Exponential Formula to Estimate Airplane Emissions
Common Approach with Three Plateaus
Most Carbon Calculators use three levels of CO2e per mile (or per kilometer), for "short," "medium," and "long" Airplane flights.
They use the highest level of CO2e per passenger mile for short flights, since takeoff burns a lot of fuel and is a bigger fraction of short flights. However this approach under-estimates emissions of the shortest flights in each level, where takeoffs are a bigger fraction than average for the level.
They use a medium level of CO2e per passenger mile for long flights, since long flights have to carry a huge amount of fuel at the beginning of the flight, so they burn extra fuel to carry the extra fuel. This approach under-estimates emissions of the longest flights, where carrying fuel is an even bigger burden.
They use the lowest level of CO2e per passenger mile for medium flights, which are the most efficient.
The dropping and rising levels of CO2e per passenger mile create anomalies where a slightly longer flight is assigned much less or much more CO2e than a slightly shorter flight. These anomalies show up in the comparative Graphs at the bottom of the "Air" tab of xls.CO2List.org
A linear equation with a constant per takeoff can give a very good approximation, especially for the large majority of short and medium flights. However it slightly over-estimates medium flights and/or under-estimates the longest.
Airplanes use above average fuel per mile for takeoff and climb, while they use below average fuel per mile for descent and landing. The constant per takeoff actually represents the net difference for the flight as a whole, plus taxiing.
An exponential equation has the best theoretical basis, since it reflects rising fuel burdens of longer flights, and it can also have a constant per takeoff. The weight of the airplane continuously drops as fuel is burned, so less fuel is needed each mile.
Wm = Weight in last mile, including plane, load and final reserves of fuel
Wm * R = Weight of fuel burned in last mile "m"
Where R is a constant representing the ratio of the weight of fuel being burned to total weight being flown at that moment
The weight of fuel for that last mile has to be carried to that point, so weight in the preceding period is bigger by that amount:
Wm-1 = Wm + ( Wm * R )
Wm-1 = Wm * ( 1 + R )
In each previous period weight has to be yet bigger by the same ratio:
Wm-2 = Wm * ( 1 + R )2
W0 = Wm * ( 1 + R )m
Where W0 is starting weight
The total fuel burned in a flight is the starting weight minus ending weight, or
W0 - Wm = Wm * [ ( 1 + R )m -1 ]
So we have this functional form:
CO2e per passenger during the whole flight = A + B [ CMILES - 1 ]
Where A is a constant per takeoff, while B and C relate to fuel consumption during the flight
More data would allow more confidence in the coefficients, but this estimate is well-behaved, rising noticeably for very short and long flights. From about 200 to 5,000 miles (300-8,000 km), the exponential and linear formulas are close. The exponential is higher at both ends.
The derivation lets A, B, and C include other effects not discussed here. For example A includes all effects which are independent of flight length, such as airport usage and some probability of weather diversions at the destination. B and C include all effects related to flight length, such as maintenance and size of plane.