) always yields T_{s}/T_{na} ≥ 1.0. The key difference 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
pressure. In essence, Eq. (10b) could be viewed as a predictor of the
reference temperature To in the Poisson formula. Thus, while qualitatively
similar, Equations (10a) and (13) are quantitatively rather different. Both
functions describe effects of pressure on temperature but in the context of
disparate physical systems. Therefore, 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 profiles 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 (**Figure 7**).

Equation (10a) also shows remarkable similarity to the SB law
relating the equilibrium skin temperature of an isothermal blackbody
(Tb, K) to the electromagnetic radiative flux (I, W m^{-2}) absorbed/
emitted by the body’s surface, i.e., T_{b}=(I ⁄ σ)^{0.25}. Dividing each side of this
fundamental relationship by the irreducible temperature of deep Space
T_{c}=2.725 K and its causative CMBR R_{c}=3.13 × 10^{-6} W m^{-2} respectively,
yields T_{b}⁄T_{c} =(I ⁄ R_{c} )^{0.25}. Further, expressing the radiative fluxes I and R_{c} 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 R_{c}=cP_{c} , leads to the
following alternative form of the SB law:

(14)

where P_{c}=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 ratio on photon pressure P_{ph}.

It is evident from **Figures 4 and 7** that formulas (10a), (13) and (14)
describe qualitatively very similar responses in quantitatively vastly
different systems. The 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 (defined 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. The direct effect
of pressure on a system’s temperature is thermodynamically described
by adiabatic processes. The pressure-induced thermal enhancement
on a planetary level portrayed in **Figure 4** and accurately quantified by
Eq. (10a or 11) is analogous to a compression heating, but not fully
identical to an adiabatic process. The latter is usually characterized by
a limited duration and oftentimes only applies to finite-size parcels of
air moving vertically through the atmosphere. Equation (11), on the
other hand, describes a surface thermal effect that is global in scope and
permanent in nature as long as an atmospheric mass is present within
the planet’s gravitational field. Hence, the planetary RATE (T_{s}/T_{na} 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 flux 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 identified
and may provide a valuable new perspective on the working of planetary
climates.

Nevertheless, important differences exist between Eq. (10a) and these other simpler pressure-temperature relations. Thus, while the Poisson formula and the SB radiation law can mathematically be derived from ‘first 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. This 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 the long-term (30 years) equilibrium GMATs of planetary bodies and does not predict inter-annual global temperature variations caused by intrinsic fluctuations of cloud albedo and/or ocean heat uptake. Thus, the observed 0.82 K rise of Earth’s global temperature since 1880 is not captured by our model, since this warming was likely 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 shift of Earth’s 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 significant drop in surface air pressure since the early Eocene. Pleistocene fluctuations 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.

The 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 icealbedo
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 [103-107]. The
only significant forcing remaining in the present paleo-climatological
toolbox to explicate the Pleistocene cycles are variations in greenhousegas
concentrations. Hence, it is difficult to explain, from a standpoint
of the current climate theory, the high accuracy of Eq. (11) describing
the relative thermal effect 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 Earth’s climate on geological time scales?
We address the potential role of greenhouse gases in more 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 amplification
during the Last Glacial Maximum indicate that such a hypothesis is not
unreasonable. However, conclusive findings from this research will be
discussed elsewhere.

According to the present understanding, Earth’s atmospheric pressure has remained nearly invariant during the Cenozoic era (i.e., 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. The 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 study, allowing for the possibility that atmospheric pressure on Earth might have varied significantly 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**). This 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]. The
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. Therefore,
the higher the pressure, the larger the infrared optical depth of an
atmosphere, and the stronger the expected greenhouse effect would
be. According to the present climate theory, pressure only indirectly
affects global surface temperature through the atmospheric infrared
opacity and its presumed constraint on the planet’s LW emission to
Space [9,107].

There are four plausible explanations for the apparent lack of a close relationship between GMAT and atmospheric greenhouse gasses in our results: 1) The 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) The analysis lumped all greenhouse gases together and did not take into account differences in the infrared spectral absorptivity of individual gasses; 3) The effect 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 effect 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 planet’s surface temperature rather than the atmospheric infrared optical depth, which merely correlates with temperature due to its co-dependence on pressure.

Based on evidences, 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 definition,
the infrared optical depth of an atmosphere is a dimensionless
quantity that carries no units of force or energy [3,4,9]. Therefore, it
is difficult 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) defined as the product of gas pressure (P,
Pa) and gas volume (V, m^{3}), i.e., E (J)=PV. Hence, the direct effect of
pressure on a system’s internal energy and temperature follows straight
from fundamental parameter definitions in classical thermodynamics.
Generally speaking, kinetic energy cannot exist without a pressure
force. Even electromagnetic radiation has pressure.

In climate models, the effect of infrared optical depth on surface temperature is simulated by mathematically decoupling radiative transfer from convective heat exchange. Specifically, the LW radiative transfer is computed in these models without simultaneous consideration of sensible- and latent heat fluxes 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 greenhousegas concentrations. These 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. This 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. This 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 differences in the case of sensible heat flux and on simple vapor pressure gradients in the case of latent heat flux [4]. The latent heat transport is in turn a function of a solvent’s saturation vapor pressure, which increases exponentially with temperature [3]. Thus, the superposition principle cannot be employed in energy budget calculations. The artificial decoupling between radiative and convective heat-transfer processes adopted in climate models leads to mathematically and physically incorrect solutions with regard to surface temperature. The 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 efficient 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 different temperature solutions in response to a changing atmospheric composition than the ones obtained by current climate models. Specifically, 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. This topic requires further investigation beyond the scope of the present study.

The direct effect 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.

**Theoretical implications of the new interplanetary
relationship**

The hereto discovered pressure-temperature relationship quantified
by Eq. (10a) and depicted in **Figure 4** has broad theoretical implications
that can be summarized as follows:

**Physical nature of the atmospheric ‘greenhouse effect’:** According
to Eq. (10b), the heating mechanism of planetary atmospheres is
analogous to a gravity-controlled adiabatic compression acting upon
the entire surface. This 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 effect’ is a pressure-induced thermal enhancement
independent of atmospheric composition. This mechanism is
fundamentally different from the hypothesized ‘trapping’ of LW
radiation by atmospheric trace gases first 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 effect 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. The latter has later
been coded into algorithms that describe the surface temperature as a
function of atmospheric infrared optical depth (instead of pressure) by
artificially decoupling radiative transfer from convective heat exchange.
Note also that the Ideal Gas Law (PV=nRT) forming the basis of
atmospheric physics is indifferent to the gas chemical composition.

**Effect of pressure on temperature:** Atmospheric pressure
provides in and of itself only a relative thermal enhancement (RATE)
to the surface quantified by Eq. (11). The absolute thermal effect 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 Earth’s atmospheric thermal effect derived by
Volokin et al. [1] using a different 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 effect of the atmosphere would drop to about 29.2 K
due to a vastly reduced solar flux. In other words, the absolute effect of
pressure on a system’s temperature depends on the background energy
level of the environment. This 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 offsetting
the effect 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 largely
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 effect rather
than a cause for it. This explains the broad variation in the size of the
observed down-welling LW flux among celestial bodies irrespective
of the amount of absorbed solar radiation. Therefore, a change in this
thermal flux brought about by a shift 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
opposite shift 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 Thermodynamics. However, current climate models do not simulate this compensatory effect of sensible and latent heat fluxes due
to an improper decoupling between radiative transfer and turbulent
convection in the estimation of total energy exchange.

**Role of planetary albedos:** The 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 reflectivity 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. The hereto proposed
passive nature of planetary albedos does not imply that the global
cloud cover could not be influenced by an external forcing such as solar
wind, galactic cosmic rays, and/or gravitational fields of other celestial
objects. Empirical evidence strongly suggests that it can [108-113], but
the magnitude of such influences 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 reflectivity 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 flux 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 fluctuations 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. This implies a relatively stable climate for a planet
such as Earth absent significant shifts 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
buffered against sudden changes.

**Effect of oceans and water vapor on global temperature:** The new
model shows that the Earth’s 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 Earth’s GMAT. This finding 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. The 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. These 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. Thus, 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 often cited as a product of a ‘runaway
greenhouse’ scenario [117,118] fits perfectly within the pressuredependent
climate continuum described by Equations (10b) and (11).

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 offering an opportunity
to validate our planetary temperature model against independent
measurements.

The MESSENGER spacecraft launched in 2004 completed the first
comprehensive mapping of Mercury in March 2013 (https://messenger.jhuapl.edu/). Among other things, the spacecraft also took infrared
measurements of the planet’s surface using a special spectrometer
[119] that should soon become available. The New Horizons spacecraft
launched in January 2006 [120] reached Pluto in July of 2015 and
performed a thermal scan of the dwarf planet during a flyby. The
complete dataset from this flyby 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 after year 2020. It envisions exploring Europa’s
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. The 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 body’s 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 T_{na} in Eq.
(10b) must be calculated from Eq. (4a), which requires knowledge of
the actual values of αe, η_{e}, and R_{g}. We assumed that Mercury had R_{g}=0.0
W m^{-2}, α=0.068 [123] and Moon-like thermo-physical properties of the
regolith (η_{e}=0.00971). Input data for Europa and Callisto were obtained
from Spencer et al. [124], Moore et al. [125], respectively. Specifically, to
calculate η_{e} 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 a figure along with a theoretical formula for computing
the average nighttime surface temperature T at the equator based on
the SB law, i.e.,

(15)

where S(1-α) η_{e} is the absorbed solar flux (W m^{-2}) stored as heat
into the subsurface. The geothermal heat flux 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 η_{e}=0.085for
Europa. A similar computational procedure was employed for Callisto
using α=0.11 and equatorial surface temperature data in Moore et al.
[125]. This produced Rg=0.5 W m^{-2} and η_{e}=0.057. Using these values
in Eq. (15) correctly reproduced Callisto’s nighttime equatorial surface
temperature of ≈ 86.0 K. The much higher η_{e} estimates for Europa and
Callisto compared to η_{e}=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 et al. [1]
showed that GMAT of airless bodies is highly sensitive to η_{e}.

**Table 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 NASA’s 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 Europa’s GMAT, 99.4 K, agrees well with the ≈ 100
K estimate reported for this moon by Sotin et al. [127]. Our estimate
of Pluto’s 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. [94] reported
initial results from the flyby observations of Pluto taken by the Radio
Experiment (REX) instrument aboard the New Horizons spacecraft 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) our model predicts an average global temperature of 36.7 K for
Pluto. Stern et al. [128] reported a near-surface temperature of ≈ 38
K. However, this value was calculated from pre-flyby 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 Pluto’s true global temperature. We
will know more about this in 2017 when spatially resolved thermal
measurements obtained by New Horizons become available.

Surface Atmospheric Pressure (Pa) | α_{e}(fraction)
η_{e}(fraction)
R_{g} (W m^{-2}) |
Predicted Average Global Surface Temperature at Specific Orbital Distances from the Sun | |||
---|---|---|---|---|---|

Aphelion | Semi-major Axis | Perihelion | |||

Mercury |
5 × 10^{-10} |
α_{e}=0.068η _{e}=0.00971R _{g}=0.0 |
296.8 K (0.459 AU) |
323.3 K (0.387 AU) |
359.5 K (0.313 AU) |

Europa |
10^{-7} |
α_{e}=0.62 η _{e}=0.085R _{g}=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} |
α_{e}=0.11 η _{e}=0.057R _{g}=0.5 |
101.2 K (5.455 AU) |
103.2 K (5.203 AU) |
105.4 K (4.951 AU) |

Pluto |
1.05 | α_{e}=0.132η _{e}=0.00971R _{g}=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. [94]. Solar irradiances required
by Eq. (10b) were calculated from reported orbital distances as explained. Values of α_{e}, η_{e} and R_{g} for Europa and Callisto were estimated from observed data by Spencer
et al. [124] and Moore et al. [125] respectively (see text for details).

One should use caution when comparing results from Eq. (10b) to remotely sensed ‘average temperatures’ commonly quoted for celestial bodies with tenuous atmospheres such as the moons of Jupiter and Neptune. Studies oftentimes report the so-called ‘brightness temperatures’ retrieved at specific wavelengths that have not been subjected to a proper spherical integration. As pointed out by Volokin et al. [1], due to Hölder’s inequality between integrals, calculated brightness temperatures of spherical objects can be significantly 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.

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 flux. This conceptualized continuous absorption and downward reemission of thermal radiation enabled by certain trace gases known to be transparent to solar rays but opaque to electromagnetic long-wavelengths has been likened to the trapping of heat by glass greenhouses, hence the term ‘atmospheric greenhouse effect’. 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 effect’ stuck in science.

The 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. This is because,
in gaseous systems, heat is primarily transferred (dissipated) by
convection (i.e., through fluid motion) rather than radiative exchange.
If gases of high LW absorptivity/emissivity such as CO_{2}, 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 IR absorptivity/emissivity and correspondingly high thermal
reflectivity such as aluminum foil. These 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 were built on
the premise that the atmosphere warms Earth by limiting radiant heat
losses of the surface through to the action of IR absorbing gases aloft.

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. Thus, 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 flux 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]. This fact cannot be explained via re-radiation of absorbed outgoing thermal emissions by gases known to supply no additional energy to the system. The desire to explicate the sizable energy surplus evident in the tropospheres of some terrestrial planets provided the original 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 effect. 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 stellar irradiance and total
surface atmospheric pressure (Eq. 10b with T_{na} computed from Eq. 4c).
Furthermore, the relative atmospheric thermal enhancement (RATE)
defined 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**) quantified by Eq. (10a)
appears to describe an emergent macro-level thermodynamic property
of planetary atmospheres heretofore unbeknown to science. The
physical significance 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 effect of pressure on
temperature albeit in the context of a different macro-physical system.
To our knowledge, this is the first 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.

The planetary temperature model consisting of Equations (4a), (10b), (11) has several fundamental theoretical implications, i.e.,

• The ‘greenhouse effect’ 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;

• The 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;

• The 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. This does not mean that the cloud albedo cannot be influenced by external forcing such as solar wind or galactic cosmic rays. However, the magnitude of such influences is expected to be small due to the stabilizing effect of negative feedbacks operating within the system. This novel understanding explains the observed remarkable stability of planetary albedos;

• The 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 buffered against sudden changes and has no tipping points;

• The 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.

The hereto reported findings point toward the need for a paradigm
shift in our understanding of key macro-scale atmospheric properties and
processes. The 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 scientific 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 (T_{s}, T_{r}, S, P_{x}, P_{r} and ρ_{x}) 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 π_{i} products per
set. To derive the π_{i} variables we employed the following objective
approach. First, we hypothesized that a planet’s GMAT (T_{s}) is a
function of all 5 independent variables listed in **Table 1**, i.e.

(B.1)

This unknown function is described to a first approximation as a simple product of the driving variables raised to various powers, i.e.

(A.2)

where a, b, c, d and e are rational numbers. In order to determine the power coefficients, Eq. (A.2) is cast in terms of physical dimensions of the participating variables, i.e.

(A.3)

Satisfying the requirement for dimensional homogeneity of Eq. (A.2) implies that the sum of powers of each fundamental dimension must be equal on both sides of Eq. (A.3). This allows us to write four simultaneous equations (one per fundamental dimension) containing five unknowns, i.e.

(A.4)

System (A4) is underdetermined and has the following
solution:α=1, b=2e, and c=3e+d. Note that, in the DA methodology,
one oftentimes 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 effect on the derivation of the
sought dimensionless π_{i} products.

Substituting the above roots in Eq. (A.2) reduces the original five unknowns to two: d and e, i.e.

(A.5a)

These solution powers may now be assigned arbitrary values,
although integers such as 0, 1 and -1 are preferable, for they offer the
simplest solution leading to the construction of proper π_{i} variables.
Setting d=0 and e=-1 reduces Eq. (A.5a) to

(A.5B)

providing the first pair of dimensionless products:

(A.6)

The second pair of π_{i} variables emerges upon setting d=-1 and e=0
in Eq. (A.5a), i.e.

(A.7)

Thus, the original function (A.1) consisting of six dimensioned variables has been reduced to a relationship between two dimensionless
quantities, i.e.π_{1}=f (π_{2}). This 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 after 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]. The 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]. The VL1 record covers about half of a Martian year,
while the VL2 series extends to nearly 1.6 years. The 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
(22.5° N). The arithmetic average of the series is 207.3 K (**Fig. B1a**).
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 significant cloud cover and perceptible
water, temperature fluctuations near the surface of Mars are tightly
coupled to diurnal, seasonal and latitudinal variations in incident solar
radiation. This 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 (**Figure B.1**).

**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: https://www-k12.atmos.washington.edu/k12/
resources/mars_data-information/data.html). Black dashed lines mark the arithmetic average (T_{mean}) 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 significantly lower than 210 K.

At 22.5° 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 summerfall
temperature at that latitude. Since the mean surface temperature
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. The mean temperature at the VL2 site located at ~48° N
latitude and 3,000 m below Datum elevation is 191.1 K (**Fig. B.1b**). The
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 42° absolute
latitude, the above calculations suggest that Mars’ GMAT is likely
between 184 K and 201 K.

**Figure B2:** Mean annual surface air temperatures at five 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 represents a third-order polynomial (Eq. B.1) fitted through the latitudinal
temperature means via non-linear regression. Mars’ GMAT, T_{M}=190.56 K (marked
by a horizontal gray dashed line) was calculated via integration of polynomial
(B.1) using formula (B.2).

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 effective radiating
temperature produced by Eq. (3) [78,6], which is T_{e} ≈ 212 K for Mars.
This presumption is rooted in the a priori assumption that Te represents
a planet’s average surface temperature in the absence of atmospheric
greenhouse effect. However, Volokin et al. [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
effective radiating temperature computed from the globally integrated
absorbed stellar flux. 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 effective radiating
temperature. Therefore, 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 the Viking landers and
Curiosity Rover, and remotely by the Mars Global Surveyor (MGS)
spacecraft. The 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 five
absolute Martian latitudes (gray dots) along with their standard errors
(vertical bars). The equatorial MAT was calculated from Curiosity Rover
observations; temperatures at absolute latitudes 0.392 rad (22.48°) and
0.837 rad (47.97°) were derived from VL measurements, while these
at latitudes 1.117 rad (64°) and 1.396 rad (80°) were estimated from
MGS-RST data. The black curve represents a third-order polynomial
fitted through the latitudinal temperature averages and described by the
polynomial:

(B.1)

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 after MGS in
2006 [136]. Shirley et al. [136] showed that, although separated in
time by 2-5 years, MCS temperature profiles 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 profiles over the Mars’ northern and southern Polar
Regions, respectively at about 75° absolute latitude. The average winter
surface temperature between the two Hemispheres for this latitude
is about 148.5 K. This compares favorably with 156.4 K produced by
Eq. (B.1) for 75° (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 profiles retrieved
by MGS-RST and MCS over lowlands (165° – 180° E) and highlands
(240° - 270° E) of the Mars’ equatorial region (8° N - 8° 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:

(B.2)

(**Figure B.2**) where 0 ≤ cosL ≤ 1 is a polar-coordinate area-weighting
factor. The result is TM=190.56 ± 0.7 K (**Figure B.2**). This estimate, while
significantly 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]. Thus, 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 et al. [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 effect [9], it makes a physical sense that the
Red Planet would be on average cooler than the Moon (i.e. TM<197.3K).
Moreover, if the average temperature 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 [6,32,78,80,124,125]. Since this interval was too broad for the target precision of our study, we employed MGSRST 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.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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