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Why do planes need air start?
Short answer
Most jet engines have no electric starter: the compressor must be spun to around 20-25% of rated speed before fuel is lit and the turbine can sustain itself. Airliners do this with an air turbine starter fed by the APU, a running engine or a ground air start unit; only a few newer designs, such as the Boeing 787, start electrically.

Starting a turbine is not like starting a car
A piston engine can be turned over by an electric motor and will run from the first firing stroke. A gas turbine cannot. Its combustion chamber only works when there is a steady flow of compressed air passing through it, and that air comes from the compressor, which is driven by the turbine, which needs the combustion to be running. The cycle has to be spun up from outside until it can sustain itself. On a large turbofan that means rotating the high-pressure spool to somewhere around 20 to 25 percent of its rated speed before fuel and ignition are introduced, and continuing to assist it until the engine reaches idle. Spinning a compressor of that size fast enough demands a great deal of power in a short burst, more than a conventional electric starter of reasonable weight could deliver on aircraft designed in the second half of the twentieth century.
Where the air comes from
The standard answer is a pneumatic starter, a small air turbine on the engine's accessory gearbox. It needs compressed air at roughly 3 bar, and the aircraft has three ways to get it. The usual one is the auxiliary power unit, a small turbine in the tail that also runs the air conditioning and electrics on the ground. The second is cross-bleed: once one engine is running, air from its compressor can be routed through the pneumatic ducting to start the next. The third is external, the ground air start unit, a trailer-mounted compressor plugged into a socket on the belly of the aircraft. Ground carts are what keep a flight going when the APU is unserviceable, and they are why an aircraft with a failed APU can still depart from a well-equipped airport but not from a remote one.
The electric alternative
Smaller turbines have long used electric starters, and the Boeing 787 extended that principle to a wide-body. Its engines have no bleed-air system at all; large starter-generators on each engine act as motors to spin the core, then switch to generating electricity once it runs. Airbus retained pneumatic starting on the A350, but the wider trend toward more-electric aircraft is likely to continue as power electronics get lighter.
Air start in the other sense
Pilots also use the term for relighting an engine that has flamed out in flight. In that case the airflow through the engine keeps the core windmilling, and if the aircraft is within the relight envelope of speed and altitude the crew simply reintroduces fuel and ignition; below that windmilling speed the starter is engaged, fed by the APU or by cross-bleed from a running engine.
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