CO2 emission budget for 1.5 °C increase

The Special Report on Global Warming of 2018 gives some numbers on the CO₂ emission budgets. One important number is 420 Gt of CO₂ (see, in German). It's the amount that we, the people of the earth, are allowed to emit to not break the 1.5 °C temperature increase by a 66% chance (in this post called TEB, total emission budget). The amount of CO₂ looks huge, but if you know that we emit around 35 Gt to 40 Gt a year it's not that much.

As I'm trying to get the facts on climate change a little more comprehensible I thought up some new ideas how to interpret the 420 Gt.

Like my post about ozone and CO₂ I like the idea to use the height of the CO₂ layer and its change caused by additional emissions. So how thick is the additional layer of CO₂ that is "allowed"? And how much mass per square meter is this? Or what area does one ton of CO₂ of that budget cover?

Here are the results:

$$h_{TEB}=\frac{m_{TEB}}{\rho_{CO_{2}}$$A_{E}}=\frac{42010^{12}~\mathrm{kg}}{1.977~\mathrm{kg/m^3}51007500010^{6}~\mathrm{m^{2}}}=0.4165~\mathrm{m}

This doesn't sound like a lot! One calculation that I found even more interesting is the CO₂ per square meter:

$$\frac{m_{TEB}}{A_{E}}=\frac{420$$10^{12}~\mathrm{kg}}{51007500010^{6}~\mathrm{m^{2}}}=0.8234~\mathrm{kg/m^{2}}=823~\mathrm{g/m^{2}}

That is an impressive number! Basically by emitting more than 823 g of CO₂ you get 1 square meter of the earths surface breaking the 1.5 °C barrier! To get an idea of what 823 g means: A modern car emits around 100 g to 150 g per kilometer!

The last calculation is the area one ton of CO₂ will push to the 1.5 °C increase of temperature.

$$\frac{A_{E}}{m_{TEB}}=\frac{510075000$$10^{6}~\mathrm{m^{2}}}{42010^{9}~\mathrm{t}}=1214.46~\mathrm{m^{2}/t}\approx 34.85^{2}~\mathrm{m^{2}/t}

In Germany the average emission of CO₂ per person and year is around 9 tons (source). That means 10930.14 square meter (roughly (100 m)^2) reach the 1.5 °C increase per year and person.