CONTENTS

Tables with Data

Calculations

Notes

Vournas Research/Full Paper

From Science Review Journal

A New Universal Equation for Planet's Mean Surface Temperature

by Christos Vournas

Contents[hide]

Summary

A new universal equation for calculating a planet's mean surface temperature is developed here, to provide better estimates than the simple "blackbody" equation which was based on simplifying assumptions. Recognizing that a real planet does not match the assumptions for an idealized blackbody, Vournas developed an expanded equation with four additional terms to better represent a planet's actual conditions, particularly considering planet axial rotation. The derivation of the new equation from the planet energy balance is shown below, followed by a description for each of the four new terms in the new equation, including rotation (N), specific heat capacity (cp), solar light reflection and dispersion (Φ, a), and a new universal constant (β) determined empirically. This new Vournas equation results are compared for 19 solar system bodies (planets and moons), with the equation's calculated temperature closely matching the data, the NASA satellite measured temperature in Table 1.

Literature Review

This section gives a brief summary of what has been published on the topics relevant to estimating a planet's mean surface temperature.

Definition of Terms

• Bond albedo-- "The fraction of incident solar radiation reflected back into space without absorption, dimensionless. Also called planetary albedo."[1]. . . symbol isain units dimensionless.
• Geometric albedo-- "The ratio of the body's brightness at a phase angle of zero to the brightness of a perfectly diffusing disk with the same position and apparent size, dimensionless."[2]. . . symbol is none (not used in this paper).
• Emissivity-- "The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation.... that may include both visible radiation (light) and infrared radiation..." "Quantitatively, emissivity is the ratio of the thermal radiation from a surface to the radiation from an ideal black surface at the same temperature as given by the Stefan–Boltzmann law. The ratio varies from 0 to 1", with the surface of a perfect black body having an emissivity of 1.[3]. . . symbol isεin units dimensionless.
• Black-body temperature-- "Equivalent black body temperature is the surface temperature the body would have if it were in radiative equilibrium and had no atmosphere, but the same albedo, in Kelvin."[4]This is also called the 'planetary equilibrium temperature' which is "a theoretical temperature that a planet would have when considered simply as if it were a black body being heated only by its parent star.... theequilibrium temperatureis calculated purely from a balance with incident stellar energy."[5]. . . . symbol is Tein units degrees K (Kelvin).
• Mean Temperature-- "This is the average temperature over the whole planet's surface (or for the gas giants at the one bar level) in degrees C (Celsius or Centigrade) or degrees F (Fahrenheit). For Mercury and the Moon, for example, this is an average over the sunlit (very hot) and dark (very cold) hemispheres and so is not representative of any given region on the planet, and most of the surface is quite different from this average value. As with the Earth, there will tend to be variations in temperature from the equator to the poles, from the day to night sides, and seasonal changes on most of the planets."[6]. . . symbol is Tmeanin units converted to degrees K (Kelvin) for this paper.

NASA Measurements of Planet's Physical Characteristics

NASA Measurement Procedure for Planet Surface Temperature--

The NASA published values for mean planet surface temperature (in last Column of Table 1) are based on satellite measurements.

NASA Interpretation of Their Data--

(How does NASA explain why some planets or satellites have higher surface temperatures than the blackbody temperature?)

Methods for Estimating Planet Surface Temperature

The five types of methods for estimating a planet's mean surface temperature are listed here, with a brief description of published research for each one.

• Simple Blackbody Equation-- based on a simple energy balance between the incoming solar energy absorbed by the planet and the outgoing infrared energy emitted by the planet. The derivation of a simple blackbody equation (shown below for Eqn.2) for calculating a planet's effective radiating temperature (Te) was shown in a 1981 paper by James Hansen et. al.[7], based on the energy balance for a planet, under simplifying assumptions that the planet conforms to blackbody conditions (required for applying the Stefan-Boltzmann equation to determine outgoing energy emitted). The planet's effective radiating temperature (Te) as calculated by this simple blackbody equation occurs at the mean radiating level (Re), which can be above the planet's surface if the planet has an atmosphere. Hansen et. al. (1981) estimated the mean radiating level for Mars (1 km), Earth (6 km), and Venus (70 km) to be above the surface of those planets.
• Modified Blackbody Equation-- based on improving the assumptions from the simple blackbody equation, by modifying the equation to better represent some (but not all) actual conditions for a real planet, such as non-uniform temperature distribution. Wikipedia[8]shows an equation for estimating the surface temperature of a planet, based on modifying the effective temperature equation to account for emissivity and temperature variation, by adding a term for emissivityεand a new term Aabs/Arad, to represent fraction of planet surface area involved in energy absorption/radiation, which varies between 1/4 (for a rapidly rotating body) to 1/2 (for a slowly rotating body). . Also, North-Cahalan-Coakley (1981) give a summary of energy balance climate models (equations) based on better modifications for the blackbody equation. .[9]. .[10]
• Three modifications are considered in this paper. First, this paper derives and presents an improvement for the simple blackbody equation, based on assuming two hemispheres (one sunlit, one dark) each with its mean temperature, instead of the uniform whole planet temperature. A second improvement was derived to calculate mean temperature by considering the non-uniform spatial temperature distribution between the planet's equator and poles. A third modification corrects for the reduced total solar irradiation absorbed by the planet as a sphere (rather than as a disk in the blackbody equation), due to low sun angle and higher light reflection near the poles[11]and at twilight, by adding a Φ-factor to the blackbody equation to account for this reduced solar radiation absorption.
• Analytical Equation-- based on integral of the energy balance for each differential area over the planet's surface. So far, no analytical equation has been found in the published literature. An analytical equation would allow better consideration of spatial temperature distribution, by latitude, as well as temporal temperature distribution, by diurnal cycle considering planet spin speed and heat storage. But irregular spatial variations in planet physical characteristics, such as albedo and specific heat capacity, would not be considered (except in the 5th method: the computer simulation model). Also see:North (1981).
• Empirical Equation-- based on finding the form and coefficients for an equation that is the best fit to the data for the measured values of planet mean surface temperature, as published by NASA. So far, no empirical equation has been found already in the published literature. The purpose of my paper here is to develop such an empirical equation, for planets having no atmosphere.
• Computer Simulation Model-- NASA uses computer simulation models for estimating planet temperature. ( . . .add references . . . )

The Simple Blackbody Equation for Planet Effective TemperatureDerivation of the Simple Blackbody Equation

"The temperature of a planet's surface is determined by the balance between the heat absorbed by the planet from sunlight, heat emitted from its core, and thermal radiation emitted back into space." . . .[12]. If the planet's surface is in thermal equilibrium, then the incoming energy (from the sun and from the planet's hot core) equals the outgoing energy (from infrared emission of the planet's warm surface). By making some additional assumptions, a simple equation (1) for the planet's energy balance can be written as:

• (input) the total solar radiation absorbed by the planet over its projected area (disk of radius Ri) = ( π Ri²) (1 - a) So
• (input) the heat emitted from the planet's core to the surface, assumed to be negligible = 0
• (output) the total infrared emission from the planet's emitting surface (sphere of radius Re) = (4 π Re²) σ Te⁴

( π Ri²) (1 - a) So = ( 4 π Re²) σ Te⁴(Eqn.1 -- planet energy balance)

where:

Ri = the radius of the planet (as incident disk) for receiving incoming solar energy,

Α = the total planet surface area (m²),

A = 4 π R²(m²), is the area of a spherical surface, where "R" – is the sphere's radius,

a = the albedo of the planet,

So = the flux of solar radiation,

σ = the Stefan-Boltzmann constant, and

Te = the planet's "effective temperature" (with blackbody assumptions) at the mean emitting radius, Re.

Assuming that the planet's disk radius and sphere radius are equal ( Ri = Re ), then
Equation (1) can be simplified to Equation (2), which is the simple "blackbody equation" here:

Te = [ So (1 - a) / 4 σ ]¹∕⁴(Eqn.2 -- Simple Blackbody Equation)

The Simple Blackbody Equation is Not Accurate

Although the simple blackbody equation (Eqn.2) has been used to provide a very rough first estimate for planet temperature, this estimate is not accurate as it differs substantially from the NASA measured mean planet temperatures. Table 1 shows a comparison of the simple blackbody equation (Eqn.2) predicted effective temperatures versus the NASA's measured mean planet temperatures, with the "blackbody predicted" values Te in Column 9 and the "NASA measured" values Tsat in Column 11 (last column), for 15 planets and moons. The calculated effective temperatures (Te) do not closely match the NASA's measured mean temperatures (Tsat) for several reasons, including that real planets do not match the blackbody and other assumptions used in developing Equation (2).

One consequence of using an inaccurate simple equation is making an inaccurate assessment of planets' physical characteristics, including the greenhouse effect.

Assumptions for Simple Blackbody

1. . Blackbody absorbs the entire incident radiation on its surface, with no reflection.
2. . Has stable and spatially uniform surface temperature.
3. . Has stable and instantaneous equilibrium emission temperature Te when irradiated.
4. . When irradiated, the blackbody's surface has emission temperature according to the Stefan-Boltzmann blackbody emission law, Je = σ*Τe⁴
5. . Blackbody's emission temperature depends only on the quantity of the incident radiative energy per unit area.
6. . No blackbody's surface accumulates energy.
7. . Blackbody has only surface physical properties (without "body"), has no mass and no specific heat capacity.
8. . The blackbody's surface has an infinitive conductivity. All the radiative energy incident on the blackbody's surface is instantly and evenly distributed across the entire blackbody's surface.

Other Assumptions

1. . The planet has thermal equilibrium, so that . . . Power in = Power out
2. . The planet's geothermal flux to the surface is negligible.
3. . The solar flux is from only one sun (no other heat sources such as nearby hot planets).
4. . The solar flux from the sun is constant.
5. . The solar flux is only attenuated by the planet's distance from the sun (not attenuated by intervening cosmic particles).
6. . The planet's distance from the sun is constant (with no orbital eccentricity).
7. . The planet's albedo is constant and uniform over the planet's surface. (Actually, albedo is temperature dependent.)
8. . The planet's functional disk and sphere radii are equal, Ri = Re.
9. . If Ri = Re = Rs (where Rs is the radius of the planet to its solid surface), then Te = Ts (the mean surface temperature).

Estimation Errors Caused by the Blackbody Assumptions

But for a real planet, many of the blackbody assumptions and other assumptions do not apply exactly, which causes the blackbody equation based on these assumptions to give inaccurate results (in Table 1).

Uniform Temperature Assumption:For example, a rotating real planet does not meet the blackbody assumption for "stable and spatially uniform surface temperature". A planet with very slow rotation (axial spin) has a warmer hemisphere facing the sun, and a cooler hemisphere on the dark side. If the planet spins faster, the solar radiation is more evenly distributed over both sides, so that the two hemispheres have more similar temperatures. The faster spinning planet, which has more similar temperatures on both sides, has a higher mean surface temperature (the average of the warm side and the cool side) than a slower rotating planet, which has a larger temperature difference between the two sides. This conclusion is demonstrated mathematically inAppendix B, based on results from the "Two-Hemisphere Equation", which is a modified version of the blackbody equation derived from an energy balance for two hemispheres (each with its own mean temperature).

The results calculated from the Two-Hemisphere Equation show three important conclusions.

First, for a planet with no spin, the planet's mean temperature is 59% of the blackbody temperature Te. For a planet with some spin, the planet's mean temperature increases as the spin speed increases (represented by the decrease in the temperature difference between the two sides). For a planet with infinitely fast spin, the planet's mean temperature is equal to 100% of the blackbody temperature Te, because the two hemispheres then have no temperature difference, so theoretically have uniform temperature conforming to the blackbody assumption. (Actually, though, the planet temperature is still non-uniform because there is anequator-to-pole temperature variation, still not accounted for in the Two-Hemisphere Equation nor in the blackbody equation.) For planets with spin between zero and infinity, the planet's mean temperature increases from 59% of Te (for a planet with no spin) up to 100% of Te (for a planet with infinitely high spin speed).

The second conclusion from the Two-Hemisphere Equation is relevant for answering the question: Why do some planets have a NASA measured temperature higher than the blackbody temperature? Can this NASA measured excess temperature be caused by a planet's fast spin? Maybe. The faster spin for a planet cannot make the planet's mean temperature be higher than the blackbody temperature, based solely on consideration of its causing more even temperature distribution. However, a faster spin might have a second effect -- (this is an hypothesis) it may increase the rate of heat transfer by increased mixing (conduction or convection) from the planet's hot interior to the planet's surface, and by this second effect then the planet's spin might contribute to the planet's temperature being higher than the blackbody temperature.

The third conclusion here is that the planet's spin speed (N, in rotations per day) does matter for the purpose of calculating the planet's mean surface temperature. For an improved equation for planet mean surface temperature, the new term for planet spin speed (N) should be a factor in the new equation.

Stored Heat Assumption:For another example: The rotating real planet's surface, when it turns to the sunlit side, is an already warm at some temperature, from the previous day. The planet surface stores heat, and does not match the assumption that "no blackbody's surface accumulates energy". A new term to indicate planet surface stored heat is needed in an improved equation for calculating planet mean surface temperature.

Improved Equation and Its New Terms for Better Assumptions

Recognizing that a real planet does not match the assumptions for an idealized blackbody, Vournas developed an expanded equation with four additional terms to better represent a planet's actual conditions, particularly considering planet rotation. The derivation of the new equation from the planet energy budget is shown below, followed by a description for each of the four new terms in the new equation, including rotation (N), specific heat capacity (cp), solar light reflection and diffusion (Φ, a), and a new universal constant (β).

Tmean = [ Φ (1 - a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴(Eqn.3 -- new Vournas Equation)

Planet Energy Balance for the New Equation

The new equation (Eqn.3) was derived from the planet energy balance (incoming energy = outgoing energy), but considering four additional terms to better account for the thermal conditions for a real planet, as follows:

Incoming-- Solar energy absorbed by a Hemisphere with radius "r" after reflection and dispersion:

Jabs = Φ * π r² S (1-a)(W)

a -- is the planet's average albedo

Φ -- is the dimensionless Solar Irradiation accepting factor

Φ = 0,47 for a smooth spherical surface

(1 - Φ + Φ*a) S -- is the reflected fraction of the incident on the planet solar flux

Φ (1 - a) S -- is the absorbed fraction of the incident on the planet solar flux

S -- is the Solar Flux at the top of the atmosphere (W/m²)

Outgoing-- Total energy emitted to space from the entire spherical planet:

Jemit = A * σΤmean⁴ / (β*N*cp)¹∕ ⁴(W)

A = 4 π r² is surface area (m²) of a sphere, where "r" is the planet's radius

Tmean -- is the Planet's Mean Temperature (°K)

(β*N*cp)¹∕ ⁴ -- dimensionless, is the Rotating Planet Surface Solar Irradiation Warming Ability

Then, assuming the planet is in thermal equilibrium, where the incoming energy = outgoing energy:

Jabs = Jemit

Φ * S * (1-a) = 4 σTmean⁴ /(β*N*cp)¹∕ ⁴

Solving for Tmean we obtain the mean temperature (°K):

Tmean = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴(Eqn.3 -- new Vournas Equation)

Where:

β = 150 days*gr*°C/rotation*cal -– is the Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant

N rotations/day, is the planet’s sidereal rotation spin

cp -– is the planet's surface specific heat

σ = 5,67*10⁻⁸ W/m²K⁴, the Stefan-Boltzmann constant.

Φ Factor-- Planet is irradiated as a sphere, not a disk

Φ -- is the dimensionless Solar Irradiation accepting factor for a spherical surface.

The Φ Factor term in the new equation is introduced to overcome the old assumption that the planet absorbs solar radiation like a flat disk. The planet reflects and absorbs solar irradiation like a sphere and not like a disk. The absorbed solar energy by the hemisphere's area of 2π r² is:

Jabs = 0.47*( 1 - a) π r² S

because a hemisphere of radius "r" absorbs only the 0.47 fraction of the Solar irradiation directly incident on the disk of the same radius "r". In spite of hemisphere having twice the area of the disk, it absorbs only the 0.47 part of the Solar irradiation directly incident on the disk.

where Φ = 0.47 for smooth without atmosphere planets,
and Φ = 1 for gaseous planets, as Jupiter, Saturn, Neptune, Uranus, Venus, Titan.
Gaseous planets do not have a surface to reflect radiation. The solar irradiation is captured in the thousands of kilometers gaseous abyss. The gaseous planets have only the albedo "a".
And Φ = 1 for heavy cratered planets, as Calisto and Rhea ( not smooth surface planets, without atmosphere ). The heavy cratered planets have the ability to capture the incoming light in their multiple craters and canyons. And the heavy cratered planets have only the albedo "a".

The light dispersed on the surface also escapes into outer space, but it is a secondary reflection and it is described by surface's Albedo "a". The direct reflection is described by the (1-Φ), where "Φ" is the Spherical Surface Solar Irradiation Accepting Factor.[14]

N-- a faster rotating Planet is warmer

N -- is the planet's rotation speed, in rotations per Earth-day.

The faster a planet rotates, the higher the planet's mean surface temperature. .[15]

It is well known that when a planet rotates faster, its daytime maximum temperature lessens and the night time minimum temperature rises. But something else very interesting happens -- when a planet rotates faster, it is a warmer planet. This happens because Tmin↑↑ grows higher than T↓max goes down. The understanding of this phenomenon comes from a deeper knowledge of the Stefan-Boltzmann Law. It happens so because when rotating faster, the planet's surface has emission temperatures the new distribution to achieve.

Cp-- Specific Heat Capacity

Cp -- is the planet's specific heat capacity, in units of cal/gr.oC.

"The specific heat capacity of a substance is . . (informally) the amount of energy that must be added, in the form of heat, to one unit of mass of the substance in order to cause an increase of one unit in its temperature."[16]

For developing the new equation for planet surface temperature, the Cp term was selected as a parameter to characterize the irradiated surface because it is a well-known physical term that mathematically represents the surface material and solar energy interaction.

A higher Cp value means that the irradiated surface has more molecules and atoms per unit area to interact with the incoming solar radiation, and in effect, functions as a blackbody surface of bigger dimensions. And when the blackbody surface dimensions are bigger, for the same solar flux on the unit area, the surface develops IR emission of lower intensity, thus developing a lower effective temperature (Te) on the spot.

β-- Numerical Value of the Universal Constant β

β = 150 days*gr*°C/rotation*cal –- is the Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant.

The β value is a constant used in the new equation's term, (β*N*cp)¹∕ ⁴ that indicates the rotating planet's warming response to solar irradiation.

(β*N*cp)¹∕ ⁴ is a dimensionless term, indicating a "Rotating Planet Surface Solar Irradiation Warming Ability".

The numerical value for β was found empirically, as a best fit to the data in Table 1 for the planets with no atmosphere.? ?

Validating the Improved EquationMethods

Selecting the Celestial Bodies to Study
This study included all the planets, and all moons in our Solar System having a radius over 1000 km, to use in evaluating the predictions of the new equation (in Table 1).

• Mercury
• Venus
• Earth (with 1: Moon)
• Mars
• Jupiter (with 4: Io, Europa, Ganymede, Callisto )
• Saturn (with 3: Titan, Enceladus, Tethys )
• Uranus
• Neptune (with 1: Triton )
• Pluto (with 1: Charon )

Obtaining the Planet's Physical Data:
The numerical values for the planets' and moons' physical characteristics all were taken from published values by NASA (see: NASA Factsheets of Planets .[17]) or from other sources, as shown in Table A1 in Appendix A. These physical characteristics include:

• a = the planet's average albedo,
• So = a Solar Flux at the top of the planet's atmosphere ( W/m² ),
• D = the planet's distance from the sun (AU),
• N = Number of the planet's axial spin rotations per Earth day,
• cp = specific heat capacity of the planet's emitting surface ( cal/gr.oC ),
• The composition of the planet's surface ("Type" column in Table 1).

Results

Mercury:-- The Tsv = 325,83 K is calculated for Mercury's Semi-major axis which is 0,387 AU. But half of the time, Mercury comes closer to the sun at its Perihelion of 0,307 AU. The fact Mercury's orbit has high eccentricity e = 0,205 partly explains the difference between the calculated Tvs = 325,83 K and the measured Tsat = 340 K .

Venus:-- In the last Column of Table 1, the NASA measured temperature for Venus is shown for two vertical locations: (1) the temperature at the same place (the 1 bar level) as for the other planets that are also classified as "deep atmosphere planets"[18]which are: Venus, Jupiter, Saturn, Uranus, Neptune; and (2) at the solid surface of the planet, which is about 70 km below the mean radiating level, Re.(Hansen et. al., 1981). "The planet Venus has an atmosphere that is two orders of magnitude more dense at its surface than that of the Earth." (p.542, Janssen, 1993 -[19])

Earth:

Mars:

Uranus:-- Uranus has an unusual sideways axis orientation, with the axis of spin being nearly parallel with the plane of its 84-year orbit, instead of nearly perpendicular. This causes the planet's poles to each have 42 years of darkness, followed by 42 years of continuous sunlight. Thus, for half of the time, only one hemisphere of the planet is irradiated, which corresponds to the assumption in the simple blackbody equation. For this reason, the predicted surface temperature (Tsv*) for Uranus shown in Table 1 is calculated as the average of Tsv and Te for the simple blackbody equation.

. . . . . . . . . . .(go to edit page for Table here )

Table 1. Comparison of Predicted vs. Measured Temperature for All Planets

NameSolar
Distance
( AU )Solar
Flux
( W/m² )Φ
Factor
( . )Albedo
(Bond)
( . )Axial Spin
( rotations
/ day )TypeSpec.Heat
capacity
( cal/gr.°C )Warming
Ability
( . )Te
( °K )
{ with Φ }Tsv
( °K )
.Tsat
( °K )
.Planets
and Moons:Ncp(β*N*cp)¹∕ ⁴blackbody
predictedVournas
predictedNASA
measured
aMercury0,3879082,70,470.0680,00568basalt rock0,200,64250439,6{364,0}325,83340bVenus0,7232601,310,770,24691
60/243gases,
96,5% carbon dioxide (CO₂)
and 3,5% nitrogen (N₂)0,191,6287226,6255,98
737
348at 1 barcEarth
.1,013610,470,3061,0ocean13,4996254{210}287,74288.dMoon
. . . .of Earth1,013610,470,110,0339regolith0,190,99141270,04{224}222220eMars1,524586,40,470,250,9747rock,
prevalent iron oxide Fe₂O₃0,182,26495209,8{174}213,42210fJupiter5,2050,3710,5032,417gases,
89% ± 2.0% hydrogen (H₂),
10% ± 2.0% helium (He),
0.3% ± 0.1% methane (CH₄)2,955,7187102158,4165
at 1 bar levelgIo
. . . .of Jupiter5,2050,3710,630,5559rock
(heavy cratered)0,145
( sulfur and
sulfur dioxide frost)1,864795,16111,55110hEuropa
. . . .of Jupiter5,2050,370,470,630,2816rock1
( water ice crust)2,549495,16 (78,83)99,56102iGanymede
. . . .of Jupiter5,2050,370,470,430,1398rock0,676
( 60% water ice crust
and 40% rocky land)1,9404106,07 (87,83)107,91110jCalisto
. . . .of Jupiter5,2050,3710,220,0599rock
(heavy cratered)1
( water ice crust)1,7313114,66131,52134 ± 11kSaturn9,5814,8410,3422,273gases,
96.3%±2.4% hydrogen (H₂),
3.25% ± 2.4% helium (He),
0.45% ± 0.2% methane (CH₄)3,0745,689881125,07134
. . . .of Saturn9,5814,8410,81 ± 0,04 (0,85)0,7299rock1
water ice crust3,234755,9775,0675nTethys
. . . .of Saturn9,5814,8410,80 ± 0,15 (0,70)0,52971rock1
water ice crust2,985666,5587,4886 ± 1pTitan
. . . .of Saturn9,5814,8410,220,06289gases
(atmos. 95% nitrogen N₂
and 5% methane CH₄)0,4980
liquid methane surface1,4722384,5293,1093,7qUranus19,223,68710,301,389gases,
83 ± 3% hydrogen (H₂),
15 ± 3% helium (He),
2.3% methane (CH₄)2,8034,915958MM *
72,2976
at 1 bar levelrNeptune30,331,4810,291,493gases,
80%±3.2% hydrogen (H₂),
19% ± 3.2% helium (He),
5% ± 0.5% methane (CH₄)2,7534,983046,469,3172
at 1 bar levelsTriton
. . . .of Neptune30,331,480,470,760,17021rock0,4116
55% N₂, 25% water ice,
20% CO₂ crust1,80035,4 (29,29)33,9238tPluto39,480,87410,500,1565rock
( N₂ crust heavy cratered)0,2481,55333741,644uCharon
. . . .of Pluto39,480,87410,20,1565rock
( H₂O crust heavy cratered)12,201441,9051,0453
REFERENCES[1][2][3][4][5][6][7][8][9][10]

Table Footnotes:

(*) . An alternate Uranus temperature (Tsv *) was calculated as the average of the blackbody temperature (Te) and that for the new equation (Tsv).
(**) .An alternate Venus measured temperature (Tsat **) is shown at the effective radiative radius (70 km above the surface).

Table Reference Numbers for Data Sources:
[1] - [8] = NASA data
[9] and [19] = calculated from Vournas Equation
[10] and [20] = NASA data
[11] - [18] = NASA data for Solar System moons[20], except Enceladus.

HERE is thefirst draftof the revised Table 1
(which will soon be made tidy, and the duplicate table removed)

TABLE 1B -- with addition of REFERENCES from Christos.....

Table 1. Comparison of Predicted vs. Measured Temperature for All Planets

NameSolar
Distance
( AU )Solar
Flux
( W/m² )Φ
Factor
( . )Albedo
(Bond)
( . )Axial Spin
( rotations
/ day )TypeSpec.Heat
capacity
( cal/gr.°C )Warming
Ability
( . )Te
( °K )
{ with Φ }Tsv
( °K )
.Tsat
( °K )
.Planets
and Moons:Ncp(β*N*cp)¹∕ ⁴blackbody
predictedVournas
predictedNASA
measured
aMercury0,3879082,70,470.068 (G)0,00568 (G)basalt rock (W)0,200,64250364,0 (calculated)325,83340 (W)bVenus0,7232601,310,77 (G)0,2462
60/243 (60 /243,69 = 0,2462 rot/day) (G)gases,
96,5% carbon dioxide (CO₂)
and 3,5% nitrogen (N₂)(G)0,19 (for Venus' surface)1,6287226,6 (G)255,98
737
348at 1 bar(464 oC = 737 K) (G)cEarth
.1,013610,470,306 (G)1,0 (G)ocean1 (for water)3,4996211 (calculated)288.288.(G)dMoon
. . . .of Earth1,013610,470,11 (G)0,0339 (G)regolith (W)0,190,99141224 (calculated)222220 (W)eMars1,524586,40,470,25 (G)0,9747 (G)rock,
prevalent iron oxide Fe₂O₃0,182,26495174 (calculated)213,11208 (G)fJupiter5,20450,2610,503 (W)2,415 (G)gases,
89% ± 2.0% hydrogen (H₂),
10% ± 2.0% helium (He),
0.3% ± 0.1% methane (CH₄)(G)2,95 (calculated)5,7187102 (G)158,04165 (G)
at 1 bar levelgIo
. . . .of Jupiter5,20450,2610,63 (W)0,565 (G)rock
(heavy cratered)(W)0,145 (calculated)
( sulfur and
sulfur dioxide frost)(W)1,864795,16 (calculated)111,55110 (W)hEuropa
. . . .of Jupiter5,20450,260,470,63 (W)0,2816 (G)rock (W)1 (for water)
( water ice crust)(W)2,549478,83 (calculated)99,56103 (G)iGanymede
. . . .of Jupiter5,20450,260,470,43 (W)0,1398 (G)rock (W)0,676 (calculated)
( 60% water ice crust
and 40% rocky land)(W)1,940487,83 (calculated)107,91113 (G)jCalisto
. . . .of Jupiter5,20450,2610,22 (W)0,0599 (G)rock
(heavy cratered)(W)1 (for water)
( water ice crust)(W)1,7313114,66 (calculated)131,52134 ± 11 (W)kSaturn9,58214,8210,342 (G)2,25 (G)gases,
96.3%±2.4% hydrogen (H₂),
3.25% ± 2.4% helium (He),
0.45% ± 0.2% methane (CH₄)(G)3,074 (calculated)5,689881 (G)125,04134 (G)
. . . .of Saturn9,58214,8210,81 ± 0,04 (0,85) (W)0,7299 (W)rock (W)1 (for water)
water ice crust(W)3,234755,97 (calculated)75,0675 (W)nTethys
. . . .of Saturn9,58214,8210,80 ± 0,15 (0,70) (W)0,52971 (W)rock (W)1 (for water)
water ice crust(W)2,985666,55 (calculated)87,4886 ± 1 (W)pTitan
. . . .of Saturn9,58214,8210,22 (W)0,06271 (G)gases
(atmos. 95% nitrogen N₂
and 5% methane CH₄)(W)0,4980
liquid methane surface1,4722384,52 (calculated)96,0393,15 (G)qUranus19,2013,6910,30 (G)1,393 (G)gases,
83 ± 3% hydrogen (H₂),
15 ± 3% helium (He),
2.3% methane (CH₄)(G)2,803 (calculated)4,915958,1 (G)MM *
72,2978 (G)
at 1 bar levelrNeptune30,0471,50810,29 (G)1,490 (G)gases,
80%±3.2% hydrogen (H₂),
19% ± 3.2% helium (He),
5% ± 0.5% methane (CH₄)(G)2,753 (calculated)4,983046,6 (G)69,6373 (G)
at 1 bar levelsTriton
. . . .of Neptune30,0471,5080,470,76 (W)0,17021 (G)rock (W)0,4116 (calculated)
55% N₂, 25% water ice,
20% CO₂ crust(G)1,80029,2 (calculated)33,9238 (W)tPluto39,4820,87310,50 (W)0,1566 (G)rock
( N₂ crust heavy cratered)(W)0,2481,553337,5 (G)41,644 (W)uCharon
. . . .of Pluto39,4820,87310,25 (G)0,1566 (G)rock
( H₂O crust heavy cratered)(W)1 (for water)2,201441,22 (calculated)50,2153 (W)COLUMN:
REFERENCE:[2]
[G] all[3]
[G] all[4]
[C] all[5][6][7][8][9][10][11]

Table References Codes:

[G]-- data from Goddard
[W]-- data from Wikipedia
[C]-- calculated value, using equation shown in the preceding text.

Column [#]:
[2] = all numerical values for "Solar Distance" are taken from Goddard.
[3] = all numerical values for "Solar Flux" are taken from Goddard.
[4] = all numerical values are calculated (as 0,47 or 1,0) -- based upon the planet's surface composition.
[5] =
[6] =
<bar>[9] and [19] = calculated from Vournas Equation
[10] and [20] = NASA data
[11] - [18] = NASA data for Solar System moons[21], except Enceladus.

Table Footnotes:

(*) . An alternate Uranus temperature (Tsv *) was calculated as the average of the blackbody temperature (Te) and that for the new equation (Tsv).
(**) .An alternate Venus measured temperature (Tsat **) is shown at the effective radiative radius (70 km above the surface).

Discussion and Recommendations

Go to subpage with . . .Vournas Research/Notes for Discussion Section(click here).

Confirming the Orderly Universe.The new equation shows that the NASA satellite measurements are very precise. Also it helps to explain some inconsistencies that scientists had noticed, in planet's and moon's temperature behavior.

When a new planet at different solar system is discovered, scientists speculate whether it is habitable or not. They were counting only on the star's irradiation flux and on the distance from the star. Now it is possible to have a much more precise estimation of the planet's temperature, in cases when the planet's spin is already known.

The new equation brings order in the field of planets' temperatures measurements.

Revising the Understanding of Planet Properties.

The new equation provides some new information about the physical properties of planets that changes established understanding. The new equation can accurately predict a planet's mean surface temperature, without using any terms in the equation to account for atmospheric properties or for internal heat, and consequently that implies that the planet's mean surface temperature is not affected by the atmospheric composition or the internal heat. The new equation shows that there is no greenhouse effect on Titan (the Saturn's satellite, which has an atmosphere of 95% N₂ and 5% methane - a very strong greenhouse gas). The 5% methane gas is not enough to create a measurable greenhouse effect on Titan. This changes the established understanding that Titan, similarly to Earth, has a strong greenhouse effect.

And also there is the consequence that the gaseous planets Jupiter, Saturn, Uranus and Neptune do not have any inner source of energy as it is wrongly assumed.

Revising the Understanding of the Greenhouse Effect.

Hansen et. al., (1981) gave an early estimate for the magnitude of the greenhouse effect as 33°C. This 33°C estimate was obtained by using the simple blackbody Equation (2) to calculate the Earth's effective radiating temperature (255 K), then comparing that to the NASA mean measured temperature (288 K), and assuming that the difference (the excess temperature of 33°C = 288 - 255 ) was entirely caused by the greenhouse effect.[22]
"Using values for planet Earth (with albedoa~ 0.3 and solar flux So = 1367 watts per square meter), this equation calculates that Te ~ 255 K."[23]
Notice that this calculated temperature of 255 K is less than the NASA's measured mean temperature of Tsat ~ 288 K by a difference of 33°C, and that this difference has been attributed to the greenhouse effect: . . . According to[24]| Hansen et. al. (1981)] . . . "The excess, Ts - Te, is the greenhouse effect of gases and clouds, which cause the mean radiating level to be above the surface."

However, attributing all of this difference (33°C = 288 - 255 ) entirely to the greenhouse effect is tantamount to assuming that the blackbody Equation (2) is perfect and has no error due to making simplifying assumptions -- which is unlikely. This is demonstrated in Table 1, which shows significant differences (Ts - Te) even for planets and moons having no atmosphere. Hence, the difference (Ts - Te) can be caused entirely or partly by factors other than a greenhouse effect.

The improved equation for the planet's surface temperature (Eqn.3) includes some additional factors to mathematically represent the planet's actual conditions more appropriately than the simplifying blackbody assumptions. Using the new equation, the Earth's mean surface temperature (with no atmosphere) is calculated to be 288°K, which closely matches the NASA measured mean temperature (with atmosphere) of 288°K, leaving no error term (formerly 33°C) to attribute to a postulated atmospheric "greenhouse" effect.

General References

• "Chapter 10. Properties of the Deep Atmospheres of the Planets from Radio Astronomical Observations," by Samuel Gulkis and Michael A. Janssen, in the book Atmospheric Remote Sensing by Microwave Radiometry, by Michael A. Janssen, 1993, publ. John Wiley & Sons. . .[25]

• "Titan Surface Temperatures During the Cassini Mission", by D.E. Jennings et. al., 20 May 2019,The Astrophysical Journal Letters; p. L8; (ISSN 2041-8205; e-ISSN 2041-8213); 877; 1 . . [doi:10.3847/2041-8213/ab1f91] . .[26]

Appendix A. Data and Sources

'Online Table .[27]
NASA Online Table -- Planet Compare . . {https://solarsystem.nasa.gov/planet-compare/]
NASA Energy Budget for the Earth .[28]
NASA Planetary Fact Sheet (Table of values) . . .[29]
NASA Solar System Small Worlds Fact Sheet (table of data for 7 of the 8 moons in Table 1, missing Saturn's moon Enceladus)[30]

Click here to go to original Table A1:Vournas Research/ Table A1 for Data and Sources

Click here to go to theedit pagefor the new Table A2:Vournas Research/ Table A2 for Data and Sourcesshowing only NASA data.

Table A1 -- Data and Sources

SymbolValueUnitsDefinitionSource or ReferenceALL PLANETS:So1362W/m²solar constant"Satellite observations of total solar irradiance". acrim.com.σ5,67*10⁻⁸W/m²K⁴Stefan-Boltzmann constant"2018 CODATA Value: Stefan–Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.β150days*gr*°C/rotation*calRotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant--Φ0,47planet's surface solar irradiation accepting factorEARTH:R.earth1AUEarth's distance from the sun, semi-major axisSimon, J.L.; Bretagnon, P.; Chapront, J.; Chapront-Touzé, M.; Francou, G.; Laskar, J. (February 1994). "Numerical expressions for precession formulae and mean elements for the Moon and planets". Astronomy and Astrophysics. 282 (2): 663–83. Bibcode:1994A&A...282..663S.a.earth0,306-Earth's albedoWilliams, David R. (16 March 2017). "Earth Fact Sheet". NASA/Goddard Space Flight Center. Retrieved 26 July 2018.1/N.earth0.99726968days/rotationEarth's (day-night) rotation periodAllen, Clabon Walter; Cox, Arthur N. (2000). Allen's Astrophysical Quantities. Springer. p. 296.ISBN 978-0-387-98746-0. Retrieved 17 August 2010.Tmean.earth287,16KelvinEarth's surface mean temperatureKinver, Mark (10 December 2009). "Global average temperature may hit record level in 2010". BBC. Retrieved 22 April 2010.cp.earth1cal/gr.°CEarth's surface average specific heat capacity (water)https://www.engineeringtoolbox.com/specific-heat-capacity-water-d_660.htmlTmean.planet°KTmean.planet = [ Φ (1-a) So (1/R)² (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴MOON:R.moon1AUMoon's distance from sunSimon, J.L.; Bretagnon, P.; Chapront, J.; Chapront-Touzé, M.; Francou, G.; Laskar, J. (February 1994). "Numerical expressions for precession formulae and mean elements for the Moon and planets". Astronomy and Astrophysics. 282 (2): 663–83. Bibcode:1994A&A...282..663S.a.moon0,136-Moon's albedoMatthews, Grant (2008). "Celestial body irradiance determination from an underfilled satellite radiometer: application to albedo and thermal emission measurements of the Moon using CERES". Applied Optics. 47 (27): 4981–4993. Bibcode:2008ApOpt..47.4981M. doi:10.1364/AO.47.004981.PMID 18806861.1/N.moon29,5days/rotationMoon's (day-night) rotation periodAllen, Clabon Walter; Cox, Arthur N. (2000). Allen's Astrophysical Quantities. Springer. p. 296.ISBN 978-0-387-98746-0. Retrieved 17 August 2010.Tmean.moon220KelvinMoon's surface mean temperatureA.R. Vasavada; D.A. Paige & S.E. Wood (1999). "Near-Surface Temperatures on Mercury and the Moon and the Stability of Polar Ice Deposits". Icarus. 141 (2): 179–193. Bibcode:1999Icar..141..179V. doi:10.1006/icar.1999.6175.cp.moon0,19cal/gr.oCMoon's surface specific heat capacity (regolith as dry soil)https://www.engineeringtoolbox.com/specific-heat-solids-d_154.htmlMERCURY:R.mercury0.387098AUMercury's distance from the sun, semi-major axisYeomans, Donald K. (April 7, 2008). "HORIZONS Web-Interface for Mercury Major Body". JPL Horizons On-Line Ephemeris System. Retrieved April 7, 2008. – Select "Ephemeris Type: Orbital Elements", "Time Span: 2000-01-01 12:00 to 2000-01-02". ("Target Body: Mercury" and "Center: Sun" should be defaulted to.) Results are instantaneous osculating values at the precise J2000 epoch.a.mercury0,088-Mercury's albedoMallama, Anthony (2017). "The spherical bolometric albedo for planet Mercury". arXiv:1703.02670 [astro-ph.EP].1/N.mercury175,938days/rotationMercury's (day-night) rotation periodMunsell, Kirk; Smith, Harman; Harvey, Samantha (May 28, 2009). "Mercury: Facts & Figures". Solar System Exploration. NASA. Retrieved April 7, 2008.Tmean.mercury340KelvinMercury's surface mean temperatureVasavada, Ashwin R.; Paige, David A.; Wood, Stephen E. (February 19, 1999). "Near-Surface Temperatures on Mercury and the Moon and the Stability of Polar Ice Deposits" (PDF). Icarus. 141 (2): 179–193. Bibcode:1999Icar..141..179V. doi:10.1006/icar.1999.6175. Figure 3 with the "TWO model"; Figure 5 for pole.cp.mercury0,19cal/gr.oCMercury's surface specific heat capacity (regolith as dry soil)https://www.engineeringtoolbox.com/specific-heat-solids-d_154.html----------Tmean.earth = [ Φ (1-a) So (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴------Tmean.earth = [ Φ (1-a) So (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴MARS:R.mars1,525AUMars' distance from the sun, semi-major axisSimon, J.L.; Bretagnon, P.; Chapront, J.; Chapront-Touzé, M.; Francou, G.; Laskar, J. (February 1994). "Numerical expressions for precession formulae and mean elements for the Moon and planets". Astronomy and Astrophysics. 282 (2): 663–683. Bibcode:1994A&A...282..663S.a.mars0,25-Mars' albedoWilliams, David R. (September 1, 2004). "Mars Fact Sheet". National Space Science Data Center. NASA. Archived from the original on June 12, 2010. Retrieved June 24, 2006.1/N.mars1,02596days/rotationMars' (day-night) rotation periodLodders, Katharina; Fegley, Bruce (1998). The Planetary Scientist's Companion. Oxford University Press. p. 190.ISBN 978-0-19-511694-6.Tmean.mars210KelvinMars' surface mean temperatureWilliams, David R. (September 1, 2004). "Mars Fact Sheet". National Space Science Data Center. NASA. Archived from the original on June 12, 2010. Retrieved June 24, 2006.cp.mars0,18cal/gr.oCMars' surface specific heat capacity (prevail Fe2O3)https://www.engineeringtoolbox.com/specific-heat-solids-d_154.html--------------------