The Numerical Temperature Effect Of The Two Greenhouse Gasses CO2 And H2O With Earth 2013 As An Example, Including Appendices For Americans

The following document is public domain, free for use by all. Time of writing 2013.05.24. This document can serve as a climate model for the greenhouse effect of CO2 and H2O on an Earth-like planet (Venus or Earth or Mars). In this case, the planet is Earth 2013, but providing the appropriate CO2 and H2O levels and the appropriate solar constant and the appropriate absorption coefficient will give a temperature estimate within a +6 to -2 Celsius spread for Venus and Earth and Mars. Given some uncertainty in absorption coefficients and cooling mechanisms of Venus and Earth and Mars, a 8 degree Celsius spread (an estimate of 6 Celsius too hot to 2 Celsius too low) is as much as can be hoped for as a test for this model. This zero-dimensional model does not include cooling from dust, so will be biased toward the high side. This model is an adjustment of the model used by Svante Arrhenius in his 1896 April climate change paper in the scientific journal "Philosophical Magazine", refined with satellite data which of course was not available to Arrhenius back in 1896.


In a planet's atmosphere, a value known as infrared (IR) optical thickness can create a significant change in that planet's temperature in a way that can be calculated accurately. For the planets Venus, Earth and Mars, taking into account H2O (water vapor) and CO2 (carbon dioxide) are sufficient for determining greenhouse gas effect. In technical parlance, a greenhouse gas is a gas that increases an atmosphere's effective radiative IR optical thickness, thereby increasing the temperature of the surface of the planet underneath that atmosphere. Counteracting IR optical thickness are somewhat investigated cooling mechanisms such as evaporation, sublimation, convection, and atmospheric absorption. The cooling mechanisms as an approximation make Venus 33 Celsius cooler, Earth 28 Celsius cooler, and Mars cooler by 19 degrees Celsius. Cooling from volcanic and duststorm particulates are excluded, meaning that the cooling estimates are probably too low. These cooling mechanisms even when excluding particulate cooling are important, because imagine planet Earth 28 degrees Celsius (50.4 degrees Fahrenheit) hotter than its current temperature.

Initial temperature calculation for Earth using the Stefan-Boltzmann equation,
without taking into account the greenhouse effect.

Let 1365 watts per square meter for Earth be the solar constant. Divide by 4.0001 since Earth departs slightly from being spherical due to a moderate rate of rotation. Multiply by 0.694 for the absorption coefficient of Earth, and we have 236.821 watts per square meter for the climate flux of Earth. Dividing 236.821 by the Stefan-Boltzmann coefficient of 5.67037 * 10^-8 (5.67037 times ten to the minus 8 power) and dividing by the emissivity value of 0.95, and then raising to the 0.25 power we get a temperature of 257.496 degrees Kelvin, much colder than the freezing point of water at 273.15 degrees Kelvin. (See appendix A for this version of the Stefan-Boltzmann equation.)

Explaination of IR optical thickness calculation from surface pressures of greenhouse gases CO2 and H20.

For infrared (IR) optical thickness value from a greenhouse gas on a planet, raise the surface pressure in Pascals for that gas to the 0.559 power, and then multiply by the appropriate coefficient. The reason for using 0.559 is that every doubling of CO2 multiplies a planet's effective IR optical thickness by about 1.47 (not 1.41), as revealed by 20th Century Russian and American satellite data from the planet Venus. From laboratory experiments, H2O (water vapor) has about 3 times the greenhouse power as CO2, so the coefficient of H20 is about 3 times as large as CO2.

IROptThickCO2 = 0.0199 * PressureCO2 ^ 0.559;
IROptThickH20 = 0.0597 * PressureH20 ^ 0.559;

Adding the optical thickness vectors is actually somewhat similar to adding two perpendicular vectors.
Since 1/0.559 = 1.78890,
IROptThickTotal =
IROptThickCO2 ^ 1.78890 +
IROptThickH20 ^ 1.78890
) ^ 0.559;
Note the similarity to a distance formula involving the square root of the sum of the squares. In this case 1.78890 is used instead of 2.0 (square), and 0.559 is used instead of 0.50 (square root). This variation on the distance formula ensures that an added pascal of H2O vapor pressure will result in 3 times the IR optical thickness change as the addition of one pascal of CO2 pressure, in keeping with experimental result.

For Earth in 2013, PressureCO2 is 40 pascals. PressureH20 is effectively about 396 pascals, about 23 percent of saturation water vapour pressure at Earth's mean temperature in accordance with an approximation by Arrhenius that was recently verified by the Hadley Centre to be valid across a broad temperature range, comparable to the temperature range that occurred in the Cenozoic Era. On Earth, the ratio of CO2 to H2O varies greatly with altitude as well as with latitude and longitude, unlike Venus, but things have a way of averaging out despite regional and altitudinal variation on Earth. Locally, there are exceptions. For example, over most of Earth, H2O vapour is the chief greenhouse gas, but in the Arctic and Antarctic in the Winter, CO2 is the chief greenhouse gas, and rapidly rising CO2 concentration as calculated below sets a rapidly rising floor of about 0.156 on Earth's atmospheric IR optical thickness at sea level.
IROptThickCO2 = 0.0199 * 40^0.559 = 0.156;

The remainder of overall IR optical thickness calculations for Earth's atmosphere follow:
IROptThickH20 = 0.0597 * 396^0.559 = 1.690;
IROptThickTotal = 0.156^1.78890 + 1.690^1.78890)^0.559
= (0.036 + 2.558)^0.559
= 1.703;

Applying total IR optical thickness to get a temperature estimate.

GreenhouseTemperature = NonGreenhouseTemperature * (1 + 0.75 * IROptThickTotal)^0.25;

For Earth,
GreenhouseTemperature = 257.496 * (1 + 0.75 * 1.703)^0.25
= 257.496 * (2.277)^0.25 (See Appendix B for the significance of 2.277 number)
= 257.496 * 1.228
= 316.337 Kelvin

Subtracting the estimated cooling mechanism value for Earth of 28 degrees Celsius gives 288.337 Kelvin (15.187 Celsius or 59.336 Fahrenheit) for the calculated temperature of Earth, in comparison to the observed temperature from worldwide weather records and from Russian and American spacecraft of 288 Kelvin (14.85 Celsius or 58.73 Fahrenheit) for Earth. (See Appendix C for how to convert Kelvin to Fahrenheit or Celsius.) The match between calculation and observation is very close for Earth, as it is for Mars and Venus if you wish to make the calculations using the appropriate inputs. Moreover, it can be seen with precision how it is that science books say that carbon dioxide and water vapour turns Earth from what would be a sub-freezing planet to a place where mammalian life forms thrive: the amount of CO2 and water vapour multiplies the climate flux temperature by more than 1.2 due to the calculated IR optical absortion thickness value of a little bit over 1.7, more than enough of a multiplication to both offset atmospheric cooling effects and to raise Earth's temperature above freezing.

Summarization of the effect of greenhouse gases on Venus, Earth, and Mars.

For Venus and Earth and Mars, after taking into account different albedos (reflectivities) and a common emissivity of 0.95, and their individual solar constants, there is a slight overestimate for all 3 planets for the greenhouse temperature -- a necessity for a good model since cooling mechanisms such as convection and atmospheric absorption and evaporation and duststorms and daily temperature range are not taken into account by the application of the optical thickness equation. A subsequent subtraction of cooling mechanism numbers (cooling from dust still not taken into account) in degrees Kelvin (or Celsius) for each planet then gives a very close approximation of observed temperature for each planet, with an error spread of less than 8 Celsius for all three Earthlike planets in the solar system (Earth, Venus, and Mars).


Appendix A.

Scientists can use the Stefan-Boltzmann equation to calculate a temperature value in degrees Kelvin, or alternatively a heat emitted value in watts per square meter. If one value is known than the other can be calculated. An approximate emissivity value of 0.95 is often used for a planet as a whole. As such, if temperature is known, then heat emitted can be calculated. The heat emitted is proportional to the fourth power of the temperature. More specifically,

HeatEmitted = 0.95 * 5.67037*10^-8 * Temperature^4;

Likewise, if the heat emitted by a planet is known, then temperature can be calculated from a second form of the Stefan-Boltzmann equation. In this form, temperature is proportional to the 0.25 power of heat emitted.

Temperature = (HeatEmitted / 0.95 / (5.67037*10^-8))^0.25;

It is this second form that is used to compute a non-greenhouse temperature for Earth. In this case, HeatEmitted represents the climate flux of Earth.

Appendix B.

IREmissivityDivisor = (1 + 0.75 * IROpticalThickness);

The number is an emissivity divisor obtained from the IR optical absorption thickness of Earth's atmosphere. Applying the second form of Stefan-Boltzmann from Appendix A, the infrared emissivity divisor if raised to the 0.25 power becomes a multiplier to the non-greenhouse temperature to obtain a raw greenhouse temperature. If the emissivity divisor obtained from the IR optical thickness is 16, the raw greenhouse temperature is 2 times the non-greenhouse temperature, just as if all except 1/16 of every square meter of the planet were covered by a substance that was perfectly transparent in the visible light range but a perfect opaque insulator in the infrared range. If the emissivity divisor is 81, then the raw greenhouse temperature is 3 times the non-greenhouse temperature as if all except 1/81 of the planet were covered with such a substance. The calculated emissivity divisor of 2.277 results in a temperature multiplier of 1.228, since 2.277^0.25 = 1.228 .

Appendix C.

The observed temperature of Earth from Russian and American spacecraft is about 288 degrees Kelvin. In terms of Fahrenheit,

Fahrenheit = 1.8*Kelvin - 459.67;

Since Earth is 288 Kelvin in temperature
EarthTemperatureFahrenheit =
= 1.8*288 - 459.67
= 518.4 - 459.67
= 58.73 Fahrenheit;

To get from Kelvin to Celsius, Celsius = Kelvin - 273.15;
EarthTemperatureCelsius =
288 - 273.15 =
m108rfd m108rfd
56-60, M
Jun 6, 2013