Reading:
NOTE: Different vertical scales! The size of a scale height in each case compared with the height of the troposphere  the troposphere is about 23 scaleheights.
How do we QUANTIFY this greenhouse effect? How do we MODEL the absorption/radiation?
First start with a very, very simple model..... see Chapter 3 pages 5261 of Goody & Walker
NOTE  Te = T_{eff} is the EFFECTIVE temperature  that's the "temperature" that is derived by measuring all the (blackbody  or thermal radiation) IR flux from a planet
The OPAQUE SLAB MODEL  sometimes called "grey model"  does not depend on wavelength.
Assumptions:
Starting at the top of the atmosphere and working down layer by layer, equating the flux into each layer with the flux out
Layer 0  T_{1} = T_{eff}
Layer 1  sT_{2}^{4 }= sT_{1}^{4 }+ sT_{1}^{4 } ....so, T_{2}^{4 }= 2T_{1}^{4 }
Layer 2  sT_{3}^{4 }+ sT_{1}^{4 } = sT_{2}^{4 }+ sT_{2}^{4 } ....so, T_{3}^{4 }= 2T_{2}^{4 } T_{1}^{4 } = 4T_{1}^{4 } T_{1}^{4}= 3T_{1}^{4 }
.....^{ }
Layer N  T_{N}^{4 }= NT_{1}^{4 }
So, if you imagine you keep going down the layers until you reach the surface of the planet  where the temperature is Tg  then you can envisage an expression
Tg^{4 } = (1+t) T_{eff4 }
where we call t = OPTICAL THICKNESS or OPTICAL DEPTH
(why (1+t) rather than t? Good question. I have no idea  probably due to some alternative derivation).
Earth: t = (Tg/Teff)^{4 } 1 = (288K/255K)^{4 } 1 = 0.6  the Earth's atmosphere is thin  not even one totally absorbing layer.
Venus: t = (Tg/Teff)^{4 } 1 = (750K/238K)^{4 } 1 = 98  Venus atmosphere is very thick
(G&W quotes a surface temperature of 700 K and gets t = 68, I've seen different values for Teff too).
The next step would be to add heating and cooling due to absorption and radiation... and make a much more complicated model.
Rates are in units of Kelvin per day
In reality.... it's further complicated....by the fact that there are horizontal variations  some places have clouds, some do not. And there are vatiations with latitude.... etc. But, when you average over the globe, for Earth, this is what you get for a NET RADIATION BUDGET  visible light in (sunlight)  and IR light out (thermal radiation). The units are Watts per square meter
But we just want a simple comparison of Earth, Venus and Mars 
Isn't there a simpler way of looking at this?
For a THIN atmosphere  e.g. Earth, Mars  we can use a simpler model. The book has this...
We can go simpler....
Think of the atmosphere as a single slab. We want to build a model similar to the colored radiation budget diagram above for the Earth but much simpler. So, starting at the top, we have the Flux of visible sunlight coming in
Fin = 1368/(4 a^{2}) Watts m^{2} where a is the distance from the Sun in AU.
(Whoa!! where did the 4 come from? Why does the color diagram above have 342 Watts m^{2} coming into the atmosphere? This is the AVERAGE flux. The net flux is 1368 x p R^{2 }which must be averaged over the total area of 4p R^{2 } making an average of 1368/4 = 342 Watts m^{2} hitting the earth).
Of this a fraction of the is reflected  the fraction being the Albedo  A. So, (1A) of Fin goes into the planet (either into the atmosphere or the ground  does not strictly matter for this simple model).
Now let's look at the infrared side  the Flux of energy going outwards.
From the ground a flux of s Tg^{4 }Watts m^{2} is emitted (where s is the StefanBoltzmann constant of 5.67 x 10^{8} in SI units).
Some of this is absorbed by the atmosphere  we are going to calculate how much. So, look at the net amount that is radiated to space  it must be the same as the net amount that is absorbed  Fout = 1368/(4 a^{2}) (1A) Watts m^{2 }  but this must be the same as the flux of energy that is radiated by the atmosphere = s Teff ^{4 }
So  what happens in the middle? Let's define the fraction that gets absorbed by the atmosphere to be the OPACITY = e
Thus, the Flux that is absorbed is e s Tg^{4 }Watts m^{2 } and the amount that is radiated to space is then (1e) s Tg^{4 }= s Teff ^{4 } Watts m^{2}
This then means that we can define
OPACITY e = = 1  (Teff/Tg)^{4}
which is not to be confused with OPTICAL DEPTH t = (Tg/Te)^{4 }1
Opacity is useful for a THIN atmosphere where only a small fraction of the outgoing IR radiation is absorbed (e.g. Mars). Optical depth is useful for a THICK atmosphere where most of the outgoing IR radiation is absorbed (e.g. Venus).
E  V  M  
Distance from Sun, a (AU)  1  0.723  1.5 
Albedo, A  0.39  0.59  0.25 
Tground  288  750  223 
Teffective  255  264  213 
OPACITY e = 1  (Teff/Tg)^{4}  0.39  0.98  0.17 
OPTICAL DEPTH t = (Tg/Te)^{4 }1 
0.63  64  0.20 
So...40% of the energy is absorbed on its way from the ground out to space on Earth  at Venus this is 98%! but only 17% at Mars  seems a little but still half of the percentage of Earth even though Mars' atmospheric pressure is only 0.6% of a bar  shows how CO2 is an effective absorber.
You could now draw your own energy budget diagrams for Earth, Venus and Mars  with appropriate colors, objects on the surface, etc  showing the AVERAGE flow of energy through the system.
For all planets temperature decreases with altitude for the tropospheric layer, the first couple of scaleheights  WHY?
ADIABATIC LAPSE RATE  dT/dz
What does ADIABATIC mean?
dT/dz = g/cp
g= gravitational acceleration cp = Specific Heat Capacity at Constant Pressure
cp = squiggle R / M
M = mean molecular mass =<amu> 0.001  e.g. air = 29 x 0.001 kg / mol
R = gas constant = 8.31 J K^{1} mol^{1}
squiggle = 5/2 for monoatomic gas, = 7/2 for diatomic gas, = 9/2 for triatomic gas
UNITS: cp  (kg m^{2} s^{2} K^{1} mole^{1} )/(kg mole^{1}) = m^{2} s^{2} K^{1}
dT/dz = g/cp  m s^{2} /(m^{2} s^{2} K^{1}) = K m^{1}
What's cp for E, V, M? What's dT/dz for Earth, Venus Mars?
Gas  M kg/ mole 
cp m^{2} s^{2} K^{1} 
g m s^{2} 
dT/dz K / km 

V  CO_{2}  44 x 0.001  850  8.87  10.4 
E  O_{2} , N_{2}  29 x 0.001  1003  9.81  9.8 
M  CO_{2}  44 x 0.001  850  3.71  4.4 
So, Earth and Venus about the same  down in the troposphere  but at Mars the adiabatic temperature profile drops off more slowly with height.
SO WHAT?
     = adiabatic lapse rate, White line = actual lapse rate
 if cools faster than adiabatic  stable (cool air sinks)
 if cools slower than than adiabatic  unstable (warm air rises)
..... which leads us to circulation patterns  next class.