On the average temperature of airless spherical bodies and the magnitude of Earth’s atmospheric thermal effect

The presence of atmosphere can appreciably warm a planet's surface above the temperature of an airless environment. Known as a natural Greenhouse Effect (GE), this near-surface Atmospheric Thermal Enhancement (ATE) as named herein is presently entirely attributed to the absorption of up-welling long-wave radiation by greenhouse gases. Often quoted as 33 K for Earth, GE is estimated as a difference between planet's observed mean surface temperature and an effective radiating temperature calculated from the globally averaged absorbed solar flux using the Stefan-Boltzmann (SB) radiation law. This approach equates a planet's average temperature in the absence of greenhouse gases or atmosphere to an effective emission temperature assuming ATE ≡ GE. The SB law is also routinely employed to estimating the mean temperatures of airless bodies. We demonstrate that this formula as applied to spherical objects is mathematically incorrect owing to Hölder's inequality between integrals and leads to biased results such as a significant underestimation of Earth's ATE. We derive a new expression for the mean physical temperature of airless bodies based on an analytic integration of the SB law over a sphere that accounts for effects of regolith heat storage and cosmic background radiation on nighttime temperatures. Upon verifying our model against Moon surface temperature data provided by the NASA Diviner Lunar Radiometer Experiment, we propose it as a new analytic standard for evaluating the thermal environment of airless bodies. Physical evidence is presented that Earth's ATE should be assessed against the temperature of an equivalent airless body such as the Moon rather than a hypothetical atmosphere devoid of greenhouse gases. Employing the new temperature formula we show that Earth's total ATE is ~90 K, not 33 K, and that ATE = GE + TE, where GE is the thermal effect of greenhouse gases, while TE > 15 K is a thermodynamic enhancement independent of the atmospheric infrared back radiation. It is concluded that the contribution of greenhouse gases to Earth's ATE defined as GE = ATE - TE might be greater than 33 K, but will remain uncertain until the strength of the hereto identified TE is fully quantified by future research.