Bob Armstrong

|(| Thu.Feb,20090212 |)|

Basic Temperature Physics of Radiantly Heated Balls


It is commonly asserted that the basic physics of planetary temperature is well understood. However, it is surprisingly difficult to find any exposition of that basic physics. This lack contributes to the most egregious impossibilities being touted as ultimate horrors we might face if we don't mend our ways. On the skeptic side, it has even been claimed that the notion of a planetary mean is meaningless. I present below my current understanding. I find that altho my master's thesis had integrals which filled a page, having spent an adulthood in the arrays of the most powerful languages for modeling actual finite data , I'm terminally inept at expressing ideas in traditional maths notation. I have, however, instantiated the basic formulas for anisotropically irradiated anisotropically shaded gray balls in a succinct model can be understood by anyone familiar with array capable programming languages.

The complete paper and model is linked at .

There are several interesting results from the analysis. Of most consequence is that the ubiquitous explanation of a "greenhouse effect" is simply the need to fill a gap created by an incorrect equation. Also notable is that the temperature of Venus, commonly cited as what "runaway" warming could do to us, is twice as hot as the sun can possibly heat any object in its orbit. That is, Venus must have some internal source of heat because it is radiating much more energy than it is receiving from the sun.

Black Bodies

I will not consider fluid motion of the atmosphere because our interest is with mean temperature averaged over the sphere over time. Stefan-Boltzmann & observed temperatures of inner planets Our interest is in the constraints on the temperature of radiantly heated balls like our earth, in a sense looking from the outside rather than looking up at the complexities of the atmosphere.

The fundamental law whose most superficial application explains all but at most a couple of percent of the earth's temperature is the Stefan-Boltzmann law. Applying this basic law for black bodies to the surface temperature of the sun and the solid angle it makes in the sky explains the mean temperatures of each of the inner planets, except for Venus, quite closely as shown in the graph [0] .

The Stefan-Boltzmann law is a simple equation with a complicated constant derived from fundamentals which states that the Power radiated by a body is its Emissivity times the stefan-boltzmann constant times its Temperature raised to the 4th power.

        Psb : {[ T ]  sb * T ^ 4 }       | For a theoretical blackbody E equals 1 and is omitted here.

The "Black Body" Wikipedia page was the first place I saw the derivation of the Earth's temperature from the Sun's based on this equation. It simply equates the power radiated by the sun and intercepted by the disk of the earth with the power radiated by the earth in all directions:[1]

( ( pi * Re ^ 2 )  *  ( sb * Ts ^ 4 ) * ( 4 * pi * Rs ^ 2 )
  %   ( 4 * pi * D ^ 2 ) )
area of earth facing sun * radiance of sun * surface area of sun   %   area of earth orbit sphere
( ( 4 pi * Re ^ 2 ) * ( sb * Te ^ 4 ) ) | e1 |
surface area of earth * radiance of earth

Canceling terms and rearranging a little, we get

        ( ( Ts ^ 4 ) * ( Rs ^ 2 ) % ( 4 * D ^ 2 ) ) = ( Te ^ 4 )       | e2 |

and taking 4th roots:

        Te = ( ( Rs % 2 * D ) ^ % 2 ) * Ts       | e3 |

we get a function which says the temperature of a body in orbit will simply be proportional to the square root of half the ratio of the sun's radius to the distance between the object and the sun. Note that the radius of the Earth has dropped out; thus a point is equivalent to a sphere, and only the solid angle subtended by the Sun matters. 

Plugging in values of 6.96e8 and 1.496e11 meters for the radius of the Sun and the distance from the Earth to the Sun gives the result the result that the temperature of a black body in Earth's orbit has a temperature of 0.04823 or about 1/21 the surface temperature of the Sun . This is the computation which produces the monotonic line in the plot above. An estimated effective sun surface temperature of 5778k gives an earth mean temperature of about 279k ; 289k for a sun temperature of 6000k .

A couple of additional observations : 

0 ) the diference between the perihelion and aphelion temperatures is about 1 percent. This should be easily confirmed.

1 ) The mean temperature of a radiantly heated ball in the orbit of Venus has a temperature .0567 the sun's or about 328k .

Gray Bodies

But, of course, real planets are not black. For a "gray" body, there is an associated parameter Emissivity, constant across the spectrum, which like a gray filter in front of a radiating black body, uniformly reduces the power at each frequency from what would be expected from a black body. There is a complementary parameter Absorptivity, ranging from black to totally reflective, which quantifies the tendency of a substance to absorb radiant energy.

150 years ago in 1859, Gustav Kirchhoff realized that at equilibrium, the emissivity of an object must equal its absorptivity, E = A . Thus equation e2 above would become

        ( Ae * ( Ts ^ 4 ) * ( Rs ^ 2 ) % ( 4 * D ^ 2 ) ) = ( Ee * Te ^ 4 )       | e5 |

with terms Ae * and Ee * added to the left and right sides for the Absorptivity and Emissivity of the earth respectively. But, by Kirchoff, these terms cancel out at equilibrium.[2] In the middle of 2008, the Wikipedia Black Body page was modified adding the Ae to the left side of this equation but failing to add Ee to the right side.

The most common explanation of the need for a "greenhouse" effect is that because the reflectivity, ( 1 - Ae ) , or albedo ( unfortunately also beginning with "A" ) of the earth is about 0.3, and therefore its absorptivity about 0.7 , its mean temperature should be only about the 4th root of 0.7 , or about 0.91 the temperature calculated above. The approximately 30c difference between this value and the earth's actual temperature which is quite near the black body value is asserted to be due to the effect of the "greenhouse" gases. Thus the putative greenhouse effect is not explained thru explicit quantitative physical assertions, rather simply an assertion of the need to fill a gap created by an incorrect equation.

Judging from Wikipedia references, this explanation of the greenhouse effect seems to be common in texts on global warming. But this explanation is simply wrong. If it were the case, one would expect a ball coated with Magnesium Oxide with an albedo of about 0.9 to come to an equilibrium temperature of about -120c in a vacuum bottle sitting in room temperature surroundings. Venus has the highest reflectivity of all the inner planets, about 0.75 . This would imply a temperature about 30% below the calculated black body temperature for its orbit, or about 233k versus 328k . However, of course, Venus is radiating energy at a temperature of about 735k, both on the side facing and away from the Sun -- and its day is slightly longer than its year. The most heating this greenhouse theory could predict is raising the temperature back to the black body temperature. No "runaway" effect could raise the temperature beyond that. On the other hand, according to this theory, as snow with an albedo which can be nearly as high as MgO covered the continents during the ice ages, the Earth should have spiraled down to a permanent snowball. That it didn't is one of the first facts which made me question the AGW orthodoxy. It is notable that, so far as I know, there is no laboratory demonstration of their claimed phenomenon. 

From this point on , I use actual expressions from my K implementation of the Stefan-Boltzmann Law for the simple case of a point in its surrounding sphere . The equilibrium temperature of a ball will vary if the absorptivity/emissivity parameter is different in directions with different temperatures.  We can express this as a function on 3 vectors and a scalar : a partition of the celestial sphere, SfeerPart , the temperature of each partition of the celestial sphere , Tcs , the absorptivity/emissivity of each partitian , AE , and the temperature of the ball considered to be uniform , Tp . ( This is the parameter upon which it is a point . ) We will just deal with a partition of the sphere into ( SunDisk  ; DaySide ; NightSide ) . 

See for the complete definitions and computations. Here we just skip to the results .

Here are the parameters for earth . You can see the Sun only makes up about a 5 millionths of the celestial sphere , with all the rest at the cosmic microwave background of 3 degrees.

SfeerPart Tcs AE
SunDisk   5.4113742e-006   5778   1
DaySide   0.49999459   3   1
NightSide   0.5   3   1

We can use K's secant descent search function to find the point temperature balancing its surrounding sphere temperature distribution by searching for a difference of zero .  The "?" function takes a third argument which is a starting guess to keep the aritmetic computable .  300 will work here .

 PgrayBall : { +/ AE * SfeerPart * Psb x }  / total radiated power from shaded gray ball
 Tdif : { -/ PgrayBall @/: ( Tcs ; x ) }    / Difference between celestial sphere & ball

 ?[ Tdif ; 0.0 ; 300 ] />/ 278.67912

The 278.68 result matches that found above .

Now we can calculate temperatures for some other interesting cases :

For a point totally surround by a uniform temperature , 
 AE : 1 1 1  ;  Tcs : 3 # Sun.temp  ;  ?[ Tdif ; 0.0 ; 6000 ] />/ 5778.0
 AE : 1 1 0  ;  Tcs : 3 # Sun.temp  ;  ?[ Tdif ; 0.0 ; 6000 ] />/ 5778.0
it makes no difference how it's shaded .

A disk in earth orbit , black on the day side ,
 totally reflective and non emissive on the night side .

 AE : 1 1 0   ;   ?[ Tdif ; 0.0 ; 300 ] />/ 331.40719
This is the maximum any object in earth orbit can get .

Note , there is an observed effect of the interaction of the season with the greater albedo of the southern icecap of about a degree centigrade .

Similarly , the temperature of an object , black on the day side , white on the night , in Venus orbit  is found to be 390k confirming that Venus is much hotter than any simply radiantly heated object in its orbit could be . There must be some other internal source of heat .

I do not understand how James Hansen could possibly have claimed that Venus's mean temperature could be due to heat trapping of any sort . I understand even less how such claims could have survived the most cursory "peer" review .

To complete the analysis for colored bodies , the spectrum needs to be "unfolded" like direction has here . Then , the effect of the couple of notches in CO2's specturm can be calculated . A priori , the specturm representing only about .08 of the sun's , and already rather saturated at levels at which plants barely survive , the effect on mean temperature is likely to be de minimis . The effect on reducing diurnal variance , tho , ought be calculated .

[0] The planets are, of course, Mercury, Venus, Earth, and Mars. The distances are in percents of Earth's mean orbital radius. NASA's temperature measurements of the planets ( at ) other than earth are much less precise than one might expect. It should be noted that the total temperature variation of the earth over the last century is on the order of one part in three hundred.

[1] My notation is informed by an adulthood living in APL languages. The rational is summarized here . The language used here is Arthur Whitney's language , K . Perhaps the only non obvious symbol is % for division and reciprocal. In APL languages f/ can be read as "f across", for instance +/  is "sum".  "

[2] I've heard complaints that the Earth and Sun are not in equilibrium, but I believe that over a cycle, like Earth's yearly cycle, it can be proved that for a conservative quantity like energy, the outcome is equivalent.
I have seen it pointed out that Kirchhoff's equating of absorption and emission only applies at equilibrium. However absorption and emission spectra are generally rather constant for a substance over a given physical state. For instance, it is when water changes to snow that it's albedo changes drastically.

    Note : I reserve the right to post all communications I receive or generate to CoSy website for further reflection .
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