I have mentioned Black Bodies, especially the Black Body Cavity concept a few times. Most will calculate the Black body temperature based on TSI(1-albedo)/4, that assumes a fixed albedo which I think is the incorrect way of doing business. The Black Body temperature should be TSI/4 which agrees with the Planck response. A simple way of looking at Planck response is TSI/T(s) which is basically how sensitive the surface temperature T(s) is to a change in energy. We are after all trying to determine climate sensitivity, why not start with sensitivity aka Planck response.
Since sensitivity is in units K/Wm-2 or Wm-2/K, for a 1 degree change in absolute temperature there would be a corresponding change in energy based on the Stefan-Boltzmann equation. So sensitivity for a surface at T degrees would be, T(s)/(5.67e-8*((T(s)+.05)^-(T(s)-0.5)^4) which increases exponentially with T(s). TSI/T(s) decreases exponentially with T(s), so you have a nice intersection if you want to solve it graphically.
Imbalances versus TOA post I used a simple model that has a BB efficiency factor as a variable. Since temperature and energy are related by T^4, that factor is (0.926)^.25=0.98 in case you were wondering.
The chart above is for 1400 Wm-2 TSI, but for Earth a TSI of 1361Wm-2 is currently in vogue though 1365-6 is still used quite often. Using 1361.1 Wm-2 for TSI, the Planck response T(s) would be 278.3K ignoring the BB fudge factor and 283.7 with the full fudge factor. Both are a long way from 255K based on assumed albedo and simplistic flat Earth geometry.
If you look around you may find that the Planck response of Earth is supposed to be 3.3Wm-2/K. That is based on ~254K degrees as the effective radiant layer temperature. With 3.3Wm-2/K and a forcing of 3.7Wm-2 you would have the no feedback climate sensitivity of ~1.12 C degrees. No feedback sensitivity temperature response has bounced around from a low of 0.8 C to 1.5C degrees.
Based on the generic Planck response, it should be 3.7/~5.0= 0.74 C degrees per 3.7 Wm-2 of forcing.
By comparing the 41km Stratosphere absolute temperature as estimated by the AQUA satellite data with the TOA seasonal TSI variation as determine by the Total Irrandiance Monitor data provided by SORCE, the average Planck response is ~5.4Wm-2 per K with a range of roungly +/-0.25 Wm-2 per K degrees.
Radiant physics provides plenty of references, but none in the middle of a black body cavity which does not depend on the geometry of the cavity.
If you care to try this yourself, the AQUA data is at Discover AMSU Temperatures in daily format and the TIM data is at the lasp SOlar Radiation & Climate Experiment (Sorce) website. One note you might wan t to make is that the Stratospause temperature at ~49 km is ~273K or zero C due to water vapor and that with ~88 Wm-2 of latent heat providing that ~315Wm-2 at ~0 C degrees, the energy is equivalent to a surface at 315+88= 403 Wm-2 or roughly 290 K degrees, right in the range estimated with the AQUA and TIM data.
This is beginning to look like there are serious model problems.