Thermal Gradients on Planetary Bodies and the Molar Mass Ideal Gas Law
Robert IH*
Faculty of Science and Engineering, Federation University, Mt Helen, Ballarat, Australia*

Corresponding author: Robert IH, Faculty of Science and Engineering, Federation University, Mt Helen, Ballarat,

Australia, Tel: 61418352801; E-mail: robertholmes@students.federation.edu.au

Received: October 20, 2017; Accepted: February 08, 2018; Published: February 10, 2018

Abstract
It has always been complicated mathematically, to calculate the average near surface atmospheric temperature on planetary bodies with athick atmosphere. Usually, the Stefan Boltzmann (S-B) black body law is used to provide the effective temperature, then debate arises about
the size or relevance of additional complicating factors, including the albedo and the greenhouse effect. Presented here is a simple and reliable
method of accurately calculating the average near surface atmospheric temperature on planetary bodies which possess a surface atmosphericpressure of over 10 KPa. The formula used is the molar mass version of the ideal gas law. This method requires a gas constant and themeasurement of only three gas parameters; the average near-surface atmospheric pressure, the average near surface atmospheric density
and the average mean molar mass of the near-surface atmosphere. This indicates that all information on the effective plus the residual near-surface atmospheric temperature on planetary bodies with thick atmospheres, is automatically ‘baked-in’ to the three mentioned gas
parameters. It is known that whenever an atmospheric pressure exceeds 10 KPa, convection then dominates over radiative interactions as themain method of energy transfer, and a rising thermal gradient is formed. This rising thermal gradient continues on down, (if there is a
depression or a mine shaft) to even below the average surface level. Given this thermodynamic situation, it is very likely that no one gas has
an anomalous effect on atmospheric temperatures that is significantly more than any other gas. In short; there is unlikely to be any
significant warming from the greenhouse effect on any planetary body in the parts of atmospheres which are >10 KPa. Instead, it is proposedthat the residual temperature difference between the S-B effective temperature and a measured near-surface temperature (the atmosphericeffect) is a thermal enhancement which is actually caused by auto-compression.

Keywords: Greenhouse effect; Temperatures of planets; Earth temperature; Venus temperature; Auto-C; Climate sensitivity; Global climatechange; Global warming; Atmospheric thermal gradientIntroduction
It is a postulate of this work that any warming effects arising from atmospheric greenhouse gases such as CO₂, are subjected
to a 100% rate of negative feedbacks which are inherent in the climate system. The detail of these feedbacks will not be

Citation: Robert IH. Thermal Gradients on Planetary Bodies and the Molar Mass Ideal Gas Law. J Phys Astron. 2018; 6(1):134© 2017 Trade Science Inc.

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www.tsijournals.com | January-2018
outlined here, but the open nature of the atmosphere, coupled with the following thermodynamic arguments, indicate that thispostulate makes sense.

It is known that planetary bodies which have thick atmospheres, naturally set up a rising thermal gradient in that part of theatmosphere which is higher than a pressure of 10 KPa, (0.1bar) until that bodies’ surface is reached [1] FIG. 1. Less wellknown is that this rising temperature gradient continues even below the surface [2] making it difficult to attribute to thegreenhouse effect. In this denser part of the atmosphere, (on Earth, the troposphere) convection and adiabatic autocompression effects rule over radiative or ‘greenhouse’ effects in the determination of atmospheric temperatures and thethermal gradient. However, higher up in the atmosphere, once the atmospheric pressure drops below 10 KPa (0.1 bar) thenradiative effects dominate. This is because the atmosphere there is too thin to initiate convection or any warming due to autocompression. Although the term ‘auto-compression’ may be unfamiliar to some, this can be seen as simply an engineeringterm for what meteorologists call the ‘lapse rate’ and in astronomy is called the ‘Kelvin-Helmholtz’ contraction. Under thelatter, the contraction and so compression of a large inter-stellar molecular gas cloud under gravity, achieves such hightemperatures that nuclear fusion initiates, and a star is born [3].

FIG 1. A thermal gradient appears in all planetary atmospheres >10 KPa (0.1 bar) [1]

Using this knowledge, an exacting yet simple method is introduced, which enables the average near-surface atmospherictemperature of any planetary body with an atmospheric pressure of over 10 KPa, to become easily and quickly calculated. A
molar version of the ideal gas law is utilised formulas 5 and 6, which consists of one gas constant and three basic atmosphericgas parameters; the average near-surface atmospheric pressure, the average near-surface atmospheric density and the meanmolar mass of the near-surface atmosphere.

This formula proves itself here, to be not only more accurate than any other method heretofore used, but is far simpler tocalculate. It requires no input from parameters previously thought to be essential for the calculation of atmospherictemperatures; for example, solar insolation, albedo, greenhouse gas content, ocean circulation and cloud cover among manyothers. The reason these are not required, is because they, (and all others) are already automatically ‘baked-in’ to the three2

www.tsijournals.com | January-2018
gas parameters mentioned. Note: although terms for insolation intensity and auto-compression are not used in the formula, itis proposed that these two are still what virtually determine the average near-surface planetary atmospheric temperature.

Venus is the planet which has been hard to explain
There has always been difficulty in explaining, or in formulating a simple method to satisfactorily explain or calculate thevery high surface atmospheric temperature of the planet Venus using conventional mathematical means or by employing thegreenhouse gas hypothesis. Here, the molar mass version of the ideal gas law will be used to simply and accurately determine
the surface temperature of this planet, by the measurement of just three variable gas parameters and the knowledge of one
fixed gas constant.

Molar mass version of ideal gas law calculates planetary surface temperatures
The ideal gas law may be used to more accurately determine surface temperatures of planets with thick atmospheres than theS-B black body law [4], if a density term is added; and if kg/m³ is used for density instead of gms/m³, the volume term V may
be dropped. This formula then may be known as the molar mass version of the ideal gas law.

The ideal gas law is; PV = nRT(1)

Convert to molar mass; PV = m/M.RT(2)

Convert to density; PM/RT = m/V = ρ(3)Drop the volume term; ρ = P/(R.T/M)(4)

Find for temperature;

T

P

 

 R 
M(5)

V= Volume
m= Massn= Number of moles
T= Near-surface atmospheric temperature in Kelvin
P= Near-surface atmospheric pressure in KPa
R= Gas constant (m³, KPa, Kelvin⁻¹, mol⁻¹) = 8.314
ρ= Near-surface atmospheric density in kg/m³
M= Near-surface atmospheric mean molar mass (gm/mol⁻¹)
Alternatively, the molar mass version of the ideal gas law can be written thus

T  PM

R

(6)

Methodology involves calculating average near-surface temperature of planets
Formula 5 is here used throughout.
Using the properties of Venus [5],3

www.tsijournals.com | January-2018

T

9200
65 

 8.314 43.45 


Venus calculated surface temperature = 739.7K
Using the properties of Earth from Wiki, [6]

T

101.3
1.225 
 8.314 
28.97 


Earth calculated surface temperature = 288.14K

Venus is calculated at 739.7K, which is given by NASA as ~740K. Earth is calculated at 288K, currently its quoted byNASA [7] at 288K. It will be noted that the average temperature of the surface of Titan was measured by the Voyager 1, andby the Huygens lander [8] and was probably used as an input to find the surface density; (the independently-measured surface
density on Titan could not be found in the literature). The 94K will therefore come out of the below formula, since it is arearrangement of formula 1. This could be seen as a circular argument. However, it is unlikely that if and when the density ofTitan is directly measured, for instance by the use of a dasymeter or similar, it will be significantly different from the
5.25kg/m³ stated here.

Calculate for Titan, data [9];

T

146.7
5.25 

 8.314 
28.0 


T

68.13
1.06 

 8.314 
28.97 

Titan calculated surface temperature = 93.6K
Calculate for Earth’s South Pole, data [10];Earth’s South Pole average calculated temperature = 224K (-49°C) Calculate for Mars;

T

0.69
0.02 

 8.314 43.34 T

0.9
0.02 

 8.314 43.34 
Mars calculated surface temperature = 180K to 234K

The average temperature on Mars is 210K; as suspected from other work [1] this method is inaccurate for Mars, due to thevery low and variable atmospheric pressure. Pressures here were measured at the Viking 1 landing site and varied between
690Pa and 900Pa according to the season. It is only in atmospheres with a pressure of over 10 KPa (0.1bar) that strong
convection and a troposphere/tropopause is formed, and its associated thermal gradient. Mars is included to demonstrate the4www.tsijournals.com | January-2018validity of the >10 KPa rule. For Mars, the mid-point between the summer and the winter pressures is used, which results in atemperature of 207K. The gas giants will now be assessed; note that these planets do not have a defined surface like theterrestrials planets have, so here they are given a ‘surface’ by using the Earth’s surface pressure of 101.3 KPa (1 atm) as alevel to use for this calculation.

Calculate for Jupiter [7];

T

101.3
0.16 

 8.314 

2.2 


Jupiter calculated temperature at 1atm of pressure = 167K
Calculate for Saturn [7];

T

101.3
0.19 

 8.314 
2.07 


Saturn calculated temperature at 1atm of pressure = 132.8K
Calculate for Uranus [7];

T

101.3
0.420 
 8.314 
2.64 


Uranus calculated temperature at 1atm of pressure = 76.6K
Calculate for Neptune [7];

T

101.3
0.450 
 8.314 
2.53 
T

101.3
0.450 
 8.314 
2.69 
In the case of Neptune, NASA gave two values for mean molar mass; 2.53 and 2.69, this necessitated two separate
calculations to give a high and a low of calculated temperatures. Neptune’s calculated temperature at 1atm of pressure =68.5K to 72.8K. The temperature on Neptune at 1atm of pressure is 72K; this lies quite between the two calculated
temperatures (TABLE 1).

TABLE 1. Comparison of calculated and actual average surface temperatures.
Planetary body
Venus
Earth
South Pole of Earth
Titan
Mars (low pressure)
Jupiter
SaturnUranus
NeptuneCalculated temperature Kelvin
739.7
288.14224
93.6
180 to 234
167
132.8
76.6
68.5 to 72.8

Actual temperature Kelvin
740
288
224.5
94
210
165
134
76
72

Error
0.04%
0.00%
0.20%
0.42%
1.40%
1.20%
0.89%
0.79%
0.00%
5

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Analysis about the postulate and hypothesis being presented herein
If this simple relationship between surface atmospheric density, pressure and molar mass is an accurate method of predicting
surface temperatures on bodies with a thick atmosphere, it will necessarily be informative about what actually determines
these planetary surface temperatures, and will have important implications for climate sensitivity.
In short, a postulate being put forward here, is that in the case of Earth, solar insolation provides the ‘first’ 255 Kelvin – inaccordance with the black body law [11]; this being the ‘effective’ or the ‘base’ level. Then gravitationally induced autocompression provides the ‘other’ 33 Kelvin, termed the ‘residual’, to arrive at the known and measured average global
temperature of 288 Kelvin. The ‘other’ 33 Kelvin is not hypothesised to be provided by the greenhouse effect, because if itwas, the molar mass version of the ideal gas law would not then work to accurately calculate real planetary temperatures in
the case of small incremental changes to gas levels, as it clearly does here.

Temperature in a gas is a measure of the average kinetic energy of the particles in the gas. When atmospheric gas pressure
exceeds 10 KPa, a temperature gradient is set up from that pressure level [1], down to the planetary surface (this thermal
gradient is known and measured to continue even below the surface, if there is for example, a mine shaft). It is postulated
here, that the cause of this thermal gradient is gravity-induced auto-compression. In general terms, the surface temperaturesets up convective overturning of the troposphere, which is adiabatic through much of the convection cycle [2], and thiscombines with gravitationally induced atmospheric auto-compression to create the observed tropospheric thermalenhancement and temperature gradient.
The origins of this thermal effect on gases go back to James Maxwell, who, in his 1872 book ‘Theory of Heat’ [12]demonstrated that the formation of the thermal gradient from the tropopause downwards is assisted by convection and moreparticularly, the increasing atmospheric pressure, which itself is a result of a combination of the Earth’s gravitational field
and the atmospheric density.
“In the convective equilibrium of temperature, the absolute temperature is proportional to the pressure.” [12].

The idea of a thermal gradient naturally forming in any column of gas in a gravitational field was first proposed in the 1860’s
by Loschmidt [13]. At the time, Maxwell thought that this idea violated the second law of thermodynamics, yet as has beenshown here, derivations of Maxwell’s own ideal gas law is an excellent predictor of temperatures – when the atmosphere is
thick enough to be compressed by a gravitational field.

The controversy between Loschmidt on one side, with Maxwell and Boltzmann on the other, raged for some time and was
finally experimentally tested in 2007, with the results published by Graeff [14]. Graeff’s experiments concluded that agravitationally-induced temperature gradient does spontaneously develop in sealed columns of both air and water – thebottom of the column being warmer than the top. The theoretical amounts of warming according to Graeff should be 0.07K/m
and 0.04 K/m respectively. Graeff’s experimental apparatus reported 0.07K/m and 0.05K/m – so basically confirming
6

www.tsijournals.com | January-2018
Loschmidt’s predictions. The thermal gradient appeared, despite the reverse gradient being prevalent in the immediate
environment of the experiment. Loschmidt originally has said that the second law of thermodynamics needed to be re-statedto include the effects of gravitational fields on fluids.

The adiabatic auto-compression hypothesis enunciated herein, states that convection/pressure/lapse rate effects dominate overradiative effects in regions of all planetary atmospheres >0.1 bar and a temperature gradient is naturally formed. In effect,
gravity formes a density and a temperature gradient; pressure is a corollary.

Auto-compression is known and used daily in mining
Auto-compression is well known in underground mining, and is used by ventilation engineers to calculate how hot the mine
air will get, so that they know how much cooling air to provide at each level. The effect of auto-compression can becalculated by the following relationship;
Pe = Ps exp(gH/RT)
Where;
Pe = Absolute pressure at end of column (KPa)
Ps = Absolute pressure at start of column (KPa)g = Acceleration due to gravity (m/s²)
H = Vertical depth (m)As can be clearly seen, this effect primarily relies on pressure and gravity, which will be different for each planetary body.

Mechanism is adiabatic
Note that we are examining an adiabatic process, and when a gas parcel expands adiabatically, as it does when rising in a
gravitational field, it does positive work – and the kinetic energy drops and so the temperature drops. However, when a gas
parcel is compressed, as it is when is descends in a gravitational field, then it does negative work, and its kinetic energy rises
and so its temperature goes up. Why does the kinetic energy of the gas rise when descending? It’s because some of its
potential energy is converted to enthalpy, so producing an increase in pressure, specific internal energy and hence,
temperature in accordance with the following equation;
H = PV + U
Where;
H = Enthalpy (J/kg)
P = Pressure (Pa)V = Specific volume (m³)
U = Specific internal energy (kinetic energy)

Discussion on Maxwell vs. Arrhenius and the ‘Greenhouse effect’Work in this area of gas physics was detailed in the 19th century. However, there is a strong difference between the work andthe views of the researchers Maxwell and Arrhenius. Maxwell’s work shows that temperatures in the lower troposphere ofEarth are primarily determined by convection and the atmospheric mass/pressure/gravity relationship. Arrhenius’s later work7

www.tsijournals.com | January-2018[15] completely ignored this and determined that temperatures in the lower troposphere of Earth are caused by the radiative
effects of greenhouse gases. There have been papers critical of Arrhenius’s radiative effects ideas since 1909 [16]. Who iscorrect is critical to the present, since if Arrhenius is correct, then there should be some concern about CO₂ emissions, if theclimate sensitivity is high enough. If Loschmidt’s version of Maxwell’s work is correct, then more CO₂ will have nomeasurable effect on tropospheric atmospheric temperatures.

What do atmospheric measurements actually show? Measurements [17] of the effects of more CO₂ in the atmosphere appear
to strongly support Maxwell. At pressures above 0.1 bar, “the extra CO₂ merely replaces water vapour” and little differenceis seen in temperatures – but at pressures below 0.1 bar more CO₂ is measured to cause strong cooling. One of the main
problems with the Arrhenius view, appears to be that convection is virtually ignored as a mode of heat transfer; later workshows that only 11% of heat transfer in the troposphere is actually carried by radiation [18]. Whether this can causesignificant net warming in an open atmosphere is debatable. A recent paper has supported the Arrhenius view somewhat byquantifying a forcing due to increased atmospheric CO₂ [19]. But there remains a lack of any paper in the literature, whichquantifies any warming attributable to increasing atmospheric CO₂ concentrations.

The accuracy and implications of formulas 5 and 6
It is apparent that this simple formula calculates the ‘surface’ temperatures of many planetary bodies in our Solar Systemaccurately FIG 2. Specifically, those which have atmospheres thick enough to form a troposphere (i.e. possessing an
atmospheric pressure of over 10 KPa or 0.1bar). These are; Venus, Earth, Jupiter, Saturn, Titan, Uranus and Neptune. All
calculated temperatures are within 1.2% of the NASA reported ‘surface’ temperature (except for Mars, which is excluded
because it has a much lower atmospheric pressure than 10 KPa). This accuracy is achieved without using the S-B black bodylaw, or the need to include terms for such parameters as TSI levels, albedo, clouds, greenhouse effect or, for that matter,
adiabatic auto-compression. All that is required to be able to accurately calculate the average near-surface atmospherictemperature, is the relevant gas constant and the knowledge of three variable gas parameters.

FIG. 2. Actual temperature vs. calculated temperature of 8 planetary bodies and the South Pole8

www.tsijournals.com | January-2018

Discussion on the implications for climate sensitivity to CO₂
The reporting of climate sensitivity in the literature has been steadily reducing for decades, with some recent papers pointing
to a very low sensitivity, of much less than 1°C [20-22]. A careful reading of these papers, (for example the most recent one)clearly indicates that the 0.6°C cited, is in fact an absolute maximum. This present work, if not invalidated by subsequent
work, clearly points to a climate sensitivity so low that it would not be possible to measure it in the real atmosphere; it could
even be negative (i.e., more CO₂ creates cooling). This work leads directly to the conclusion that a small change in any single
atmospheric gas, such as a doubling of the CO₂ level, (from the so-called ‘pre-industrial’ 0.03% to 0.06%) can have nomeasurable positive or negative effect on atmospheric temperatures.

Probable implications for the climate sensitivity to CO₂Some reflection upon the simplicity and accuracy of these results, combined with knowledge of significant other factors suchas the thermal gradient, should bring an unbiased person to some probable implications of this work. These are that theresidual near-surface atmospheric temperatures on planetary bodies with thick atmospheres are not mainly determined by thegreenhouse effect, but instead most likely by an effect from fluid dynamics, namely; auto-compression. This leads directly tothe conclusion that the climate sensitivity to, for example, a doubling of the atmospheric carbon dioxide concentration has tobe not only operating instantaneously, but also must be extremely low. Under this scenario, logic spells out that thetemperature change caused by the addition of 0.03% of CO₂ in the lower troposphere, cannot be very different to the additionof a similar quantity of any other gas.
To be clear, formulas 5 and 6 appear to rule out any possibility that 33°C of global warming from a ‘greenhouse effect’ of thetype proposed by the IPCC in their reports [23] can exist in the real atmosphere. The reason is that the IPCC state in theirreports that a 0.03% increase in atmospheric CO₂, which represents a doubling from pre-industrial levels, must result in a
global lower tropospheric near-surface temperature rise of ~3°C; (within a range of 1.5°C to 4.5°C) [24,25] and an evengreater temperature rise in the upper troposphere.

This reported level of climate sensitivity to a doubling of atmospheric CO₂, has not changed significantly in the regular IPCC
reports since 1990. Anything like this magnitude of warming caused by such a small change in gas levels appears to becompletely ruled out by the molar mass version of the ideal gas law. The reason is that inserting any reasonable changes tothe three gas paraments caused by the small 0.03% of extra CO₂ into the formula, leads to almost no change in atmospherictemperatures. A reasonable expectation would be that a 0.03% increase in atmospheric CO₂, which is a relatively heavy gas,
would result in the following changes in the three gas parameters;

Pressure:

An increase of 0.03%

Density:

An increase of 0.05%

Molar Mass:

An increase of 0.05%

Calculate for a doubling of CO₂ from the pre-industrial level of 0.03% (by volume);9

www.tsijournals.com | January-2018

T

101.331.2256  8.314 
28.984 Calculated temperature after doubling of CO₂ to 0.06% ≈ 288.23K
Climate sensitivity to CO₂ ≈ 288.23 - 288.14 ≈ 0.09K

The change would in fact be extremely small and difficult to estimate exactly, but would be of the order 0.09°C. That is,thirty-three times smaller than the stated ‘likely’ climate sensitivity of 3°C cited in the IPCC’s reports. Even that small
number would likely be a maximum change, since if fossil fuels are burned to create the emitted CO₂, then atmospheric O₂
will also be consumed, reducing that gas in the atmosphere – and offsetting any temperature change generated by the extra
CO₂. This climate sensitivity is already so low that it would be impossible to detect or measure in the real atmosphere, even
before any allowance is made for the consumption of atmospheric O₂.

Conclusion
It has always been complicated mathematically, to calculate the average near surface atmospheric temperature on planetary
bodies with a thick atmosphere. Usually, the (S-B) black body law is used to provide the effective temperature, then debate
arises about the size or relevance of additional factors such as the greenhouse effect. Here is presented a simple and reliable
method of calculating the average near surface atmospheric temperature on planetary bodies which possess a surface
atmospheric pressure of over 10 KPa. This method requires knowledge of the gas constant and the measurement of only threeatmospheric gas parameters; average near- surface atmospheric pressure, average near surface atmospheric density and themean molar mass of the atmosphere.

The formula used is the molar mass version of the ideal gas law. It is here demonstrated that the information contained in justthese three gas parameters alone is an extremely accurate predictor of average near-surface atmospheric temperatures, inatmospheres >10 KPa. Therefore, all information on the effective plus the residual near-surface atmospheric temperature onplanetary bodies with thick atmospheres; residual meaning the difference between the effective, (that predicted by S-B black
body law), and the measured actuality, must be automatically ‘baked-in’ to the three mentioned gas parameters.

This leads directly to the conclusion that a small change in any single atmospheric gas, not only has little effect on
atmospheric temperatures, but has a very similar effect to the same percentage change in any other atmospheric gas. It is seen
therefore, that no one gas particularly affects atmospheric temperatures more than any other gas; so, there can be no
significant ‘greenhouse warming’ caused by ‘greenhouse gases’ on Earth, or for that matter on any other planetary body.
Instead, it is hypothesised that the residual temperature differences, and the tropospheric thermal gradient observed onplanetary bodies, are actually caused by auto-compression.

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Acknowledgement and Funding
Acknowledgement of non-specific Australian government support through “The Australian Government Research Training
Program Scholarship”.

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