Development of Method A


  • δTs = -0.30 * δH, where H is ModTran output for earth’s radiation to space
    and Ts = earth surface temperature, deg C.
  • The Present Greenhouse Effect is 36.7% or 139 w/m2 or 31C.

This method allows anyone with a background in differential calculus to understand the computation of change in average earth’s surface temperature Ts using baseline Ts and the heat H radiated by earth to space before and after altering the baseline. H is calculated by simulating conditions with ModTran.  Effect of Albedo is addressed under the albedo tab above.

NOTE: Upper case delta Δ represents the change in a variable, such as (T).  Lower case delta (δ) represents a small change.  A small finite change in temperature δT represents a finite enlargement of the infinitesimal change in temperature dT (from differential calculus).

NOTE: This method does not allow for variation in sunlight reflected from the earth’s surface; i.e. albedo.   The total temperature change is the sum of the δT’s for each independent variable:

δT = ∑ (òT/òxi)*δxi + òT/òA)*δA,

where xi are components of the atmosphere, whose effect on surface temperature can be computed via ModTran.  A is albedo.  The value of òT/òA*δA must be calculated separately and added to the effect of changing concentrations of atmospheric components (primarily CO2).

  • compute the annual average temperature of the earth’s surface Ts
  • compute the change in Ts with environmental conditions, ΔTs
  • compute the Greenhouse Effect as heat flux, flux as a fraction, and as ΔTs.

Derive formula for the earth’s surface temperature Ts corresponding to earth’s heat balance:

πr2 * So * (1 – A) = 4πr2εσTs* (1 – B)
(heat in)               =         (heat out)


  • r = radius of earth = Intensity of Solar Radiation at Earth Orbit = 1367 w/m2
  • A = Earth’s Albedo (overall reflectivity) ≈ 30%
  • ε = Earth’s Emissivity, approximately 97% (blackbody = 100%)
  • σ = Stefan-Boltzmann Law constant = 5.67E-08 w/m2K4
  • Ts = Annual Average   Temperature of the Earth’s Surface = 288K
  • B = Greenhouse Coefficient = (presently 0.367)

Divide by earth’s surface area 4πr2 yields, and assuming equilibrium between heat in and heat out (heat loss is negative):

H = So * (1 – A) / 4  =  -εσTs* (1 – B)  = |240 w/m2 of Earth’s Surface|

Note: H = 240 w/mwas measured by satellite (average annual earth surface temp.)

Compute “B” and Greenhouse Effect

  • Define: Q as heat loss to space if there were no greenhouse effect, w/m2
  • H = actual heat lost to space, = solar heat gained = 240 w/m2

The greenhouse effect lowers the amount of heat lost to space, causing earth’s temperature to rise to maintain loss equaling solar input, therefore

Greenhouse Effect B is Q – H, w/m2 .

Current average annual earth surface temperature Ts = 288.2K.

Q = 0.97 * 5.67E-08 * (288.2K4) = 379.4 w/m2


B = Q – H = 379.4 w/m2 – 240 w/m2 = 139.4 w/m2 

or as a fraction

139.4 w/m2  / 379.4  w/m0.367
and the term (1-B) = 0.633

Compute Temperature Change as Function of Modtran Output

Remember, H = infrared emitted to space, computed by Modtran.

Differentiate  equation H = εσTs* (1 – B)  with respect to T:

 dH = -4εσTs* (1 – B) * dT

approximating:                           δH =  -4εσTs* (0.633) * δT = -3.33 * δT


δT = 3.0 * δH 

Compute Greenhouse Effect in Terms of Temperature Change

  • The solar input = So * (1 – A)  =  εσTs* (1 – B) = 240, doesn’t change.
  • For no greenhouse effect, B = 0 and  εσTs = 240
  • Rearranging and solving for T gives T = 257K
  • Since current average annual earth surface temperature is 288 K, the greenhouse effect, in terms of temperature, would be

 288K – 257K = 31K = 31C