There is a much stronger theoretical confirmation of the 255k. The greenhouse gas (ghg) effect can be said to be the
difference between the Earth-surface emission of longwave energy (IR) and the outgoing longwave radiation (OLR) at the top of the atmosphere (TOA)
In the long run heat only leaves the Earth-system as radiation to space. Incoming solar energy must equal outgoing longwave energy from the Earth to balance the temperature. If the incoming energy is larger than the outgoing energy the Earth will heat until it is hot enough to radiate more, to the point where the radiative balance is restored. Hence, the radiation from the Earth surface should equal incoming solar radiation and the outgoing longwave radiation at TOA. But it doesn’t, the Earth surface is much warmer. This is the most intuitive and fundamental description of the greenhouse effect.
The difference between the Earth surface OLR and OLR at TOA is the magnitude of the greenhouse effect. With an average ES temperature of 285k the average emission of infrared radiation from the surface is 333 watt/m2. But the OLR at TOA is measured to be 233 watt/m2. 233 watt/m2 approximately corresponds to the famous equilibrium temperature of 255k. Hence, the equilibrium temperature of the Earth is 255k and the difference from the actual Earth temperature is the greenhouse effect.
The OLR at TOA is measured by the Ceres satellite. According to their FAQ it’s measured in the stratosphere, 20 km above the surface of the Earth.
What is TOA? : ANSWER The TOA, Top-of-the-Atmosphere, is a surface approximately 20 km above the Earth surface. Specifically, the TOA is an ellipsoid x2/a2 + y2/a2 + z2/b2 = 1 ; where a = 6408.1370 km and b = 6386.6517 km
Unlike the troposphere the stratosphere has an increasing temperature gradient with height. That’s because ozone is the dominant gas, absorbing UV-light from the sun, whereas the troposphere mainly consist of water vapour and co2 absorbing heat from the Earth surface. Since ozone emits in the long range spectrum according to the SB-law (t**4) it is plausible that the OLR is higher at the stratopause then at 20 km.
Where should TOA be measured? In a system where multiple gases, clouds, and water vapour are entangled in the different layers of the atmosphere it’s hard to define the boundry of the system. The periphery must be a point where there are no more greenhouse gases re-radiating the longwave radiation from Earth as well as no infrared radiation from non-surface absorbers in the system.
In climate science they talk about the effective emissions height; the avarage height of the multiple layers where radiation comes from. One definition is as follows:
…the effective emission level is defined as the level at which the climatological annual mean tropospheric temperature is equal to the emission temperature: (OLR/σ)1/4, where σ is the Stefan–Boltzmann constant.
In other words, the effective emission height is not defined by the boundry of the system, but rather the level where the emission temperature is equal to the blackbody temperature of the system; the theoretical 255 k. A circular argument.
Is 255k a house number? I don’t know.