**SUMMARY:**

**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/m**^{2}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/òx_{i)*}δx_{i} + òT/òA)*δA,

where x_{i} 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:**

πr^{2} * S_{o} * (1 – A) = 4πr^{2}εσT_{s}^{4 }* (1 – B)

(heat in) = (heat out)

Where

- r = radius of earth = Intensity of Solar Radiation at Earth Orbit = 1367 w/m
^{2} - A = Earth’s Albedo (overall reflectivity) ≈ 30%
- ε = Earth’s Emissivity, approximately 97% (blackbody = 100%)
- σ = Stefan-Boltzmann Law constant = 5.67E-08 w/m
^{2}K^{4 } - T
_{s}= Annual Average Temperature of the Earth’s Surface = 288K - B = Greenhouse Coefficient = (presently 0.367)

Divide by earth’s surface area 4πr^{2} yields, and assuming equilibrium between heat in and heat out (heat loss is negative):

H = S_{o} * (1 – A) / 4 = -εσT_{s}^{4 }* (1 – B) = |240 w/m^{2 }of Earth’s Surface|

Note: H = 240 w/m^{2 }was 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/m
^{2} - H = actual heat lost to space, = solar heat gained = 240 w/m
^{2 }

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/m^{2 }.

Current average annual earth surface temperature T_{s} = 288.2K.

Then

Q = 0.97 * 5.67E-08 * (288.2K^{4}) = 379.4 w/m^{2}

Then

B = Q – H = 379.4 w/m^{2} – 240 w/m^{2 }= **139.4 w/m ^{2} **

or as a fraction

139.4 w/m^{2} / 379.4 w/m^{2 }= **0.367**

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 = εσT_{s}^{4 }* (1 – B) with respect to T:

dH = -4εσT_{s}^{3 }* (1 – B) * dT

approximating: δH = -4εσT_{s}^{3 }* (0.633) * δT = -3.33 * δT

rearranging:

**δT = 3.0 * δH **

**Compute Greenhouse Effect in Terms of Temperature Change**

- The solar input = S
_{o}* (1 – A) = εσT_{s}^{4 }* (1 – B) = 240, doesn’t change. - For no greenhouse effect, B = 0 and εσT
_{s}^{4 }= 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**