- 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) = 4πr2εσTs4 * (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 = -εσTs4 * (1 – B) = |240 w/m2 of Earth’s Surface|
Note: H = 240 w/m2 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/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/m2 = 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 = εσTs4 * (1 – B) with respect to T:
dH = -4εσTs3 * (1 – B) * dT
approximating: δH = -4εσTs3 * (0.633) * δT = -3.33 * δT
δT = 3.0 * δH
Compute Greenhouse Effect in Terms of Temperature Change
- The solar input = So * (1 – A) = εσTs4 * (1 – B) = 240, doesn’t change.
- For no greenhouse effect, B = 0 and εσTs4 = 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