**Exponential Formula to Estimate
Airplane Emissions **

**Common Approach with Three Plateaus**

Most Carbon Calculators use three levels of CO_{2}e
per mile (or per kilometer), for "short," "medium," and
"long" Airplane flights.

They use the highest level of CO_{2}e 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 CO_{2}e 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 CO_{2}e per
passenger mile for medium flights, which are the most efficient.

The dropping and rising levels of CO_{2}e per
passenger mile create anomalies where a slightly longer flight is assigned much
less or much more CO_{2}e than a slightly shorter flight. These
anomalies show up in the comparative **Graphs** at the bottom of the "Air" tab of xls.CO2List.org

**Linear Formula**

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.

**Exponential Formula**

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.

**W _{m}** = Weight in last mile, including plane, load and final reserves
of fuel

**W _{m} * 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:

**W _{m-1}** =

Simplifying:

**W _{m-1}** =

In each previous period weight has to be yet bigger by the same
ratio:

**W _{m-2}** =

**...**

**W _{0}** =

Where **W _{0}** is starting weight

The total fuel burned in a flight is the starting weight minus
ending weight, or

**W _{0} - **

So we have this functional form:

CO_{2}e per passenger
during the whole flight = **A + ****B [**** ****C**^{MILES}** ****- 1 ]**

Where **A** is a constant per takeoff, while **B** and **C** relate to fuel
consumption during the flight

A _{2}e and ** A,**** ****B****, **and** C**. Full
derivation is on the "Airplane" tab of xls.CO2List.org with graphs comparing linear
and exponential estimates.

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.