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Service ceiling is where the rate of climb drops below a prescribed value.
The service ceiling is the maximum usable altitude of an aircraft. Specifically, it is the density altitude at which flying in a clean configuration, at the best rate of climb airspeed for that altitude and with all engines operating and producing maximum continuous power, will produce a given rate of climb (a typical value might be 100 feet per minute climb or 30 metres per minute, or on the order of 500 feet per minute climb for jet aircraft). Margin to stall at service ceiling is 1.5 g.
The one engine inoperative (OEI) service ceiling of a twin-engine, fixed-wing aircraft is the density altitude at which flying in a clean configuration, at the best rate of climb airspeed for that altitude with one engine producing maximum continuous power and the other engine shut down and feathered, will produce a given rate of climb (usually 50 feet per minute).
However some performance charts will define the service ceiling as the pressure altitude at which the aircraft will have the capability of climbing at 50 ft/min with one propeller feathered.
Absolute ceiling is : the height at which rate of climb drops to zero / when the aircraft can climb no higher
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The absolute ceiling is the highest altitude at which an aircraft can sustain level flight. Due to the thin air at higher altitudes, a much higher airspeed is required to generate sufficient lift on the wings. The absolute ceiling is therefore the altitude at which the engines are operating at maximum thrust, yet only generate enough lift to match the weight of the aircraft. Hence, the aircraft will not have any excess capacity to climb further. At absolute ceiling, the aircraft can no longer accelerate, since any acceleration will lead to higher airspeed and therefore excess lift. Stated technically, it is the altitude where the maximum sustained (with no decreasing airspeed) rate of climb is zero.
Most commercial jetliners have a service (or certificated) ceiling of about 42,000 ft (13,000 m) and some business jets about 51,000 ft (16,000 m). Before its retirement, the Concorde Supersonic transport (SST) routinely flew at 60,000 ft (18,000 m). While these aircraft's absolute ceiling is much higher than standard operational purposes—in Concorde's case, it was tested to 68,000 ft (21,000 m)—it is impossible to reach for most (because of the vertical speed asymptotically approaching zero) without afterburners or other devices temporarily increasing thrust. Another factor that makes it impossible for some aircraft to reach their absolute ceiling, even with temporary increases in thrust, is the aircraft reaching the "coffin corner." Flight at the absolute ceiling is also not economically advantageous due to the low indicated airspeed which can be sustained: although the true airspeed (TAS) at an altitude is typically greater than indicated airspeed (IAS), the difference is not enough to compensate for the fact that IAS at which minimum drag is achieved is usually low, so a flight at an absolute ceiling altitude results in a low TAS as well, and hence in a high fuel burn rate per distance traveled. The absolute ceiling varies with the air temperature and, overall, the aircraft weight (usually calculated at MTOW).