Editing Greenhouse effect (section)

== Basic formulas ==

=== Effective temperature ===

A given flux of thermal radiation has an associated ''effective radiating temperature'' or ''[[effective temperature]]''. Effective temperature is the temperature that a [[black body]] (a perfect absorber/emitter) would need to be to emit that much thermal radiation.<ref name="mbta">{{cite web |title=Box 2: Solar and Earth Radiation and the Greenhouse Effect (adapted from Mackenzie, 2003) |url=http://www.soest.hawaii.edu/mguidry/Unnamed_Site_2/Chapter%202/Figures/Box%202%20Solar%20and%20Earth%20Radiation%20Greenhouse%20Effect.pdf |website=Myron B. Thompson Academy |date=2006}}</ref> Thus, the overall effective temperature of a planet is given by

:<math>T_\mathrm{eff} = (\mathrm{OLR}/\sigma)^{1/4}</math>

where OLR is the average flux (power per unit area) of outgoing longwave radiation emitted to space and <math>\sigma</math> is the [[Stefan-Boltzmann constant]]. Similarly, the effective temperature of the surface is given by

:<math>T_\mathrm{surface,eff} = (\mathrm{SLR}/\sigma)^{1/4}</math>

where SLR is the average flux of longwave radiation emitted by the surface. (OLR is a conventional abbreviation. SLR is used here to denote the flux of surface-emitted longwave radiation, although there is no standard abbreviation for this.)<ref name="Haberle2013" />

=== Metrics for the greenhouse effect ===
[[File:Greenhouse Effect metrics time series.svg|thumb|upright=1.35|Increase in the Earth's greenhouse effect (2000–2022) based on NASA CERES satellite data.]]

The IPCC reports the ''greenhouse effect'', {{mvar|G}}, as being 159 W m{{sup|-2}}, where {{mvar|G}} is the flux of longwave thermal radiation that leaves the surface minus the flux of outgoing longwave radiation that reaches space:<ref name="ipcc-ar6wg1-ch7" />{{rp|968}}<ref name="ravram1" /><ref name="Schmidt2010" /><ref name="Schmidt2010paper" />

:<math>G = \mathrm{SLR} - \mathrm{OLR}\;.</math>

Alternatively, the greenhouse effect can be described using the ''normalized greenhouse effect'', {{mvar|g̃}}, defined as

:<math>\tilde g = G/\mathrm{SLR} = 1 - \mathrm{OLR}/\mathrm{SLR}\;.</math>

The normalized greenhouse effect is ''the fraction of the amount of thermal radiation emitted by the surface that does not reach space''.
Based on the IPCC numbers, {{mvar|g̃}} = 0.40. In other words, 40 percent less thermal radiation reaches space than what leaves the surface.<ref name="ipcc-ar6wg1-ch7">{{cite book |title=Climate Change 2021: The Physical Science Basis |date=2021 |publisher=IPCC |url=https://www.ipcc.ch/report/ar6/wg1/downloads/report/IPCC_AR6_WGI_Chapter07.pdf |access-date=24 April 2023 |chapter=Chapter 7: The Earth's Energy Budget, Climate Feedbacks, and Climate Sensitivity}}</ref>{{rp|968}}<ref name="ravram1">{{cite journal |last1=Raval |first1=A. |last2=Ramanathan |first2=V. |title=Observational determination of the greenhouse effect |journal=Nature |date=1989 |volume=342 |issue=6251 |pages=758–761 |doi=10.1038/342758a0 |bibcode=1989Natur.342..758R |s2cid=4326910 |url=https://www.nature.com/articles/342758a0}}</ref><ref name="ravram2">{{cite journal |last1=Raval |first1=A. |last2=Ramanathan |first2=V. |title=Observational determination of the greenhouse effect |journal=Global Climate Feedbacks: Proceedings of the Brookhaven National Laboratory Workshop |date=1990 |pages=5–16 |url=https://www.osti.gov/servlets/purl/6440147-fP92Pd/#page=10 |access-date=24 April 2023}}</ref>

Sometimes the greenhouse effect is quantified as a temperature difference. This temperature difference is closely related to the quantities above.

When the greenhouse effect is expressed as a temperature difference, <math>\Delta T_\mathrm{GHE}</math>, this refers to the effective temperature associated with thermal radiation emissions from the surface minus the effective temperature associated with emissions to space:

:<math>\Delta T_\mathrm{GHE} = T_\mathrm{surface,eff} - T_\mathrm{eff}</math>
:<math>\Delta T_\mathrm{GHE} = \left(\mathrm{SLR}/\sigma\right)^{1/4} - \left(\mathrm{OLR}/\sigma\right)^{1/4}</math>

Informal discussions of the greenhouse effect often compare the actual surface temperature to the temperature that the planet would have if there were no greenhouse gases. However, in formal technical discussions, when the size of the greenhouse effect is quantified as a temperature, this is generally done using the above formula. The formula refers to the effective surface temperature rather than the actual surface temperature, and compares the surface with the top of the atmosphere, rather than comparing reality to a hypothetical situation.<ref name="Haberle2013">{{cite journal |last1=Haberle |first1=Robert M. |title=Estimating the power of Mars' greenhouse effect |journal=Icarus |date=2013 |volume=223 |issue=1 |pages=619–620 |doi=10.1016/j.icarus.2012.12.022|bibcode=2013Icar..223..619H }}</ref>

The temperature difference, <math>\Delta T_\mathrm{GHE}</math>, indicates how much warmer a planet's surface is than the planet's overall effective temperature.

===Radiative balance===
{{Further|Earth's energy budget}}
[[File:Earth Energy Budget with GHE.svg|thumb|upright=1.35|The greenhouse effect can be understood as a decrease in the efficiency of planetary cooling. The greenhouse effect is quantified as the portion of the radiation flux emitted by the surface minus that doesn't reach space, i.e., 40% or 159 W/m<sup>2</sup>. Some emitted radiation is effectively cancelled out by downwelling radiation and so does not [[Heat transfer#Radiation|transfer heat]]. Evaporation and convection partially compensate for this reduction in surface cooling. Low temperatures at high altitudes limit the rate of thermal emissions to space.]]
Earth's top-of-atmosphere (TOA) [[Earth's energy budget|energy imbalance]] (EEI) is the amount by which the power of incoming radiation exceeds the power of outgoing radiation:<ref name="UNeei">{{cite web |title=The Earth's Energy Imbalance: Where does the energy go? |url=https://unfccc.int/sites/default/files/resource/EID%20Pres%20T1%20Mercator%20EEI.pdf |publisher=United Nations Climate Change |access-date=14 June 2023}}</ref>

:<math>\mathrm{EEI} = \mathrm{ASR} -\mathrm{OLR}</math>

where ASR is the mean flux of absorbed solar radiation. ASR may be expanded as

:<math>\mathrm{ASR} = (1-A) \,\mathrm{MSI}</math>

where <math>A</math> is the [[albedo]] (reflectivity) of the planet and MSI is the [[solar irradiance|mean solar irradiance]] incoming at the top of the atmosphere.

The [[Planetary equilibrium temperature|radiative equilibrium temperature]] of a planet can be expressed as

:<math>T_\mathrm{radeq} = (\mathrm{ASR}/\sigma)^{1/4} = \left[(1-A)\,\mathrm{MSI}/\sigma  \right]^{1/4} \;.</math>
 
A planet's temperature will tend to shift towards a state of radiative equilibrium, in which the TOA energy imbalance is zero, i.e., <math>\mathrm{EEI} = 0</math>. When the planet is in radiative equilibrium, the overall effective temperature of the planet is given by

:<math>T_\mathrm{eff} = T_\mathrm{radeq}\;.</math>

Thus, the concept of radiative equilibrium is important because it indicates what effective temperature a planet will tend towards having.<ref name="ACSPredPlanTemp">{{cite web |title=Predicted Planetary Temperatures |url=https://www.acs.org/climatescience/energybalance/predictedplanetarytemperatures.html |website=ACS Climate Science Toolkit |publisher=American Chemical Society |archive-url=https://web.archive.org/web/20230326030348/https://www.acs.org/climatescience/energybalance/planetarytemperatures.html| archive-date=26 March 2023|access-date=14 June 2023}}</ref><ref name="rrtmeeb" />

If, in addition to knowing the effective temperature, <math>T_\mathrm{eff}</math>, we know the value of the greenhouse effect, then we know the mean (average) surface temperature of the planet.

This is why the quantity known as the greenhouse effect is important: it is one of the few quantities that go into determining the planet's mean surface temperature.

=== Greenhouse effect and temperature ===

Typically, a planet will be close to radiative equilibrium, with the rates of incoming and outgoing energy being well-balanced. Under such conditions, the planet's equilibrium temperature is determined by the mean solar irradiance and the planetary albedo (how much sunlight is reflected back to space instead of being absorbed).

The greenhouse effect measures how much warmer the surface is than the overall effective temperature of the planet. So, the effective surface temperature, <math>T_\mathrm{surface,eff}</math>, is, using the definition of <math>\Delta T_\mathrm{GHE}</math>,

:<math>T_\mathrm{surface,eff} = T_\mathrm{eff} + \Delta T_\mathrm{GHE} \;.</math>

One could also express the relationship between <math>T_\mathrm{surface,eff}</math> and <math>T_\mathrm{eff}</math> using {{mvar|''G''}} or {{mvar|''g̃''}}.

So, the principle that a larger greenhouse effect corresponds to a higher surface temperature, if everything else (i.e., the factors that determine <math>T_\mathrm{eff}</math>) is held fixed, is true as a matter of definition.

Note that the greenhouse effect influences the temperature of the planet as a whole, in tandem with the planet's tendency to move toward radiative equilibrium.<ref name="Modest2021" />