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.
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.
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 StefanBoltzmann 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 StefanBoltzmann 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 stefanboltzmann
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 StefanBoltzmann 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 http://cosy.com/Science/TemperatureOfGrayBalls.htm
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 
5.4113742e006 
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 http://sse.jpl.nasa.gov/planets/
) 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.
