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OMICS International

Research Article

Environment Pollution and

Climate Change

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Nikolov and Zeller, Environ Pollut Climate Change 2017, 1:2

Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal

Keywords: Greenhouse eect; Emergent model; Planetary

temperature; Atmospheric pressure; Greenhouse gas; Mars temperature

Introduction

In a recent study Volokin and ReLlez [1] demonstrated that the

strength of Earth’s atmospheric Greenhouse Eect (GE) is about 90 K

instead of 33 K as presently assumed by most researchers [2-7]. e new

estimate corrected a long-standing mathematical error in the application

of the Stefan–Boltzmann (SB) radiation law to a sphere pertaining to

Hölder’s inequality between integrals. Since the current greenhouse

theory strives to explain GE solely through a retention (trapping) of

outgoing long-wavelength (LW) radiation by atmospheric gases [2,5,7-

10], a thermal enhancement of 90 K creates a logical conundrum, since

satellite observations constrain the global atmospheric LW absorption

to 155–158 W m-2 [11-13]. Such a ux might only explain a surface

warming up to 35 K. Hence, more than 60% of Earth’s 90 K atmospheric

eect appears to remain inexplicable in the context of the current theory.

Furthermore, satellite- and surface-based radiation measurements have

shown [12-14] that the lower troposphere emits 42-44% more radiation

towards the surface (i.e. 341-346 W m-2) than the net shortwave ux

delivered to the Earth-atmosphere system by the Sun (i.e. 240 W m-2).

In other words, the lower troposphere contains signicantly more

kinetic energy than expected from solar heating alone, a conclusion also

supported by the new 90 K GE estimate. A similar but more extreme

situation is observed on Venus as well, where the atmospheric down-

welling LW radiation near the surface (>15,000 W m-2) exceeds the

total absorbed solar ux (65–150 W m-2) by a factor of 100 or more [6].

e radiative greenhouse theory cannot explain this apparent paradox

considering the fact that infrared-absorbing gases such as CO2, water

vapor and methane only re-radiate available LW emissions and do not

constitute signicant heat storage or a net source of additional energy to

the system. is raises a fundamental question about the origin of the

observed energy surplus in the lower troposphere of terrestrial planets

with respect to the solar input. e above inconsistencies between theory

and observations prompted us to take a new look at the mechanisms

controlling the atmospheric thermal eect.

We began our study with the premise that processes controlling

the Global Mean Annual near-surface Temperature (GMAT) of Earth

are also responsible for creating the observed pattern of planetary

temperatures across the Solar System. us, our working hypothesis was

that a general physical model should exist, which accurately describes

equilibrium GMATs of planets using a common set of drivers. If true,

such a model would also reveal the forcing behind the atmospheric

thermal eect.

Instead of examining existing mechanistic models such as 3-D

*Corresponding author: Ned Nikolov, Ksubz LLC, 9401 Shooy Lane, Wellington

CO 80549, USA, Tel: 970-980-3303, 970-206-0700; E-mail: ntconsulting@comcast.net

Received November 11, 2016; Accepted February 06, 2017; Published February

13, 2017

Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the

Atmospheric Greenhouse Effect Deduced from an Empirical Planetary Temperature

Model. Environ Pollut Climate Change 1: 112.

Copyright: © 2017 Nikolov N, et al. This is an open-access article distributed under

the terms of the Creative Commons Attribution License, which permits unrestricted

use, distribution, and reproduction in any medium, provided the original author and

source are credited.

Abstract

A recent study has revealed that the Earth’s natural atmospheric greenhouse effect is around 90 K or about 2.7 times stronger

than assumed for the past 40 years. A thermal enhancement of such a magnitude cannot be explained with the observed amount

of outgoing infrared long-wave radiation absorbed by the atmosphere (i.e. ≈ 158 W m-2), thus requiring a re-examination of the

underlying Greenhouse theory. We present here a new investigation into the physical nature of the atmospheric thermal effect using a

novel empirical approach toward predicting the Global Mean Annual near-surface equilibrium Temperature (GMAT) of rocky planets

with diverse atmospheres. Our method utilizes Dimensional Analysis (DA) applied to a vetted set of observed data from six celestial

bodies representing a broad range of physical environments in our Solar System, i.e. Venus, Earth, the Moon, Mars, Titan (a moon

of Saturn), and Triton (a moon of Neptune). Twelve relationships (models) suggested by DA are explored via non-linear regression

analyses that involve dimensionless products comprised of solar irradiance, greenhouse-gas partial pressure/density and total

atmospheric pressure/density as forcing variables, and two temperature ratios as dependent variables. One non-linear regression

model is found to statistically outperform the rest by a wide margin. Our analysis revealed that GMATs of rocky planets with tangible

atmospheres and a negligible geothermal surface heating can accurately be predicted over a broad range of conditions using

only two forcing variables: top-of-the-atmosphere solar irradiance and total surface atmospheric pressure. The hereto discovered

interplanetary pressure-temperature relationship is shown to be statistically robust while describing a smooth physical continuum

without climatic tipping points. This continuum fully explains the recently discovered 90 K thermal effect of Earth’s atmosphere. The

new model displays characteristics of an emergent macro-level thermodynamic relationship heretofore unbeknown to science that

has important theoretical implications. A key entailment from the model is that the atmospheric ‘greenhouse effect’ currently viewed

as a radiative phenomenon is in fact an adiabatic (pressure-induced) thermal enhancement analogous to compression heating

and independent of atmospheric composition. Consequently, the global down-welling long-wave ux presently assumed to drive

Earth’s surface warming appears to be a product of the air temperature set by solar heating and atmospheric pressure. In other

words, the so-called ‘greenhouse back radiation’ is globally a result of the atmospheric thermal effect rather than a cause for it. Our

empirical model has also fundamental implications for the role of oceans, water vapour, and planetary albedo in global climate. Since

produced by a rigorous attempt to describe planetary temperatures in the context of a cosmic continuum using an objective analysis

of vetted observations from across the Solar System, these ndings call for a paradigm shift in our understanding of the atmospheric

‘greenhouse effect’ as a fundamental property of climate.

New Insights on the Physical Nature of the Atmospheric Greenhouse

Effect Deduced from an Empirical Planetary Temperature Model

Ned Nikolov* and Karl Zeller

Ksubz LLC, 9401 Shooy Lane, Wellington CO 80549, USA

Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary

Temperature Model. Environ Pollut Climate Change 1: 112.

Page 2 of 22

Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal

GCMs, we decided to try an empirical approach not constrained by a

particular physical theory. An important reason for this was the fact that

current process-oriented climate models rely on numerous theoretical

assumptions while utilizing planet-specic parameterizations of key

processes such as vertical convection and cloud nucleation in order

to simulate the surface thermal regime over a range of planetary

environments [15]. ese empirical parameterizations oentimes

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-specic parameter adjustments while

successfully describing the observed range of planetary temperatures

across the Solar System.

Methods and Data

In our model development we employed a ‘top-down’ empirical

approach based on Dimensional Analysis (DA) of observed data

from our Solar System. We chose DA as an analytic tool because of

its ubiquitous past successes in solving complex problems of physics,

engineering, mathematical biology, and biophysics [16-21]. To our

knowledge DA has not previously been applied to constructing

predictive models of macro-level properties such as the average global

temperature of a planet; thus, the following overview of this technique

is warranted.

Dimensional analysis background

DA is a method for extracting physically meaningful relationships

from empirical data [22-24]. e goal of DA is to restructure a set of

original variables deemed critical to describing a physical phenomenon

into a smaller set of independent dimensionless products that may be

combined into a dimensionally homogeneous model with predictive

power. Dimensional homogeneity is a prerequisite for any robust

physical relationship such as natural laws. DA distinguishes between

measurement units and physical dimensions. For example, mass is a

physical dimension that can be measured in gram, pound, metric ton

etc.; time is another dimension measurable in seconds, hours, years,

etc. While the physical dimension of a variable does not change, the

units quantifying that variable may vary depending on the adopted

measurement system.

Many physical variables and constants can be described in terms of four

fundamental dimensions, i.e. mass [M], length [L], time [T], and absolute

temperature [Θ]. For example, an energy ux commonly measured in W

m-2 has a physical dimension [M T-3] since 1 W m-2 = 1 J s-1 m-2 = 1 (kg m2

s-2) s-1 m-2 = kg s-3. Pressure may be reported in units of Pascal, bar, atm.,

PSI or Torr, but its physical dimension is always [M L-1 T-2] because 1 Pa

= 1 N m-2 = 1 (kg m s-2) m-2 = 1 kg m-1 s-2. inking in terms of physical

dimensions rather than measurement units fosters a deeper understanding

of the underlying physical reality. For instance, a comparison between

the physical dimensions of energy ux and pressure reveals that a ux is

simply the product of pressure and the speed of moving particles [L T-1],

i.e. [M T-3] = [M L-1 T-2] [L T-1]. us, a radiative ux FR (W m-2) can be

expressed in terms of photon pressure Pph (Pa) and the speed of light c (m

s-1) as FR = c Pph. Since c is constant within a medium, varying the intensity

of electromagnetic radiation in a given medium eectively means altering

the pressure of photons. us, the solar radiation reaching Earth’s upper

atmosphere exerts a pressure (force) of sucient magnitude to perturb the

orbits of communication satellites over time [25,26].

e simplifying power of DA in model development stems from the

Buckingham Pi eorem [27], which states that a problem involving n

dimensioned xi variables, i.e.

( )

12 0

n

f x , x , , x

…=

can be reformulated into a simpler relationship of (n-m) dimensionless

πi products derived from xi, i.e.

ϕ(π1, π2, …. ,πn-m) = 0

where m is the number of fundamental dimensions comprising the

original variables. is theorem determines the number of non-

dimensional πi variables to be found in a set of products, but it does not

prescribe the number of sets that could be generated from the original

variables dening a particular problem. In other words, there might be,

and oentimes is more than one set of (n-m) dimensionless products to

analyze. DA provides an objective method for constructing the sets of

πi variables employing simultaneous equations solved via either matrix

inversion or substitution [22].

e second step of DA (aer the construction of dimensionless

products) is to search for a functional relationship between the πi

variables of each set using regression analysis. DA does not disclose

the best function capable of describing the empirical data. It is the

investigator’s responsibility to identify a suitable regression model

based on prior knowledge of the phenomenon and a general expertise

in the subject area. DA only guarantees that the nal model (whatever

its functional form) will be dimensionally homogeneous, hence it may

qualify as a physically meaningful relationship provided that it (a) is

not based on a simple polynomial t; (b) has a small standard error;

(c) displays high predictive skill over a broad range of input data; and

(d) is statistically robust. e regression coecients of the nal model

will also be dimensionless, and may reveal true constants of Nature by

virtue of being independent of the units utilized to measure the forcing

variables.

Selection of model variables

A planet’s GMAT depends on many factors. In this study, we focused

on drivers that are remotely measurable and/or theoretically estimable.

Based on the current state of knowledge we identied seven physical

variables of potential relevance to the global surface temperature: 1) top-

of-the-atmosphere (TOA) solar irradiance (S); 2) mean planetary surface

temperature in the absence of atmospheric greenhouse eect, hereto

called a reference temperature (Tr); 3) near-surface partial pressure

of atmospheric greenhouse gases (Pgh); 4) near-surface mass density

of atmospheric greenhouse gases (ρgh); 5) total surface atmospheric

pressure (P); 6) total surface atmospheric density (ρ); and 7) minimum

air pressure required for the existence of a liquid solvent at the surface,

hereto called a reference pressure (Pr). Table 1 lists the above variables

along with their SI units and physical dimensions. Note that, in order to

simplify the derivation of dimensionless products, pressure and density

are represented in Table 1 by the generic variables Px and ρx, respectively.

As explained below, the regression analysis following the construction

of

π

i

variables explicitly distinguished between models involving

partial pressure/density of greenhouse gases and those employing total

atmospheric pressure/density at the surface. e planetary Bond albedo

(αp) was omitted as a forcing variable in our DA despite its known eect

on the surface energy budget, because it is already dimensionless and

also partakes in the calculation of reference temperatures discussed

be low.

Appendix A details the procedure employed to construct the πi

variables. DA yielded two sets of πi products, each one consisting of two

Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary

Temperature Model. Environ Pollut Climate Change 1: 112.

Page 3 of 22

Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal

dimensionless variables, i.e.

3

12

2

sx

rx

TP

;

T S

π= π=

ρ

and

12

sx

rr

TP

;

TP

π= π=

is implies an investigation of two types of dimensionally homogeneous

functions (relationships):

3

2

sx

rx

TP

T S

=

ρ

ƒ

(1)

and

( )

2

sx

rr

TP

f

TP

=

(2)

Note that π1 = Ts/Tr occurs as a dependent variable in both relationships,

since it contains the sought temperature Ts. Upon replacing the generic

pressure/density variables Px and ρx in functions (1) and (2) with

either partial pressure/density of greenhouse gases (Pgh and ρgh) or total

atmospheric pressure/density (P and ρ), one arrives at six prospective

regression models. Further, as explained below, we employed two

distinct kinds of reference temperature computed from dierent

formulas, i.e. an eective radiating equilibrium temperature (Te) and

a mean ‘no-atmosphere’ spherical surface temperature (Tna). is

doubled the πi instances in the regression analysis bringing the total

number of potential models for investigation to twelve.

Reference temperatures and reference pressure

A reference temperature (Tr) characterizes the average thermal

environment at the surface of a planetary body in the absence of

atmospheric greenhouse eect; hence, Tr is dierent for each body and

depends on solar irradiance and surface albedo. e purpose of Tr is

to provide a baseline for quantifying the thermal eect of planetary

atmospheres. Indeed, the Ts/Tr ratio produced by DA can physically be

interpreted as a Relative Atmospheric ermal Enhancement (RATE)

ideally expected to be equal to or greater than 1.0. Expressing the

thermal eect of a planetary atmosphere as a non-dimensional quotient

instead of an absolute temperature dierence (as done in the past)

allows for an unbiased comparison of the greenhouse eects of celestial

bodies orbiting at dierent distances from the Sun. is is because the

absolute strength of the greenhouse eect (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 eect of planetary

atmospheres.

Two methods have been proposed thus far for estimating the

average surface temperature of a planetary body without the greenhouse

eect, both based on the SB radiation law. e rst and most popular

approach uses the planet’s global energy budget to calculate a single

radiating equilibrium temperature Te (also known as an eective

emission temperature) from the average absorbed solar ux [6,9,28],

i.e.

( )

0 25

e

1

4

.

p

S

T

−

=

εσ

α

(3)

Here, S is the solar irradiance (W m-2) dened as the TOA

shortwave ux incident on a plane perpendicular to the incoming rays,

αp is the planetary Bond albedo (decimal fraction),

ε

is the planet’s

LW emissivity (typically 0.9 ≤ ε <1.0; in this study we assume ε = 0.98

based on lunar regolith measurements reported by Vasavada et al. [29],

and σ = 5.6704 × 10-8 W m-2 K-4 is the SB constant. e term S(1-αp )⁄4

represents a globally averaged shortwave ux absorbed by the planet-

atmosphere system. e rationale behind Eq. (3) is that the TOA energy

balance presumably denes 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 eect of planetary atmospheres [2-

4,6,9,28]. However, Volokin and ReLlez [1] showed that, due to Hölder’s

inequality between integrals [33], Te is a non-physical temperature for

spheres and lacks a meaningful relationship to the planet’s Ts.

e second method attempts to estimate the average surface

temperature of a planet (Tna) in the complete absence of an atmosphere

using an explicit spatial integration of the SB law over a sphere. Instead

of calculating a single bulk temperature from the average absorbed

shortwave ux as done in Eq. (3), this alternative approach rst

computes the equilibrium temperature at every point on the surface of

an airless planet from the local absorbed shortwave ux using the SB

relation, and then spherically integrates the resulting temperature eld

to produce a global temperature mean. While algorithmically opposite

to Eq. (3), this method mimics well the procedure for calculating 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 and ReLlez

[1] improved upon Rubincam’s formulation by deriving a closed-form

integral expression that explicitly accounts for the eect of subterranean

heat storage, cosmic microwave background radiation (CMBR) and

geothermal heating on the average global surface temperature of

airless bodies. e complete form of their analytic Spherical Airless-

Temperature (SAT) model reads:

Planetary Variable Symbol SI Units Physical Dimension

Global mean annual near-surface temperature (GMAT), the dependent variable TsK[Θ]

Stellar irradiance (average shortwave ux incident on a plane perpendicular to the stellar rays at the top of a planet’s

atmosphere) SW m-2 [M T-3]

Reference temperature (the planet’s mean surface temperature in the absence of an atmosphere or an atmospheric

greenhouse effect) TrK[Θ]

Average near-surface gas pressure representing either partial pressure of greenhouse gases or total atmospheric

pressure PxPa [M L-1 T-2]

Average near-surface gas density representing either greenhouse-gas density or total atmospheric density

x

ρ

kg m-3 [M L-3]

Reference pressure (the minimum atmospheric pressure required a liquid solvent to exists at the surface) PrPa [M L-1 T-2]

Table 1: Variables employed in the Dimensional Analysis aimed at deriving a general planetary temperature model. The variables are comprised of 4 fundamental physical

dimensions: mass [M], length [L], time [T] and absolute temperature [Θ].

Citation: Nikolov N, Zeller K (2017) New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary

Temperature Model. Environ Pollut Climate Change 1: 112.

Page 4 of 22

Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal

Tna =

( ) ( )

( )

( ) ( )( )

( )

( )

( )( )

54 54

14

54 54

14

11

11

2

50 754 1

0 754 1

//

e e cg Cg

/

e e

//

e e c g C g

/

e e

S R R R R

S

. S RR RR

. S

− −α + + − +

+

−η −α εσ

−α + + − +

− α εσ

η

η

η

(4a)

where αe is the eective shortwave albedo of the surface, ηe is the

eective ground heat storage coecient in a vacuum, Rc = σ 2.7254 =

3.13 × 10-6 W m-2 is the CMBR [35], and Rg is the spatially averaged

geothermal ux (W m-2) emanating from the subsurface. e heat

storage term ηe is dened as a fraction of the absorbed shortwave ux

conducted into the subsurface during daylight hour and subsequently

released as heat at night.

Since the eect of CMBR on Tna is negligible for S > 0.15 W m-2 [1]

and the geothermal contribution to surface temperatures is insignicant

for most planetary bodies, one can simplify Eq. (4a) by substituting Rc =

Rg = 0 is produces:

( ) ( )

( )

0.25

0.25 0.25

1

21 0.932

5

e

na e e

S

T

αηη

εσ

−

= −+

(4b)

where 0.932 = 0.7540.25. e complete formula (4a) must only be used if

S ≤ 0.15 W m-2 and/or the magnitude of Rg is signicantly greater than

zero. For comparison, in the Solar System, the threshold S ≤ 0.15 W m-2

is encountered beyond 95 astronomical unis (AU) in the region of the

inner Oort cloud. Volokin and ReLlez [1] veried Equations (4a) and

(4b) against Moon temperature data provided by the NASA Diviner

Lunar Radiometer Experiment [29,36]. ese authors also showed that

accounting for the subterranean heat storage (ηe) markedly improves

the physical realism and accuracy of the SAT model compared to the

original formulation by Rubincam [34].

e conceptual dierence between Equations (3) and (4b) is that Τe

represents the equilibrium temperature of a blackbody disk orthogonally

illuminated by shortwave radiation with an intensity equal to the average

solar ux absorbed by a sphere having a Bond albedo αp, while Τna is the

area-weighted average temperature of a thermally heterogeneous airless

sphere [1,37]. In other words, for spherical objects, Τe is an abstract

mathematical temperature, while Tna is the average kinetic temperature

of an airless surface. Due to Hölder’s inequality between integrals, one

always nds Τe >> Τna when using equivalent values of stellar irradiance

and surface albedo in Equations (3) and (4b) [1].

To calculate the Tna temperatures for planetary bodies with tangible

atmospheres, we assumed that the airless equivalents of such objects

would be covered with a regolith of similar optical and thermo-physical

properties as the Moon surface. is is based on the premise that, in

the absence of a protective atmosphere, the open cosmic environment

would erode and pulverize exposed surfaces of rocky planets over time

in a similar manner [1]. Also, properties of the Moon surface are the

best studied ones among all airless bodies in the Solar System. Hence,

one could further simplify Eq. (4b) by combining the albedo, the heat

storage fraction and the emissivity parameter into a single constant

using applicable values for the Moon, i.e. αe = 0.132, ηe = 0.00971 and ε

= 0.98 [1,29]. is produces:

0.25

32.44

na

TS=

(4c)

Equation (4c) was employed to estimate the ‘no-atmosphere’ reference

temperatures of all planetary bodies participating in our analysis and

discussed below.

For a reference pressure, we used the gas-liquid-solid triple point of

water, i.e. Pr = 611.73 Pa [38] dening a baric threshold, below which water

can only exists in a solid/vapor phase and not in a liquid form. e results

of our analysis are not sensitive to the particular choice of a reference-

pressure value; hence, the selection of Pr is a matter of convention.

Regression analysis

Finding the best function to describe the observed variation of

GMAT among celestial bodies requires that the πi variables generated

by DA be subjected to regression analyses. As explained in Appendix A,

twelve pairs of πi variables hereto called Models were investigated. In

order to ease the curve tting and simplify the visualization of results,

we utilized natural logarithms of the constructed πi variables rather than

their absolute values, i.e. we modeled the relationship ln (π1) = f (ln(π2))

instead of π1 = f(π2). In doing so we focused on monotonic functions

of conservative shapes such as exponential, sigmoidal, hyperbolic,

and logarithmic, for their tting coecients might be interpretable in

physically meaningful terms. A key advantage of this type of functions

(provided the existence of a good t, of course) is that they also tend

to yield reliable results outside the data range used to determine their

coecients. We specically avoided non-monotonic functions such as

polynomials because of their ability to accurately t almost any dataset

given a suciently large number of regression coecients while at the

same time showing poor predictive skills beyond the calibration data

range. Due to their highly exible shape, polynomials can easily t

random noise in a dataset, an outcome we particularly tried to avoid.

e following four-parameter exponential-growth function was

found to best meet our criteria:

( ) ( )

exp exp y a bx c d x= +

(5)

where x = ln (π2) and y = ln (π1) are the independent and dependent

variable respectively while a, b, c and d are regression coecients. is

function has a rigid shape that can only describe specic exponential

patterns found in our data. Equation (5) was tted to each one of the

12 planetary data sets of logarithmic πi pairs suggested by DA using the

standard method of least squares. e skills of the resulting regression

models were evaluated via three statistical criteria: coecient of

determination (R2), adjusted R2, and standard error of the estimate (σest)

[39,40]. All calculations were performed with SigmaPlotTM 13 graphing

and analysis soware.

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 dened

in terms of TOA solar irradiance and atmospheric properties. is

resulted in the selection of three planets: Venus, Earth, and Mars; and

three natural satellites: Moon of Earth, Titan of Saturn, and Triton of

Neptune.

Each celestial body was described by nine parameters shown in

Table 2 with data sources listed in Table 3. In an eort to minimize

the eect of unforced (internal) climate variability on the derivation

of our temperature model, we tried to assemble a dataset of means

representing an observational period of 30 years, i.e. from 1981 to 2010.

us, Voyager measurements of Titan from the early 1980s suggested

an average surface temperature of 94 ± 0.7 K [41]. Subsequent

observations by the Cassini mission between 2005 and 2010 indicated

a mean global temperature of 93.4 ± 0.6 K for that moon [42,43]. Since

Temperature Model. Environ Pollut Climate Change 1: 112.

Page 5 of 22

Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal

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) spacecra to derive representative

estimates of GMAT and atmospheric surface pressure for the Red

Planet. Some parameter values reported in the literature did not meet

our criteria for global representativeness and/or physical plausibility

and were recalculated using available observations as described below.

e mean solar irradiances of all bodies were calculated as S = SE rau

-2

where rau is the body’s average distance (semi-major axis) to the Sun

(AU) and SE = 1,360.9 W m-2 is the Earth’s new lower irradiance at 1 AU

according to recent satellite observations reported by Kopp and Lean

[49]. Due to a design aw in earlier spectrometers, the solar irradiance

at Earth’s distance has been overestimated by ≈ 5 W m-2 prior to 2003

[49]. Consequently, our calculations yielded slightly lower irradiances

for bodies such as Venus and Mars compared to previously published

data. Our decision to recalculate S was based on the assumption that the

orbital distances of planets are known with much greater accuracy than

TOA solar irradiances. Hence, a correction made to Earth’s irradiance

requires adjusting the ‘solar constants’ of all other planets as well.

We found that quoted values for the mean global temperature and

surface atmospheric pressure of Mars were either improbable or too

uncertain to be useful for our analysis. us, studies published in the

last 15 years report Mars’ GMAT being anywhere between 200 K and

240 K with the most frequently quoted values in the range 210–220

K [6,32,76-81]. However, in-situ measurements by Viking Lander 1

suggest that the average surface air temperature at a low-elevation site

in the Martian subtropics does not exceed 207 K during the summer-

fall season (Appendix B). erefore, the Red Planet’s GMAT must be

lower than 207 K. e Viking records also indicate that average diurnal

temperatures above 210 K can only occur on Mars during summertime.

Hence, all such values must be signicantly higher than the actual mean

annual temperature at any Martian latitude. is is also supported by

results from a 3-D global circulation model of the Red Planet obtained

by Fenton et al. [82]. e surface atmospheric pressure on Mars varies

appreciably with season and location. Its global average value has

previously been reported between 600 Pa and 700 Pa [6,32,78,80,83,84],

a range that was too broad for the target precision of our study. Hence

our decision to calculate new annual global means of near-surface

temperature and air pressure for Mars via a thorough analysis of available

data from remote-sensing and in-situ observations. Appendix B details

our computational procedure with the results presented in Table 2. It is

noteworthy that our independent estimate of Mars’ GMAT (190.56 ±

0.7 K), while signicantly 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-ow

measurements conducted at depths between 80 and 140 cm at the

Apollo 15 and 17 landing sites located at 26oN; 3.6oE and 20oN; 30.6oE,

respectively [88]. Due to a strong temperature dependence of the lunar

regolith thermal conductivity in the topmost 1-2 cm soil, the 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]. e

diurnal temperature uctuations completely vanish below a depth of 80

cm. At 100 cm depth, the temperature of the lunar regolith ranged from

250.7 K to 252.5 K at the Apollo 15 site and between 254.5 K and 255.5 K

at the Apollo 17 site [88]. Hence, reported Moon average temperatures

in the range 250-255 K do not describe surface conditions. Moreover,

since measured in the lunar subtropics, such temperatures do not likely

even represent Moon’s global thermal environment at these depths. On

the other hand, frequently quoted Moon global temperatures of ~270 K

have actually been calculated from Eq. (3) and are not based on surface

measurements. However, as demonstrated by Volokin and ReLlez [1],

Parameter Venus Earth Moon Mars Titan Triton

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

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

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

Average absorbed shortwave radiation, Sa = S(1-αp)/4 (W m-2)65.0 240.2 294.0 112.1 2.72 0.13

Global average surface atmospheric pressure, P (Pa) 9,300,000.0 ±

100,000 98,550.0 ± 6.5 2.96 × 10-10 ±

10-10 685.4 ± 14.2 146,700.0 ± 100 4.0 ± 1.2

Global average surface atmospheric density, ρ (kg m-3) 65.868 ± 0.44 1.193 ± 0.002 2.81 × 10-15 ±

9.4 × 10-15

0.019 ± 3.2 ×

10-4 5.161 ± 0.03 3.45 × 10-4 ± 9.2

× 10-5

Chemical composition of the lower atmosphere (% of volume)

96.5 CO2

3.48 N2

0.02 SO2

77.89 N2

20.89 O2

0.932 Ar

0.248 H2O

0.040 CO2

26.7 4He

26.7 20Ne

23.3 H2

20.0 40Ar

3.3 22Ne

95.32 CO2

2.70 N2

1.60 Ar

0.13 O2

0.08 CO

0.021 H2O

95.1 N2

4.9 CH4

99.91 N2

0.060 CO

0.024 CH4

Molar mass of the lower atmosphere, M (kg mol-1)0.0434 0.0289 0.0156 0.0434 0.0274 0.0280

GMAT, Ts (K) 737.0 ± 3.0 287.4 ± 0.5 197.35 ± 0.9 190.56 ± 0.7 93.7 ± 0.6 39.0 ± 1.0

Table 2: Planetary data set used in the Dimensional Analysis compiled from sources listed in Table 3. The estimation of Mars’ GMAT and the average surface atmospheric

pressure are discussed in Appendix B. See text for details about the computational methods employed for some parameters.

Planetary Body Information Sources

Venus [32,44-48]

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

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

Mars [32,48,60-63], Appendix B

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

Triton [48,73-75]

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

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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 and ReLlez [1] using

output from a NASA thermo-physical model validated against Diviner

observations [29].

Surprisingly, many publications report incorrect values even

for Earth’s mean global temperature. Studies of terrestrial climate

typically focus on temperature anomalies and if Earth’s GMAT is

ever mentioned, it is oen loosely quoted as 15 C (~288 K) [2-4,6].

However, observations archived in the HadCRUT4 dataset of the

UK Met Oce’s Hadley Centre [50,89] and in the Global Historical

Climatology Network [51,52,90,91] indicate that, between 1981 and

2010, Earth’s mean annual surface air temperature was 287.4 K (14.3

C) ± 0.5 K. Some recent studies acknowledge this more accurate lower

value of Earth’s absolute global temperature [92]. For Earth’s mean

surface atmospheric pressure we adopted the estimate by Trenberth et

al. [53] (98.55 kPa), which takes into account the average elevation of

continental landmasses above sea level; hence, it is slightly lower than

the typical sea-level pressure of ≈ 101.3 kPa.

e average near-surface atmospheric densities (ρ, kg m-3) of

planetary bodies were calculated from reported means of total

atmospheric pressure (P), molar mass (M, kg mol-1) and temperature

(Ts) using the Ideal Gas Law, i.e.

( )

6

s

PM

RT

ρ

= (6)

where R = 8.31446 J mol-1 K-1 is the universal gas constant. is

calculation was intended to make atmospheric densities physically

consistent with independent data on pressure and temperature utilized

in our study. e resulting ρ values were similar to previously published

data for individual bodies. Standard errors of the air-density estimates

were calculated from reported errors of P and Τs for each body using

Eq. (6).

Data in Table 2 were harnessed to compute several intermediate

variables and all dimensionless πi products necessary for the regression

analyses. e results from these computations are shown in Table 4.

Greenhouse gases in planetary atmospheres represented by the major

constituents carbon dioxide (CO2), methane (CH4) and water vapor

(H2O) were collectively quantied via three bulk parameters: average

molar mass (Mgh, kg mol-1), combined partial pressure (Pgh, Pa) and

combined partial density (ρgh, kg m-3). ese parameters were estimated

from reported volumetric concentrations of individual greenhouse

gases (Cx, %) and data on total atmospheric pressure and density in

Table 2 using the formulas:

( )

CO2 CH 4 H2O

0.044 0.016 0.018 /

gh gh

M C C CC= ++

(7)

( )

0.01

gh gh

PP C=

(8)

( )( )

0.01 /

gh gh gh

CMM

ρρ

=

(9)

where Cgh = CCO2 + CCH4 + CH2O is the total volumetric concentration

of major greenhouse gases (%). e reference temperatures Τe and Τna

were calculated from Equations (3) and (4c), respectively.

Results

Function (5) was tted to each one of the 12 sets of logarithmic πi

pairs generated by Equations (1) and (2) and shown in Table 4. Figures

1 and 2 display the resulting curves of individual regression models

with planetary data plotted in the background for reference. Table 5 lists

the statistical scores of each non-linear regression. Model 12 depicted

in Figure 2f had the highest R2 = 0.9999 and the lowest standard error

σest = 0.0078 among all regressions. Model 1 (Figure 1a) provided the

second best t with R2 = 0.9844 and σest = 0.1529. Notably, Model 1

shows almost a 20-time larger standard error on the logarithmic scale

than Model 12. Figure 3 illustrates the dierence in predictive skills

between the two top-performing Models 1 and 12 upon conversion

of vertical axes to a linear scale. Taking an antilogarithm weakens

the relationship of Model 1 to the point of becoming immaterial and

highlights the superiority of Model 12. e statistical results shown in

Table 5 indicate that the explanatory power and descriptive accuracy of

Model 12 surpass those of all other models by a wide margin.

Since Titan and Earth nearly overlap on the logarithmic scale of Figure

2f, we decided to experiment with an alternative regression for Model 12,

Intermediate Variable or Dimensionless Product Venus Earth Moon Mars Titan Triton

Average molar mass of greenhouse gases, Mgh (kg mol-1)

(Eq. 7) 0.0440 0.0216 0.0 0.0440 0.0160 0.0160

Near-surface partial pressure of greenhouse gases, Pgh (Pa)

(Eq. 8)

8,974,500.0 ±

96,500 283.8 ± 0.02 0.0 667.7 ± 13.8 7,188.3 ± 4.9 9.6 × 10-4 ± 2.9

× 10-4

Near-surface density of greenhouse gases, ρgh (kg m-3) (Eq. 9) 64.441 ± 0.429 2.57 × 10-3 ± 4.3

× 10-6 0.0 0.018 ± 3.1 ×

10-4

0.148 ± 8.4 ×

10-4

4.74 × 10-8 ± 1.3

× 10-8

Radiating equilibrium temperature, Te (K) (Eq. 3) 185.0 256.4 269.7 211.9 83.6 39.2

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

Ts/ Te3.985 ± 0.016 1.121 ± 0.002 0.732 ± 0.003 0.899 ± 0.003 1.120 ± 0.008 0.994 ± 0.026

Ts/Tna 3.181 ± 0.013 1.459 ± 0.002 1.002 ± 0.004 1.194 ± 0.004 1.473 ± 0.011 1.086 ± 0.028

ln(Ts/Te) 1.3825 ± 0.0041 0.1141 ± 0.0017 -0.3123 ± 0.0046 -0.1063 ± 0.0037 0.1136 ± 0.0075 -5.2×10-3 ±

0.0256

ln(Ts/Tna) 1.1573 ± 0.0041 0.3775 ± 0.0017 1.59×10-3 ±

0.0046 0.1772 ± 0.0037 0.3870 ± 0.0075 0.0828 ± 0.0256

ln[Pgh

3/(ρgh S2)] 28.1364 8.4784 Undened 10.7520 23.1644 -4.7981

ln[P3/(ρgh S2)] 28.2433 26.0283 +∞ 10.8304 32.2122 20.2065

ln[Pgh

3/(ρ S2)] 28.1145 2.3370 Undened 10.7396 19.6102 -13.6926

ln[Pgh/Pr] 9.5936 -0.7679 Undened 0.0876 2.4639 -13.3649

ln[P3/(ρ S2)] 28.2214 19.8869 -46.7497 10.8180 28.6580 11.3120

ln(P/Pr)9.6292 ± 0.0108 5.0820 ±

6.6×10-5

-28.3570 ±

0.3516 0.1137 ± 0.0207 5.4799 ±

6.8×10-4

-5.0300 ±

0.3095

Table 4: Intermediate variables and dimensionless products required for the regression analyses and calculated from data in Table 2. Equations used to compute

intermediate variables are shown in parentheses. The reference pressure is set to the barometric triple point of water, i.e. Pr = 611.73 Pa.

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which excludes Titan from the input dataset. is new curve had R2 =

1.0 and σest = 0.0009. Although the two regression equations yield similar

results over most of the relevant pressure range, we chose the one without

Titan as nal for Model 12 based on the assumption that Earth’s GMAT

is likely known with a much greater accuracy than Titan’s mean annual

temperature. Taking an antilogarithm of the nal regression equation,

which excludes Titan, yielded the following expression for Model 12:

0.150263 1.04193

5

na

+ 1. exp 0.17 84205 31 0 21 1

s

rr

TP P

TP P

−

=

×

(10a)

e regression coecients in Eq. (10a) are intentionally shown in

full precision to allow an accurate calculation of RATE (i.e. the Ts/

Tna ratios) provided the strong non-linearity of the relationship and

to facilitate a successful replication of our results by other researchers.

Figure 4 depicts Eq. (10a) as a dependence of RATE on the average

surface air pressure. Superimposed on this graph are the six planetary

bodies from Table 4 along with their uncertainty ranges.

Equation (10a) implies that GMATs of rocky planets can be

calculated as a product of two quantities: the planet’s average surface

temperature in the absence of an atmosphere (Tna, K) and a non-

dimensional factor (Ea ≥ 1.0) quantifying the relative thermal eect of

the atmosphere, i.e.

( )

10b

s na a

T TE=

(10b)

where Τna is obtained from the SAT model (Eq. 4a) and Ea is a function

of total pressure (P) given by:

( ) ( )

0.150263 1.04193

5

exp 0.174205 exp 1.83121 10 11

a

rr

PP

EP PP

−

= ×

(11)

Note that, as P approaches 0 in Eq. (11), Ea approaches the physically

realistic limit of 1.0. Other physical aspects of this equation are

discussed below.

For bodies with tangible atmospheres (such as Venus, Earth,

Figure 1: The relative atmospheric thermal enhancement (Ts/Tr) as a function of various dimensionless forcing variables generated by DA using data on solar

irradiance, near-surface partial pressure/density of greenhouse gases, and total atmospheric pressure/density from Table 4. Panels a through f depict six regression

models suggested by DA with the underlying celestial bodies plotted in the background for reference. Each pair of horizontal graphs represents different reference

temperatures (Tr) dened as either Tr = Te (left) or Tr = Tna (right).

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Mars, Titan and Triton), one must calculate Tna using αe = 0.132 and

ηe = 0.00971, which assumes a Moon-like airless reference surface in

accordance with our pre-analysis premise. For bodies with tenuous

atmospheres (such as Mercury, the Moon, Calisto and Europa), Tna

should be calculated from Eq. (4a) (or Eq. 4b respectively if S > 0.15

W m-2 and/or Rg ≈ 0 W m-2) using the body’s observed values of Bond

albedo αe and ground heat storage fraction ηe. In the context of this

model, a tangible atmosphere is dened as one that has signicantly

modied 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 inuence on the surface albedo and regolith

thermo-physical properties and is completely transparent to shortwave

radiation. e need for such delineation of atmospheric masses when

calculating Tna arises from the fact that Eq. (10a) accurately describes

RATEs of planetary bodies with tangible atmospheres over a wide

range of conditions without explicitly accounting for the observed large

dierences in albedos (i.e. from 0.235 to 0.90) while assuming constant

values of αe and ηe for the airless equivalent of these bodies. One possible

explanation for this counterintuitive empirical result is that atmospheric

pressure alters the planetary albedo and heat storage properties of the

surface in a way that transforms these parameters from independent

controllers of the global temperature in airless bodies to intrinsic

byproducts of the climate system itself in worlds with appreciable

atmospheres. In other words, once atmospheric pressure rises above a

certain level, the eects of albedo and ground heat storage on GMAT

become implicitly accounted for by Eq. (11). Although this hypothesis

requires a further investigation beyond the scope of the present study,

one nds an initial support for it in the observation that, according to

data in Table 2, GMATs of bodies with tangible atmospheres do not

show a physically meaningful relationship with the amounts of absorbed

shortwave radiation determined by albedos. Our discovery for the

need to utilize dierent albedos and heat storage coecients 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 dierent aspects of the same phenomenon. An

example is the distinction made in uid mechanics between laminar

and turbulent ow, where the non-dimensional Reynold’s number is

employed to separate the two regimes that are subjected to dierent

mathematical treatments.

Figure 2: The same as in Figure 1 but for six additional regression models (panels a through f).

Temperature Model. Environ Pollut Climate Change 1: 112.

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We do not currently have sucient data to precisely dene 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 signicantly below 1.0 Pa. In this

study, we use 0.01 Pa as a tentative threshold value. us, in the context

of Eq. (10b), we recommend computing Tna from Eq. (4c) if P > 10-2 Pa,

and from Eq. (4a) (or Eq. 4b, respectively) using observed values of αe

and ηe if P ≤ 10-2 Pa. Equation (4a) should also be employed in cases,

where a signicant geothermal ux exists such as on the Galilean moons

of Jupiter due to tidal heating, and/or if S ≤ 0.15 W m-2. Hence, the

30-year mean global equilibrium surface temperature of rocky planets

depends in general on ve factors: TOA stellar irradiance (S), a reference

airless surface albedo (αe), a reference airless ground heat storage fraction

No. Functional Model Coefcient of Determination (R2) Adjusted R2Standard Error σest

1

3

2

e

gh

s

gh

P

Tf

TS

ρ

=

0.9844 0.9375 0.1529

2

3

2

na

gh

s

gh

P

Tf

TS

ρ

=

0.9562 0.8249 0.1773

3

3

2

e

s

gh

TP

f

TS

ρ

=

0.1372 -2.4511 1.1360

4

3

2

na

s

gh

TP

f

TS

ρ

=

0.2450 -2.0200 0.7365

5

3

2

e

gh

s

P

Tf

TS

ρ

=

0.9835 0.9339 0.1572

6

3

2

na

gh

s

P

Tf

TS

ρ

=

0.9467 0.7866 0.1957

7

e

gh

s

r

P

Tf

TP

=

0.9818 0.9274 0.1648

8

na

gh

s

r

P

Tf

TP

=

0.9649 0.8598 0.1587

9

3

2

e

s

TP

f

TS

ρ

=

0.4488 -0.3780 0.7060

10

3

2

na

s

TP

f

TS

ρ

=

0.6256 0.0639 0.4049

11

e

s

r

TP

f

TP

=

0.9396 0.8489 0.2338

12

na

s

r

TP

f

TP

=

0.9999 0.9997 0.0078

Table 5: Performance statistics of the twelve regression models suggested by DA. Statistical scores refer to the model logarithmic forms shown in Figures 1 and 2.

Figure 3: Comparison of the two best-performing regression models according to statistical scores listed in Table 5. Vertical axes use linear scales to better illustrate

the difference in skills between the models.

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(ηe), the average geothermal ux reaching the surface (Rg), and the total

surface atmospheric pressure (P). For planets with tangible atmospheres

(P > 10-2 Pa) and a negligible geothermal heating of the surface (Rg ≈ 0),

the equilibrium GMAT becomes only a function of two factors: S and

P, i.e. Τs = 32.44 S0.25Eα(P). e nal model (Eq. 10b) can also be cast

in terms of Ts as a function of a planet’s distance to the Sun (rau, AU) by

replacing S in Equations (4a), (4b) or (4c) with 1360.9 rau

-2.

Environmental scope and numerical accuracy of the new

model

Figure 5 portrays the residuals between modeled and observed

absolute planetary temperatures. For celestial bodies participating in

the regression analysis (i.e. Venus, Earth, Moon, Mars and Triton), the

maximum model error does not exceed 0.17 K and is well within the

uncertainty of observations. e error for Titan, an independent data

point, is 1.45 K or 1.5% of that moon’s current best-known GMAT (93.7

K). Equation (10b) produces 95.18 K for Titan at Saturn’s semi-major

axis (9.582 AU) corresponding to a solar irradiance S = 14.8 W m-2. is

estimate is virtually identical to the 95 K average surface temperature

reported for that moon by the NASA JPL Voyager Mission website

[94]. e Voyager spacecra 1 and 2 reached Saturn and its moons in

November 1980 and August 1981, respectively, when the gas giant was

at a distance between 9.52 AU and 9.60 AU from the Sun corresponding

approximately to 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 Ts ≈ 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 ≤ Ts ≤ 98.1 K. Notably this range encompasses

all current observation-based estimates of Titan’s GMAT. Since both

Voyager and Cassini mission covered shorter periods than a single

Titan season (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. us, the observed 1.45 K discrepancy

between our independent model prediction and Titan’s current

best-known GMAT seems to be within the range of plausible global

temperature uctuations on that moon. Hence, further observations are

needed to more precisely constrain Titan’s long-term GMAT.

Measurements conducted by the Voyager spacecra in 1989

indicated a global mean temperature of 38 ± 1.0 K and an average

atmospheric pressure of 1.4 Pa at the surface of Triton [73]. Even

though Eq. (10a) is based on slightly dierent data for Triton (i.e. Ts =

39 ±1.0 K and P = 4.0 Pa) obtained by more recent stellar occultation

measurements [73], employing the Voyager-reported pressure in Eq.

(10b) produces Ts = 38.5 K for Triton’s GMAT, a value well within the

uncertainty of the 1989 temperature measurements.

e above comparisons indicate that Eq. (10b) rather accurately

describes the observed variation of the mean surface temperature across

a wide range of planetary environments in terms of solar irradiance

(from 1.5 W m-2 to 2,602 W m-2), total atmospheric pressure (from

near vacuum to 9,300 kPa) and greenhouse-gas concentrations (from

0.0% to over 96% per volume). While true that Eq. (10a) is based on

data from only 6 celestial objects, one should keep in mind that these

constitute virtually all bodies in the Solar System meeting our criteria

for availability and quality of measured data. Although function (5)

has 4 free parameters estimated from just 5-6 data points, there are no

signs of model overtting in this case because (a) Eq. (5) represents

a monotonic function of a rigid shape that can only describe well

certain exponential pattern as evident from Figures 1 and 2 and the

statistical scores in Table 5; (b) a simple scatter plot of ln (P/Pr) vs. ln(Ts/

Tna) visibly reveals the presence of an exponential relationship free of

data noise; and (c) no polynomial can t the data points in Figure 2f

as accurately as Eq. (5) while also producing a physically meaningful

response curve similar to known pressure-temperature relationships in

other systems. ese facts indicate that Eq. (5) is not too complicated

to cause an over-tting but just right for describing the data at hand.

e fact that only one of the investigated twelve non-linear

regressions yielded a tight relationship suggests that Model 12 describes

Figure 4: The relative atmospheric thermal enhancement (Ts/Tna ratio) as a

function of the average surface air pressure according to Eq. (10a) derived from

data representing a broad range of planetary environments in the solar system.

Saturn’s moon Titan has been excluded from the regression analysis leading

to Eq. (10a). Error bars of some bodies are not clearly visible due to their small

size relative to the scale of the axes. See Table 2 for the actual error estimates.

Figure 5: Absolute differences between modeled average global temperatures

by Eq. (10b) and observed GMATs (Table 2) for the studied celestial bodies.

Saturn’s moon Titan represents an independent data point, since it was excluded

from the regression analysis leading to Eq. (10a).

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a macro-level thermodynamic property of planetary atmospheres

heretofore unbeknown to science. A function of such predictive power

spanning the entire breadth of the Solar System cannot be just a result

of chance. Indeed, complex natural systems consisting of myriad

interacting agents have been known to sometime exhibit emergent

responses at higher levels of hierarchical organization that are amenable

to accurate modeling using top-down statistical approaches [96].

Equation (10a) also displays several other characteristics discussed

below that lend further support to the above notion.

Model robustness

Model robustness denes the degree to which a statistical

relationship would hold when recalculated using a dierent dataset. To

test the robustness of Eq. (10a) we performed an alternative regression

analysis, which excluded Earth and Titan from the input data and

only utilized logarithmic pairs of Ts/Tna and P/Pr for Venus, the Moon,

Mars and Triton from Table 4. e goal was to evaluate how well the

resulting new regression equation would predict the observed mean

surface temperatures of Earth and Titan. Since these two bodies occupy

a highly non-linear region in Model 12 (Figure 2f), eliminating them

from the regression analysis would leave a key portion of the curve

poorly dened. As in all previous cases, function (5) was tted to the

incomplete dataset (omitting Earth and Titan), which yielded the

following expression:

( )

0.150275 3.32375

15

na

exp 0.174222 5.25043 10

12a

s

rr

T

PP

TP P

−

= +×

(12a)

Substituting the reference temperature Tna in Eq. (12a) with its

equivalent from Eq. (4c) and solving for Ts produces

( )

0.150275 3.32375

0.25 15

32.44 exp 0.174222 exp 5.25043 10 12b

s

rr

PP

TS PP

−

= ×

(12b)

It is evident that the regression coecients in the rst exponent term of

Eq. (12a) are nearly identical to those in Eq. (10a). is term dominates

the Ts-P relationship over the pressure range 0-400 kPa accounting

for more than 97.5% of the predicted temperature magnitudes. e

regression coecients of the second exponent dier somewhat between

the two formulas causing a divergence of calculated RATE values

over the pressure interval 400–9,100 kPa. e models converge again

between 9,000 kPa and 9,300 kPa. Figure 6 illustrates the similarity of

responses between Equations (10a) and (12a) over the pressure range

0–300 kPa with Earth and Titan plotted in the foreground for reference.

Equation (12b) reproduces the observed global surface temperature

of Earth with an error of 0.4% (-1.0 K) and that of Titan with an error

of 1.0% (+0.9 K). For Titan, the error of the new Eq. (12b) is even

slightly smaller than that of the original model (Eq. 10b). e ability

of Model 12 to predict 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). e apparent

smoothness of this continuum for bodies with tangible atmospheres

(illustrated in Figure 4) suggests that planetary climates are well-

buered and have no ‘tipping points’ in reality, i.e. states enabling

rapid and irreversible changes in the global equilibrium temperature

as a result of destabilizing positive feedbacks assumed to operate within

climate systems. is robustness test also serves as a cross-validation

suggesting that the new model has a universal nature and it is not a

product of overtting.

e above characteristics of Eq. (10a) including dimensional

homogeneity, high predictive accuracy, broad environmental scope of

validity and statistical robustness indicate that it represents an emergent

macro-physical model of theoretical signicance deserving further

investigation. is conclusive result is also supported by the physical

meaningfulness of the response curve described by Eq. (10a).

Discussion

Given the high statistical scores of the new model discussed above,

it is important to address its physical signicance, potential limitations,

and broad implications for the current climate theory.

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

radiation law

e functional response of Eq. (10a) portrayed in Figure 4 closely

resembles the shape of the dry adiabatic temperature curve in Figure

7a described by the Poisson formula and derived from the First Law of

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

( )

/

13

p

Rc

oo

Tp

Tp

=

(13)

Here, To and po are reference values for temperature and pressure

typically measured at the surface, while T and p are corresponding scalars

in the free atmosphere, and cp is the molar heat capacity of air (J mol-1

K-1). For the Earth’s atmosphere, R/cp = 0.286. Equation (13) essentially

describes the direct eect of pressure p on the gas temperature (T) in

the absence of any heat exchange with the surrounding environment.

Equation (10a) is structurally similar to Eq. (13) in a sense that

both expressions relate a temperature ratio to a pressure ratio, or more

precisely, a relative thermal enhancement to a ratio of physical forces.

However, while the Poisson formula typically produces 0 ≤ T/To ≤ 1.0,

Eq. (10a) always yields Ts/Tna ≥ 1.0. e key dierence between the two

models stems from the fact that Eq. (13) describes vertical temperature

changes in a free and dry atmosphere induced by a gravity-controlled

pressure gradient, while Eq. (10a) predicts the equilibrium response of a

planet’s global surface air temperature to variations in total atmospheric

Figure 6: Demonstration of the robustness of Model 12. The solid black curve

depicts Eq. (10a) based on data from 5 celestial bodies (i.e. Venus, Earth, Moon,

Mars and Triton). The dashed grey curve portrays Eq. (12a) derived from data of

only 4 bodies (i.e. Venus, Moon, Mars and Triton) while excluding Earth and Titan

from the regression analysis. The alternative Eq. (12b) predicts the observed

GMATs of Earth and Titan with accuracy greater than 99% indicating that Model

12 is statistically robust.

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pressure. In essence, Eq. (10b) could be viewed as a predictor of the

reference temperature To in the Poisson formula. us, while qualitatively

similar, Equations (10a) and (13) are quantitatively rather dierent. Both

functions describe eects of pressure on temperature but in the context of

disparate physical systems. erefore, estimates obtained from Eq. (10a)

should not be confused with results inferred from the Poisson formula.

For example, Eq. (10b) cannot be expected to predict the temperature

lapse rate and/or vertical temperature proles within a planetary

atmosphere as could be using Eq. (13). Furthermore, Eq. (10a) represents

a top-down empirical model that implicitly accounts for a plethora of

thermodynamic and radiative processes and feedbacks operating in real

climate systems, while the Poisson formula (derived from the Ideal Gas

Law) only describes pressure-induced temperature changes in a simple

mixture of dry gases without any implicit or explicit consideration of

planetary-scale mechanisms such as latent heat transport and cloud

radiative forcing.

Equation (10a) also shows a remarkable similarity to the SB law

relating the equilibrium skin temperature of an isothermal blackbody

(Tb, K) to the electromagnetic radiative ux (I, W m-2) absorbed/

emitted by the body’s surface, i.e. Tb = (I ⁄ σ)0.25. Dividing each side of

this fundamental relationship by the irreducible temperature of deep

Space Tc = 2.725 K and its causative CMBR Rc = 3.13 × 10-6 W m-2

respectively, yields Tb⁄Tc = (I ⁄ Rc )0.25. Further, expressing the radiative

uxes I and Rc on the right-hand side as products of photon pressure

and the speed of light (c, m s-1) in a vacuum, i.e. I = cPph and Rc = cPc,

leads to the following alternative form of the SB law:

( )

0.25

14

ph

b

cc

P

T

TP

=

(14)

where Pc = 1.043 × 10-14 Pa is the photon pressure of CMBR. Clearly, Eq.

(10a) is analogous to Eq. (14), while the latter is structurally identical to

the Poisson formula (13). Figure 7b depicts Eq. (14) as a dependence of

the Tb/Tc ratio on photon pressure Pph.

It is evident from Figures 4 and 7 that formulas (10a), (13) and (14)

describe qualitatively very similar responses in quantitatively vastly

dierent systems. e presence of such similar relations in otherwise

disparate physical systems can fundamentally be explained by the fact

that pressure as a force per unit area represents a key component of

the internal kinetic energy (dened as a product of gas volume and

pressure), while temperature is merely a physical manifestation of this

energy. Adding a force such as gas pressure to a physical system inevitably

boosts the internal kinetic energy and raises its temperature, a process

known in thermodynamics as compression heating. e direct eect

of pressure on a system’s temperature is thermodynamically described

by adiabatic processes. e pressure-induced thermal enhancement

at a planetary level portrayed in Figure 4 and accurately quantied by

Eq. (10a or 11) is analogous to a compression heating, but not fully

identical to an adiabatic process. e latter is usually characterized by

a limited duration and oentimes only applies to nite-size parcels of

air moving vertically through the atmosphere. Equation (11), on the

other hand, describes a surface thermal eect that is global in scope and

permanent in nature as long as an atmospheric mass is present within

the planet’s gravitational eld. Hence, the planetary RATE (Ts/Tna ratio)

could be understood as a net result of countless simultaneous adiabatic

processes continuously operating in the free atmosphere. Figures 4 and

7 also suggest that the pressure control of temperature is a universal

thermodynamic principle applicable to systems ranging in complexity

from a simple isothermal blackbody absorbing a homogeneous ux of

electromagnetic radiation to diverse planetary atmospheres governed

by complex non-linear process interactions and cloud-radiative

feedbacks. To our knowledge, this cross-scale similarity among various

pressure-temperature relationships has not previously been identied

and could provide a valuable new perspective on the working of

planetary climates.

Nevertheless, important dierences exist between Eq. (10a) and

these other simpler pressure-temperature relations. us, while the

Poisson formula and the SB radiation law can mathematically be

derived from ‘rst principles’ and experimentally tested in a laboratory,

Eq. (10a) could neither be analytically deduced from known physical

laws nor accurately simulated in a small-scale experiment. is is

because Eq. (10a) describes an emergent macro-level property of

planetary atmospheres representing the net result of myriad process

interactions within real climate systems that are not readily computable

using mechanistic (bottom-up) approaches adopted in climate models

or fully reproducible in a laboratory setting.

Potential limitations of the planetary temperature model

Equation (10b) describes long-term (30-year) equilibrium GMATs

of planetary bodies and does not predict inter-annual global temperature

variations caused by intrinsic uctuations of cloud albedo and/or ocean

heat uptake. us, the observed 0.82 K rise of Earth’s global temperature

since 1880 is not captured by our model, as this warming was likely

Figure 7: Known pressure-temperature kinetic relations: (a) Dry adiabatic response of the air/surface temperature ratio to pressure changes in a free dry atmosphere

according to Poisson’s formula (Eq. 13) with a reference pressure set to po = 100 kPa; (b) The SB radiation law expressed as a response of a blackbody temperature

ratio to variations in photon pressure (Eq. 14). Note the qualitative striking similarity of shapes between these curves and the one portrayed in Figure 4 depicting the

new planetary temperature model (Eq. 10a).

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not the result of an increased atmospheric pressure. Recent analyses of

observed dimming and brightening periods worldwide [97-99] suggest

that the warming over the past 130 years might have been caused by a

decrease in global cloud cover and a subsequent increased absorption of

solar radiation by the surface. Similarly, the mega shi of 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 signicant drop

in surface air pressure since the early Eocene. Pleistocene uctuations

of global temperature in the order of 3.0–8.0 K during the last 2 My

revealed by multiple proxies [100] are also not predictable by Eq. (10b)

if due to factors other than changes in total atmospheric pressure and/

or TOA solar irradiance.

e current prevailing view mostly based on theoretical

considerations and results from climate models is that the Pleistocene

glacial-interglacial cycles have been caused by a combination of three

forcing agents: Milankovitch orbital variations, changes in atmospheric

concentrations of greenhouse gases, and a hypothesized positive ice-

albedo feedback [101,102]. However, recent studies have shown that

orbital forcing and the ice-albedo feedback cannot explain key features

of the glacial-interglacial oscillations such as the observed magnitudes

of global temperature changes, the skewness of temperature response

(i.e. slow glaciations followed by rapid meltdowns), and the mid-

Pleistocene transition from a 41 Ky to 100 Ky cycle length [103-105]. e

only signicant forcing remaining in the present paleo-climatological

toolbox to explicate the Pleistocene cycles are variations in greenhouse-

gas concentrations. Hence, it is dicult to explain, from a standpoint

of the current climate theory, the high accuracy of Eq. (11) describing

the relative thermal eect 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 details below.

Since the relationship portrayed in Figure 4 is undoubtedly real, our

model results point toward the need to reexamine some fundamental

climate processes thought to be well understood for decades. For

example, we are currently testing a hypothesis that Pleistocene glacial

cycles might have been caused by variations in Earth’s total atmospheric

mass and surface air pressure. Preliminary results based on the ability

of an extended version of our planetary model (simulating meridional

temperature gradients) to predict the observed polar amplication

during the Last Glacial Maximum indicate that such a hypothesis is not

unreasonable. However, conclusive ndings from this research will be

discussed elsewhere.

According to the present understanding, Earth’s atmospheric

pressure has remained nearly invariant during the Cenozoic era (last

65.5 My). However, this notion is primarily based on theoretical

analyses [106], since there are currently no known geo-chemical proxies

permitting a reliable reconstruction of past pressure changes in a

manner similar to that provided by various temperature proxies such as

isotopic oxygen 18, alkenones and TEX86 in sediments, and Ar-N isotope

ratios and deuterium concentrations in ice. e lack of independent

pressure proxies makes the assumption of a constant atmospheric mass

throughout the Cenozoic a priori and thus questionable. Although

this topic is beyond the scope of our present study, allowing for the

possibility that atmospheric pressure on Earth might have varied

signicantly over the past 65.5 My could open exciting new research

venues in Earth sciences in general and paleoclimatology in particular.

Role of greenhouse gasses from a perspective of the new

model

Our analysis revealed a poor relationship between GMAT and the

amount of greenhouse gases in planetary atmospheres across a broad

range of environments in the Solar System (Figures 1-3 and Table 5).

is is a surprising result from the standpoint of the current Greenhouse

theory, which assumes that an atmosphere warms the surface of a planet

(or moon) via trapping of radiant heat by certain gases controlling the

atmospheric infrared optical depth [4,9,10]. e atmospheric opacity

to LW radiation depends on air density and gas absorptivity, which in

turn are functions of total pressure, temperature, and greenhouse-gas

concentrations [9]. Pressure also controls the broadening of infrared

absorption lines in individual gases. erefore, the higher the pressure,

the larger the infrared optical depth of an atmosphere, and the stronger

the expected greenhouse eect would be. According to the current

climate theory, pressure only indirectly aects global surface temperature

through the atmospheric infrared opacity and its presumed constraint on

the planet’s LW emission to Space [9,107].

ere are four plausible explanations for the apparent lack of a

close relationship between GMAT and atmospheric greenhouse gasses

in our results: 1) e amounts of greenhouse gases considered in our

analysis only refer to near-surface atmospheric compositions and

do not describe the infrared optical depth of the entire atmospheric

column; 2) e analysis lumped all greenhouse gases together and did

not take into account dierences in the infrared spectral absorptivity of

individual gasses; 3) e eect 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 eect 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 evidence discussed earlier, we argue that option #4 is

the most likely reason for the poor predictive skill of greenhouse

gases with respect to planetary GMATs revealed in our study (Figures

1-3). By denition, the infrared optical depth of an atmosphere is a

dimensionless quantity that carries no units of force or energy [3,4,9].

erefore, it is dicult 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) dened as the product of gas pressure (P,

Pa) and gas volume (V, m3), i.e. E (J) = PV. Hence, the direct eect of

pressure on a system’s internal energy and temperature follows straight

from fundamental parameter denitions in classical thermodynamics.

Generally speaking, kinetic energy cannot exist without a pressure

force. Even electromagnetic radiation has pressure.

In climate models, the eect of infrared optical depth on surface

temperature is simulated by mathematically decoupling radiative

transfer from convective heat exchange. Specically, the LW

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radiative transfer is calculated in these models without simultaneous

consideration of sensible- and latent heat uxes in the solution matrix.

Radiative transfer modules compute the so-called heating rates (K/

day) strictly as a function of atmospheric infrared opacity, which

under constant-pressure conditions solely depends on greenhouse-

gas concentrations. ese heating rates are subsequently added to the

thermodynamic portion of climate models and distributed throughout

the atmosphere. In this manner, the surface warming becomes a

function of an increasing atmospheric infrared opacity. is approach to

modeling of radiative-convective energy transport rests on the principle

of superposition, which is only applicable to linear systems, where the

overall solution can be obtained as a sum of the solutions to individual

system components. However, the integral heat transport within a

free atmosphere is inherently nonlinear with respect to temperature.

is is because, in the energy balance equation, radiant heat transfer

is contingent upon power gradients of absolute temperatures, while

convective cooling/heating depends on linear temperature dierences

in the case of sensible heat ux and on simple vapor pressure gradients

in the case of latent heat ux [4]. e latent heat transport is in turn

a function of a solvent’s saturation vapor pressure, which increases

exponentially with temperature [3]. us, the superposition principle

cannot be employed in energy budget calculations. e articial

decoupling between radiative and convective heat-transfer processes

adopted in climate models leads to mathematically and physically

incorrect solutions with respect to surface temperature. e LW

radiative transfer in a real climate system is intimately intertwined

with turbulent convection/advection as both transport mechanisms

occur simultaneously. Since convection (and especially the moist one)

is orders of magnitude more ecient 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

dierent temperature solutions in response to a changing atmospheric

composition than the ones obtained by current climate models.

Specically, a correctly coupled convective-radiative system will render

the surface temperature insensitive to variations in the atmospheric

infrared optical depth, a result indirectly supported by our analysis as

well. is topic requires further investigation beyond the scope of the

present study.

e direct eect of atmospheric pressure on the global surface

temperature has received virtually no attention in climate science thus

far. However, the results from our empirical data analysis suggest that it

deserves a serious consideration in the future.

eoretical implications of the new interplanetary

relationship

e hereto discovered pressure-temperature relationship quantied

by Eq. (10a) and depicted in Figure 4 has broad theoretical implications

that can be summarized as follows:

Physical nature of the atmospheric ‘greenhouse eect’: According

to Eq. (10b), the heating mechanism of planetary atmospheres is

analogous to a gravity-controlled adiabatic compression acting upon

the entire surface. is means that the atmosphere does not function

as an insulator reducing the rate of planet’s infrared cooling to space as

presently assumed [9,10], but instead adiabatically boosts the kinetic

energy of the lower troposphere beyond the level of solar input through

gas compression. Hence, the physical nature of the atmospheric

‘greenhouse eect’ is a pressure-induced thermal enhancement

(PTE) independent of atmospheric composition. is mechanism

is fundamentally dierent from the hypothesized ‘trapping’ of LW

radiation by atmospheric trace gases rst proposed in the 19th century

and presently forming the core of the Greenhouse climate theory.

However, a radiant-heat trapping by freely convective gases has never

been demonstrated experimentally. We should point out that the hereto

deduced adiabatic (pressure-controlled) nature of the atmospheric

thermal eect rests on an objective analysis of vetted planetary

observations from across the Solar System and is backed by proven

thermodynamic principles, while the ‘trapping’ of LW radiation by an

unconstrained atmosphere surmised by Fourier, Tyndall and Arrhenius

in the 1800s was based on a theoretical conjecture. e latter has later

been coded into algorithms that describe the surface temperature as a

function of atmospheric infrared optical depth (instead of pressure) by

articially decoupling radiative transfer from convective heat exchange.

Note also that the Ideal Gas Law (PV = nRT) forming the basis of

atmospheric physics is indierent to the gas chemical composition.

Eect of pressure on temperature: Atmospheric pressure

provides in and of itself only a relative thermal enhancement (RATE)

to the surface quantied by Eq. (11). e absolute thermal eect 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 eect derived

by Volokin and ReLlez [1] using a dierent 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 eect of the atmosphere would drop

to about 29.2 K due to a vastly reduced solar ux. In other words, the

absolute eect of pressure on a system’s temperature depends on the

background energy level of the environment. is implies that the

absolute temperature of a gas may not follow variations of pressure

if the gas energy absorption changes in opposite direction to that of

pressure. For instance, the temperature of Earth’s stratosphere increases

with altitude above the tropopause despite a falling air pressure, because

the absorption of UV radiation by ozone steeply increases with height,

thus osetting the eect of a dropping pressure. If the UV absorption

were constant throughout the stratosphere, the air temperature would

decrease with altitude.

Atmospheric back radiation and surface temperature: Since

(according to Eq. 10b) the equilibrium GMAT of a planet is mainly

determined by the TOA solar irradiance and surface atmospheric

pressure, the down-welling LW radiation appears to be globally a product

of the air temperature rather than a driver of the surface warming. In

other words, on a planetary scale, the so-called back radiation is a

consequence of the atmospheric thermal eect rather than a cause for

it. is explains the broad variation in the size of the observed down-

welling LW ux among celestial bodies irrespective of the amount of

absorbed solar radiation. erefore, a change in this thermal ux brought

about by a shi in atmospheric LW emissivity cannot be expected to

impact the global surface temperature. Any variation in the global

infrared back radiation caused by a change in atmospheric composition

would be compensated for by a corresponding shi in the intensity of

the vertical convective heat transport. Such a balance between changes

in atmospheric infrared heating and the upward convective cooling at

the surface is required by the First Law of ermodynamics. However,

current climate models do not simulate this compensatory eect of

sensible and latent heat uxes due to an improper decoupling between

radiative transfer and turbulent convection in the computation of total

energy exchange.

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Role of planetary albedos: e fact that Eq. (10b) accurately

describes planetary GMATs without explicitly accounting for the

observed broad range of albedos, i.e. from 0.136 to 0.9 (Table 2),

indicates that the shortwave reectivity of planetary atmospheres is

mostly an intrinsic property (a byproduct) of the climate system itself

rather than an independent driver of climate as currently believed. In

other words, it is the internal energy of the atmosphere maintained by

solar irradiance and air pressure that controls the bulk of the albedo.

An indirect support for this unorthodox conclusion is provided by

the observation that the amounts of absorbed shortwave radiation

determined by albedos show no physically meaningful relationship

with planetary GMATs. For example, data in Table 2 indicate that

Venus absorbs 3.7 times less solar energy per unit area than Earth, yet

its surface is about 450 K hotter than that of Earth; the Moon receives

on average 54 W m-2 more net solar radiation than Earth, but it is

about 90 K cooler on average than our planet. e hereto proposed

passive nature of planetary albedos does not imply that the global

cloud cover could not be inuenced by an external forcing such as solar

wind, galactic cosmic rays, and/or gravitational elds of other celestial

objects. Empirical evidence strongly suggests that it can [108-113], but

the magnitude of such inuences 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 reectivity as a byproduct of the available internal

energy rather than a driver of climate can also help explain the observed

remarkable stability of Earth’s albedo [54,114].

Climate stability: Our semi-empirical model (Equations 4a, 10b

and 11) suggests that, as long as the mean annual TOA solar ux and

the total atmospheric mass of a planet are stationary, the equilibrium

GMAT will remain stable. Inter-annual and decadal variations of global

temperature forced by uctuations of cloud cover, for example, are

expected to be small compared to the magnitude of the background

atmospheric warming because of strong negative feedbacks limiting

the albedo changes. is implies a relatively stable climate for a planet

such as Earth absent signicant shis 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

buered against sudden changes.

Eect of oceans and water vapor on global temperature: e 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. is nding may sound inexplicable

from the standpoint of the radiative Greenhouse theory, but it follows

logically from the new paradigm of a pressure-induced atmospheric

warming. e presence of liquid water on the surface of a planet requires

an air pressure greater than 612 Pa and an ambient temperature above

273.2 K. ese conditions are provided by the planet’s size and gravity,

its distance to the Sun, and the mass of the atmosphere. Hence, the

water oceans on Earth seem to be a thermodynamic consequence of

particular physical conditions set by cosmic arrangements rather than

an active controller of the global climate. Similarly, the hydrocarbon

lakes on the surface of Titan [115,116] are the result of a high

atmospheric pressure and an extremely cold environment found on that

moon. us, our analysis did not reveal evidence for the existence of a

feedback between planetary GMAT and a precipitable liquid solvent on

the surface as predicted by the current climate theory. Consequently,

the hypothesized runaway greenhouse, which requires a net positive

feedback between global surface temperature and the atmospheric LW

opacity controlled by water vapor [117], appears to be a model artifact

rather than an actual physical possibility. Indeed, as illustrated in Figure

4, the hot temperature of Venus oen cited as a product of a ‘runaway

greenhouse’ scenario [117,118] ts perfectly within the pressure-

dependent climate continuum described by Equations (10b) and (11).

Model Application and Validation

Encouraged by the high predictive skill and broad scope of validity

of Model 12 (Figure 2f) we decided to apply Eq. (10b) to four celestial

bodies spanning the breadth of the Solar System, i.e. Mercury, Europa,

Callisto and Pluto, which global surface temperatures are not currently

known with certainty. Each body is the target of either ongoing or

planned robotic exploration missions scheduled to provide surface

thermal data among other observations, thus oering an opportunity

to validate our planetary temperature model against independent

measurements.

e MESSENGER spacecra launched in 2004 completed the rst

comprehensive mapping of Mercury in March 2013 (http://messenger.

jhuapl.edu/). Among other things, the spacecra also took infrared

measurements of the planet’s surface using a special spectrometer

[119] that should soon become available. e New Horizons spacecra

launched in January 2006 [120] reached Pluto in July of 2015 and

performed a thermal scan of the dwarf planet during a yby. e

complete dataset from this yby were received on Earth in October of

2016 and are currently being analyzed. A proposed joint Europa-Jupiter

System Mission by NASA and the European Space Agency is planned to

study the Jovian moons aer 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. e atmospheric pressure on

Pluto is believed to vary between 1.0 and 4.0 Pa over the course of its

orbital period as a function of insolation-driven sublimation of nitrogen

and methane ices on the surface [122]. Each 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 Tna in

Eq. (10b) must be calculated from Eq. (4a), which requires knowledge

of the actual values of αe, ηe, and Rg. We assumed that Mercury had Rg =

0.0 W m-2, αe = 0.068 [123] and Moon-like thermo-physical properties

of the regolith (ηe = 0.00971). Input data for Europa and Callisto were

obtained from Spencer et al. [124] and Moore et al. [125], respectively.

Specically, in order to calculate ηe and Rg for these moons we utilized

equatorial temperature data provided by Spencer et al. [124] in their

Figure 1, and by Moore et al. [125] in their Fig. 17.7 along with a

theoretical formula for computing the average nighttime surface

temperature T at the equator based on the SB law, i.e.

( ) ( )

0.25

1 15

0.98

eg

SR

T

αη

σ

−+

=

(15)

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where S(1-α)ηe is the absorbed solar ux (W m-2) stored as heat into

the subsurface. e geothermal heat ux on Europa is poorly known.

However, based on thermal observations of Io reported by Veeder et al.

[126], we assumed Rg = 2.0 W m-2 for Europa. Using S = 50.3 W m-2, an

observed nighttime equatorial temperature T = 90.9 K and an observed

average night-side albedo α = 0.58 [124], we solved Eq. (15) for the

surface heat storage fraction to obtain ηe = 0.085 for Europa. A similar

computational procedure was employed for Callisto using α = 0.11 and

equatorial surface temperature data from Fig. 17.7 in Moore et al. [125].

is produced Rg = 0.5 W m-2 and ηe = 0.057. Using these values in

Eq. (15) correctly reproduced Callisto’s nighttime equatorial surface

temperature of ≈ 86.0 K. e much higher ηe estimates for Europa and

Callisto compared to ηe = 0.00971 for the Moon can be explained with

the large water-ice content on the surface of these Galilean moons.

Europa is almost completely covered by a thick layer of water ice, which

has a much higher thermal conductivity than the dry regolith. Also,

sunlight penetrates deeper into ice than it does into powdered regolith.

All this enables a much larger fraction of the absorbed solar radiation to

be stored into the subsurface as heat and later released at night boosting

the nighttime surface temperatures of these moons. Volokin and ReLlez

[1] showed that GMAT of airless bodies is highly sensitive to ηe.

Tabl e 6 lists the average global surface temperatures of the four

celestial bodies predicted by Eq. (10b) along with the employed input

data. According to our model, Mercury is about 117 K cooler on average

than 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. [93] reported initial

results from yby observations of Pluto taken by the Radio Experiment

(REX) instrument aboard the New Horizons spacecra in July 2015,

when the dwarf planet was approximately at 32.9 AU from the Sun.

Using the observed surface pressure of 1.05 ± 0.1 Pa (10.5 ± 1 μbar)

[93] our model predicts an average global temperature of 36.7 K for

Pluto. Stern et al. [128] reported a near-surface temperature of ≈ 38

K. However, this value was calculated from pre-yby global brightness

measurements rather than derived via spherical integration of spatially

resolved surface temperatures (Stern, personal communication). Since

global brightness temperatures tend to be higher than spherically

averaged kinetic surface temperatures [1], our model prediction may

well be within the uncertainty of Pluto’s true global temperature. We

will know more about this in 2017 when spatially resolved thermal

measurements obtained by New Horizons become available.

One should use caution when comparing results from Eq. (10b)

to remotely sensed ‘average temperatures’ commonly quoted for

celestial bodies with tenuous atmospheres such as the moons of Jupiter

and Neptune. Studies oentimes report the so-called ‘brightness

temperatures’ retrieved at specic wavelengths that have not been

subjected to a proper spherical integration. As pointed out by Volokin

and ReLlez [1], due to Hölder’s inequality between integrals, calculated

brightness temperatures of spherical objects can be signicantly higher

than actual mean kinetic temperatures of the surface. Since Eq. (10b)

yields spherically averaged temperatures, its predictions for airless

bodies are expected to be lower than the disk-integrated brightness

temperatures typically quoted in the literature.

Conclusion

For 190 years the atmosphere has been thought to warm Earth

by absorbing a portion of the outgoing LW infrared radiation and

reemitting it back toward the surface, thus augmenting the incident

solar ux. is conceptualized continuous absorption and downward

reemission of thermal radiation enabled by certain trace gases known

to be transparent to solar rays while opaque to electromagnetic

long-wavelengths has been likened to the trapping of heat by glass

greenhouses, hence the term ‘atmospheric greenhouse eect’. 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 eect’ stuck in science.

e hypothesis that a freely convective atmosphere could retain

(trap) radiant heat due its opacity has remained undisputed since its

introduction in the early 1800s even though it was based on a theoretical

conjecture that has never been proven experimentally. It is important to

note in this regard that the well-documented enhanced absorption of

thermal radiation by certain gases does not imply an ability of such gases

to trap heat in an open atmospheric environment. is is because, in

gaseous systems, heat is primarily transferred (dissipated) by convection

(i.e. through uid motion) rather than radiative exchange. If gases of

high LW absorptivity/emissivity such as CO2, methane and water vapor

were indeed capable of trapping radiant heat, they could be used as

insulators. However, practical experience has taught us that thermal

radiation losses can only be reduced by using materials of very low LW

Surface Atmospheric

Pressure (Pa)

αe (fraction)

ηe (fraction)

Rg (W m-2)

Predicted Average Global

Surface Temperature at Specic Orbital Distances from the Sun

Aphelion Semi-major Axis Perihelion

Mercury 5 × 10-10

αe = 0.068

ηe = 0.00971

Rg = 0.0

296.8 K

(0.459 AU)

323.3 K

(0.387 AU)

359.5 K

(0.313 AU)

Europa 10-7

αe = 0.62

ηe = 0.085

Rg = 2.0

98.1 K

(5.455 AU)

99.4 K

(5.203 AU)

100.7 K

(4.951 AU)

Callisto 7.5 × 10-7

αe = 0.11

ηe = 0.057

Rg = 0.5

101.2 K

(5.455 AU)

103.2 K

(5.203 AU)

105.4 K

(4.951 AU)

Pluto 1.05

αe = 0.132

ηe = 0.00971

Rg = 0.0

30.0 K

(49.310 AU)

33.5 K

(39.482 AU)

38.6 K

(29.667 AU)

Table 6: Average global surface temperatures predicted by Eq. (10b) for Mercury, Europa, Calisto and Pluto. Input data on orbital distances (AU) and total atmospheric

pressure (Pa) were obtained from the NASA Solar System Exploration [48] website, the NASA Planetary Factsheet [32] and Gladstone et al. [93]. Solar irradiances required

by Eq. (10b) were calculated from reported orbital distances as explained in the text. Values of αe, ηe and Rg for Europa and Callisto were estimated from observed data by

Spencer et al. [124] and Moore et al. [125] respectively (see text for details).

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absorptivity/emissivity and correspondingly high thermal reectivity

such as aluminum foil. ese materials are known among engineers at

NASA and in the construction industry as radiant barriers [129]. It is

also known that high-emissivity materials promote radiative cooling.

Yet, all climate models proposed since 1800s are built on the premise

that the atmosphere warms Earth by limiting radiant heat losses of the

surface through the action of infrared absorbing gases alo.

If a trapping of radiant heat occurred in Earth’s atmosphere, the

same mechanism should also be expected to operate in the atmospheres

of other planetary bodies. us, the Greenhouse concept should be able

to mathematically describe the observed variation of average planetary

surface temperatures across the Solar System as a continuous function

of the atmospheric infrared optical depth and solar insolation. However,

to our knowledge, such a continuous description (model) does not

exist. Furthermore, measured magnitudes of the global down-welling

LW ux on planets with thick atmospheres such as Earth and Venus

indicate that the lower troposphere of these bodies contains internal

kinetic energy far exceeding the solar input [6,12,14]. is fact cannot

be explained via re-radiation of absorbed outgoing thermal emissions

by gases known to supply no additional energy to the system. e desire

to explicate the sizable energy surplus evident in the tropospheres of

some terrestrial planets provided the main impetus for this research.

We combined high-quality planetary data from the last three

decades with the classical method of dimensional analysis to search for

an empirical model that might accurately and meaningfully describe

the observed variation of global surface temperatures throughout the

Solar System while also providing a new perspective on the nature of the

atmospheric thermal eect. Our analysis revealed that the equilibrium

global surface temperatures of rocky planets with tangible atmospheres

and a negligible geothermal surface heating can reliably be estimated

across a wide range of atmospheric compositions and radiative regimes

using only two forcing variables: TOA solar irradiance and total surface

atmospheric pressure (Eq. 10b with Tna computed from Eq. 4c).

Furthermore, the relative atmospheric thermal enhancement (RATE)

dened 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) quantied by Eq. (10a)

appears to describe an emergent macro-level thermodynamic property

of planetary atmospheres heretofore unbeknown to science. e

physical signicance 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 eect of pressure on

temperature albeit in the context of a dierent macro-physical system.

To our knowledge, this is the rst model accurately describing the

average surface temperatures of planetary bodies throughout the Solar

System in the context of a thermodynamic continuum using a common

set of drivers.

e planetary temperature model consisting of Equations (4a),

(10b), and (11) has several fundamental theoretical implications, i.e.

• e ‘greenhouse eect’ is not a radiative phenomenon driven

by the atmospheric infrared optical depth as presently believed,

but a pressure-induced thermal enhancement analogous to

adiabatic heating and independent of atmospheric composition;

• e down-welling LW radiation is not a global driver of surface

warming as hypothesized for over 100 years but a product of

the near-surface air temperature controlled by solar heating

and atmospheric pressure;

• e albedo of planetary bodies with tangible atmospheres is not

an independent driver of climate but an intrinsic property (a

byproduct) of the climate system itself. is does not mean that

the cloud albedo cannot be inuenced by external forcing such

as solar wind or galactic cosmic rays. However, the magnitude

of such inuences is expected to be small due to the stabilizing

eect of negative feedbacks operating within the system. is

understanding explains the observed remarkable stability of

planetary albedos;

• e equilibrium surface temperature of a planet is bound to

remain stable (i.e. within ± 1 K) as long as the atmospheric

mass and the TOA mean solar irradiance are stationary. Hence,

Earth’s climate system is well buered against sudden changes

and has no tipping points;

• e proposed net positive feedback between surface

temperature and the atmospheric infrared opacity controlled

by water vapor appears to be a model artifact resulting from

a mathematical decoupling of the radiative-convective heat

transfer rather than a physical reality.

e hereto reported ndings point toward the need for a paradigm

shi in our understanding of key macro-scale atmospheric properties and

processes. e implications of the discovered planetary thermodynamic

relationship (Figure 4, Eq. 10a) are fundamental in nature and require

careful consideration by future research. We ask the scientic community

to keep an open mind and to view the results presented herein as a possible

foundation of a new theoretical framework for future exploration of

climates on Earth and other worlds.

Appendices

Appendix A. Construction of the Dimensionless π Variables

Table 1 lists 6 generic variables (Ts, Tr, S, Px, Pr and ρx) composed of

4 fundamental dimensions: mass [M], length [L], time [T], and absolute

temperature [Θ]. According to the Buckingham Pi theorem [27], this

implies the existence of two dimensionless πi products per set. To

derive the πi variables we employed the following objective approach.

First, we hypothesized that a planet’s GMAT (Ts) is a function of all 5

independent variables listed in Table 1, i.e.

( )

rs xr x

T T , S , P , P , = ρ

ƒ

(A.1)

is unknown function is described to a rst approximation as a simple

product of the driving variables raised to various powers, i.e.

abc d e

s r xr x

T T SPP

ρ

≈

(A.2)

where a, b, c, d and e are rational numbers. In order to determine the

power coecients, Eq. (A.2) is cast in terms of physical dimensions of

the participating variables, i.e.

[ ] [ ]

3 12 12 3

M T M L T M L T M L

− −− −− −

Θ≈Θ

b c de

a (A.3)

Satisfying the requirement for dimensional homogeneity of Eq.

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(A.2) implies that the sum of powers of each fundamental dimension

must be equal on both sides of Eq. (A.3). is allows us to write four

simultaneous equations (one per fundamental dimension) containing

ve unknowns, i.e.

[ ]

[ ]

[ ]

[ ]

1 :

0 :

3 0 :

3 2 2 0 :

a

bcd e M

cd e L

bcd T

= Θ

++ +=

−− − =

−− − =

(A.4)

System (A.4) is underdetermined and has the following solution: a

= 1, b = 2e, and c = -(3e + d). Note that, in the DA methodology,

one oentimes 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 eect on the derivation of the

sought dimensionless πi products.

Substituting the above roots in Eq. (A.2) reduces the original ve

unknowns to two: d and e, i.e.

( )

3

12

ed

e de

s r x rx

T TS P P

ρ

−+

≈

(A.5a)

ese solution powers may now be assigned arbitrary values, although

integers such as 0, 1 and -1 are preferable, for they oer the simplest

solution leading to the construction of proper πi variables. Setting d = 0

and e = -1 reduces Eq. (A.5a) to

1 23 1

s r xx

T TS P

ρ

−−

≈

(A.5b)

providing the rst pair of dimensionless products:

3

12

2

;

π= π=

sx

rx

TP

TS

ρ

(A.6)

e second pair of πi variables emerges upon setting d = -1 and e = 0 in

Eq. (A.5a), i.e.

12

; π= π=

sx

rr

TP

TP

(A.7)

us, the original function (A.1) consisting of six dimensioned

variables has been reduced to a relationship between two dimensionless

quantities, i.e. π1 = f (π2). is relationship must further be investigated

through regression analysis.

Appendix B. Estimation of Mars’ GMAT and Surface

Atmospheric Pressure

Although Mars is the third most studied planetary body in the

Solar System aer Earth and the Moon, there is currently no consensus

among researchers regarding its mean global surface temperature (TM).

TM values reported over the past 15 years span a range of 40 K. Examples

of disparate GMATs quoted for the Red Planet include 200 K [79], 202

K [82,130], 210 K [32], 214 K [80], 215 K [6,81], 218 K [77], 220 K [76],

227 K [131] and 240 K [78]. e most frequently cited temperatures fall

between 210 K and 220 K. However, a close examination of the available

thermal observations reveals a high improbability for any of the above

estimates to represent Mars’ true GMAT.

Figure B.1 depicts hourly temperature series measured at 1.5 m

aboveground by Viking Landers 1 and 2 (VL1 and VL2 respectively) in

the late 1970s [60]. e VL1 record covers about half of a Martian year,

while the VL2 series extends to nearly 1.6 years. e VL1 temperature

series captures a summer-fall season on a site located at about 1,500 m

below Datum elevation in the subtropics of Mars’ Northern Hemisphere

(22.5o N). e arithmetic average of the series is 207.3 K (Fig. B.1a).

Since the record lacks data from the cooler winter-spring season, this

value is likely higher than the actual mean annual temperature at that

location. Furthermore, observations by the Hubble telescope from the

mid-1990s indicated that the Red Planet may have cooled somewhat

since the time of the Viking mission [132,133]. Because of a thin

atmosphere and the absence of signicant cloud cover and perceptible

water, temperature uctuations near the surface of Mars are tightly

coupled to diurnal, seasonal and latitudinal variations in incident solar

radiation. is causes sites located at the same latitude and equivalent

altitudes to have similar annual temperature means irrespective of

their longitudes [134]. Hence, one could reliably estimate a latitudinal

temperature average on Mars using point observations from any

elevation by applying an appropriate lapse-rate correction for the

average terrain elevation of said latitude.

At 22.5o absolute latitude, the average elevation between Northern

and Southern Hemisphere on Mars is close to Datum level, i.e. about

1,500 m above the VL1 site. Adjusting the observed 207.3 K temperature

average at VL1 to Datum elevation using a typical near-surface Martian

lapse rate of -4.3 K km-1 [78] produces ~201 K for the average summer-

fall temperature at that latitude. Since the mean surface temperature

Figure B.1: Near-surface hourly temperatures measured on Mars by (a) Viking Lander 1 at Chryse Planitia (22.48° N, 49.97° W, Elevation: -1,500 m); and (b) Viking

Lander 2 at Utopia Planitia (47.97° N, 225.74° W, Elevation: -3,000 m) (Kemppinen et al. [60]; data downloaded from: http://www-k12.atmos.washington.edu/k12/

resources/mars_data-information/data.html). Black dashed lines mark the arithmetic average (Tmean) of each series. Grey dashed lines highlight the range of most

frequently reported GMAT values for Mars, i.e. 210–240 K. The average diurnal temperature can only exceed 210 K during the summer; hence, all Martian latitudes

outside the Equator must have mean annual temperatures signicantly lower than 210 K.

Temperature Model. Environ Pollut Climate Change 1: 112.

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of a sphere is typically lower than its subtropical temperature average,

we can safely conclude based on Figure B.1a that Mars’ GMAT is likely

below 201 K. e mean temperature at the VL2 site located at ~48o N

latitude and 3,000 m below Datum elevation is 191.1 K (Fig. B.1b). e

average terrain elevation between Northern and Southern Hemisphere

at 48o absolute latitude is about -1,500 m. Upon adjusting the VL2

annual temperature mean to -1,500 m altitude using a lapse rate of

-4.3 K km-1 we obtain 184.6 K. Since a planet’s GMAT numerically falls

between the mean temperature of the Equator and that of 42o absolute

latitude, the above calculations suggest that Mars’ GMAT is likely

between 184 K and 201 K.

A close examination of the Viking record also reveals that average

diurnal temperatures above 210 K only occur on Mars during the

summer season and, therefore, cannot possibly represent an annual

mean for any Martian latitude outside the Equator. On the other hand,

frequently reported values of Mars’ GMAT in excess of 210 K appear to

be based on the theoretical expectation that a planet’s average surface

temperature should exceed the corresponding eective radiating

temperature produced by Eq. (3) [6,78], which is Te ≈ 212 K for Mars.

is presumption is rooted in the a priori assumption that Te represents

a planet’s average surface temperature in the absence of atmospheric

greenhouse eect. However, Volokin and ReLlez [1] have shown

that, due to Hölder’s inequality between integrals, the mean physical

temperature of a spherical body with a tenuous atmosphere is always

lower than its eective radiating temperature computed from the

globally integrated absorbed solar ux. In other words, Eq. (3) yields

non-physical temperatures for spheres. Indeed, based on results from

a 3-D climate model Haberle [130] concluded that Mars’ mean global

surface temperature is at least 8 K cooler than the planet’s eective

radiating temperature. erefore, Mars’ GMAT must be inferred from

actual measurements rather than from theoretical calculations.

In order to obtain a reliable estimate of Mars’ GMAT, we calculated

the mean annual temperatures at several Martian latitudes employing

near-surface time series measured in-situ by Viking Landers and the

Curiosity Rover, and remotely by the Mars Global Surveyor (MGS)

spacecra. e Radio Science Team (RST) at Stanford University

utilized radio occultation of MGS refraction data to retrieve seasonal

time-series of near-surface atmospheric temperature and pressure on

Mars [61,62,135]. We utilized MGS-RST data obtained between 1999

and 2005. Calculated mean temperatures from in-situ measurements

were adjusted to corresponding average terrain elevations of target

latitudes using a lapse rate of -4.3 K km-1 [78]. Figure B.2 portrays

the estimated Mean Annual near-surface Temperatures (MAT) at ve

absolute Martian latitudes (gray dots) along with their standard errors

(vertical bars). e equatorial MAT was calculated from Curiosity Rover

observations; temperatures at absolute latitudes 0.392 rad (22.48o) and

0.837 rad (47.97o) were derived from VL measurements, while these

at latitudes 1.117 rad (64o) and 1.396 rad (80o) were estimated from

MGS-RST data. e black curve represents a third-order polynomial

tted through the latitudinal temperature averages and described by the

polynomial:

( ) ( )

23

202.888 0.781801 22.3673 3.16594 B.1TL L L L=−−−

with L being the absolute latitude (rad). MAT values predicted by

Eq. (B.1) for Mars’ Equatorial and Polar Regions agree well with

independent near-surface temperatures remotely measured by the

Mars Climate Sounder (MCS), a platform deployed aer MGS in

2006 [136]. Shirley et al. [136] showed that, although separated in

time by 2-5 years, MCS temperature proles 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 proles over the Mars’ northern and southern Polar

Regions, respectively at about 75o absolute latitude. e average winter

surface temperature between the two Hemispheres for this latitude

is about 148.5 K. is compares favorably with 156.4 K produced by

Eq. (B.1) for 75o (1.309 rad) latitude considering that MAT values are

expected to be higher than winter temperature averages. Figures 4 and

5 of Shirley et al. [136] portray average temperature proles retrieved

by MGS-RST and MCS over lowlands (165o – 180o E) and highlands

(240o - 270o E) of the Mars’ equatorial region (8o N - 8o S), respectively.

For highlands (≈5 km above Datum), the near-surface temperature

appears to be around 200 K, while for lowlands (≈2.5 km below Datum)

it is ≈211 K. Since most of Mars’ equatorial region lies above Datum, it

is likely that Mars’ equatorial MAT would be lower than 205.5 K and

close to our independent estimate of ≈203 K based on Curiosity Rover

measurements.

Mars’ GMAT (TM) was calculated via integration of polynomial

(B.1) using the formula:

( ) ( )

/2

0

cos B.2

M

T T L L dL

π

=

∫

(B.2)

where 0 ≤ cosL ≤ 1 is a polar-coordinate area-weighting factor.

e result is TM = 190.56 ± 0.7 K (Figure B.2). is estimate, while

signicantly lower than GMAT values quoted in recent publications,

agrees quite well with spherically integrated brightness temperatures

of Mars retrieved from remote microwave observations during the

late 1960s and early 1970s [85-87]. us, according to Hobbs et al.

[85] and Klein [86], the Martian mean global temperature (inferred

from measurements at wavelengths between 1 and 21 cm) is 190 –

193 K. Our TM estimate is also consistent with the new mean surface

temperature of the Moon (197.35 K) derived by Volokin and ReLlez

[1] using output from a validated NASA thermo-physical model [29].

Since Mars receives 57% less solar ittadiance than the Moon and has

a thin atmosphere that only delivers a weak greenhouse eect [9], it

makes a physical sense that the Red Planet would be on average cooler

than our Moon (i.e. TM < 197.3K). Moreover, if the average temperature

Figure B.2: Mean annual surface air temperatures at ve Martian absolute

latitudes (gray dots) estimated from data provided by Viking Landers, Curiosity

Rover, and the Mars Global Surveyor Radio Science Team. Each dot represents

a mean annual temperature corresponding to the average terrain elevation

between Northern and Southern Hemisphere for particular latitude. The black

curve depicts a third-order polynomial (Eq. B.1) tted through the latitudinal

temperature means using a non-linear regression. Mars’ GMAT, TM = 190.56

K (marked by a horizontal gray dashed line) was calculated via integration of

polynomial (B.1) using formula (B.2).

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Volume 1 • Issue 2 • 1000112Environ Pollut Climate Change, an open access journal

of the lunar equator (Moon’s warmest latitude) is 213 K as revealed by

NASA Diviner observations [1,29], it is unlikely that Mars’ mean global

temperature would be equal to or higher than 213 K as assumed by

many studies [6,76-78,80,131]

Published values of Mars’ average surface atmospheric pressures

range from 600 Pa to 700 Pa [6,32,78,80,83,84]. Since this interval was

too broad for the target precision of our study, we employed MGS-RST

data retrieved from multiple latitudes and seasons between 1999 and

2005 to calculate a new mean surface air pressure for the Red Planet.

Our analysis produced P = 685.4 ± 14.2 Pa, an estimate within the

range of previously reported values.

Funding Sources

This research did not receive any specic grant from funding agencies in the

public, commercial, or not-for-prot sectors.

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