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
propose a exposition of this basic physics and
present below my current understanding. I welcome review and, indeed,
coauthorship with someone with more established credentials.
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.
' Psb : {[ E ; T ] E * sb * T ^ 4 }
 e0  For a
theoretical blackbody E
equals 1 and is
usually omitted.
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]
( ( 4 * pi
* Rs ^ 2 ) 
* 
( sb
* Ts ^ 4 ) 
* 
( pi
* Re ^ 2 ) 
% 
( 4 * pi
* D ^ 2 ) ) 
surface area of sun 
* 
radiance of sun 
* 
area of earth facing
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 
or to express as a function :
` Trd : {[ R ; D ] ( ( R % D ) % 2 ) ^ % 2 }
 e4 
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. Given that the SolidAngle
suptended by the a disk of radius R is (
pi * R ^ 2 ) % ( D ^ 2 ) , equation e2
can be rewritten
( ( Ts ^ 4 ) * SAs % ( 4 * pi
) ) = (
Te ^ 4 )  e5 
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 about 0.04823 the surface temperature of the Sun . This
is the computation which produces the green line in the plot above.
But, of course, the 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 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. 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"
gasses. 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.
The equilibrium temperature will vary
if the absorptivity/emissivity parameter is different in different
directions. One way to approach the issue is to
( +/ Ae * ( Ts ^ 4 ) * SA % (
4 * pi ) )
= ( +/ Ee * Te ^ 4 )

e5 
( +/ Ae * ( Ts ^ 4 )  ( Te ^ 4 ) )
= 0
[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
. Perhaps the only non obvious symbol is %
for division and reciprocal.
[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.
This Wikipedia section has been
screwed up by the introduction of a term for absorptivity of the earth
without the countervailing term for emissivity in the expression for
radiation from the earth.
