Over at the Hockey Schtick, Michael has an interesting new angle on determining planetary surface temperature from pressure and gravity – a subject covered extensively here at the Talkshop over the last three years. Here’s an extract. Of particular interest here is his new method of using the centre of mass of the atmosphere as a reference point. Head on over to read the full post.
Step 2: Determine the height at the center of mass of the atmosphere
We are determining the temperature gradient within the mass of the atmosphere and the equilibrium temperature is thus at the center of mass. The “effective radiating level” or ERL of planetary atmospheres is located at the approximate center of mass of the atmosphere where the temperature is equal to the equilibrium temperature with the Sun. The equilibrium temperature of Earth with the Sun is commonly assumed to be 255K or -18C as calculated here. As a rough approximation, this height is where the pressure is ~50% of the surface pressure. It is also located at the approximate half-point of the troposphere temperature profile set by the adiabatic lapse rate, since to conserve energy in the troposphere, the increase in temperature from the ERL to the surface is offset by the temperature decrease from the ERL to the tropopause.
Fig 1. From Robinson & Catling, Nature, 2014 with added notations in red showing at the center of mass of Earth’s atmosphere at ~0.5 bar the temperature is ~255K, which is equal to the equilibrium temperature with the Sun. Robinson & Catling also demonstrated that the height of the tropopause is at 0.1 bar for all the planets in our solar system with thick atmospheres, as also shown by this figure, and that convection dominates over radiative-convective equilibrium in the troposphere to produce the troposphere lapse rates of each of these planets as shown above. R&C also show the lapse rates of each of these planets are remarkably similar despite very large differences in greenhouse gas composition and equilibrium temperatures with the Sun, once again proving pressure, not radiative forcing from greenhouse gases, determines tropospheric temperatures.
Step 3: Determine the surface temperature
For Earth, surface pressure is 1 bar, so the ERL is located where the pressure ~0.5 bar, which is near the middle of the ~10 km high troposphere at ~5km. The average lapse rate on Earth is 6.5 km, intermediate between the 10C/km dry adiabatic lapse rate and the 5C/km wet adiabatic lapse rate, since the atmosphere on average is intermediate between dry and saturated with water vapor.
Plugging the average 6.5C/km lapse rate and 5km height of the ERL into our equation (6) above gives
T = -18 – (6.5 × (h – 5))
Using this equation we can perfectly reproduce the temperature at any height in the troposphere as shown in Fig 1. At the surface, h = 0, thus temperature at the surface Ts is calculated as
Ts = -18 – (6.5 × (0 – 5))
Ts = -18 + 32.5
Ts = 14.5°C or 288°K
which is the same as determined by satellite observations and is ~33C above the equilibrium temperature with the Sun.
Thus, we have determined the entire 33C greenhouse effect, the surface temperature, and the temperature of the troposphere at any height, entirely on the basis of the 1st law of thermodynamics and ideal gas law, without use of radiative forcing from greenhouse gases, nor the concentrations of greenhouse gases, nor the emission/absorption spectra of greenhouse gases at any point in this derivation, demonstrating that the entire 33C greenhouse effect is dependent upon atmospheric mass/pressure/gravity, rather than radiative forcing from greenhouse gases.