High density altitude is called that because the low density air found in hot air mimics the air found at HIGH altitudes. At HIGH altitudes and HIGH temperatures, there is not only less air for an engine to "breathe", but there is less air for the wings to ride on as well.
High density altitude is also known as performance altitude. High density altitude caused by high temperatures and high humidity can occur at lower elevations but is more of a problem at airports at higher elevations such as those in California and the southwest. If you are based at an airport at 3,000 feet above sea level and the air temperature is hot (e.g. 95 degrees) your performance altitude is going to be higher. The higher the density altitude the longer runway you will need to take off at a given gross weight. This can be computed and G explains how this works:
The entire exercise of computing density altitude is used to predict aircraft performance on hot days or in high places or both. All the airplane manufactures publish charts that help you determine how much runway you'll need at a particular gross weight and a particular altitude. You add up the weights, figure the density altitude, (which tells you the altitude your airplane will act like its flying at), go to the performance charts and you find that at 8800 ft and full gross weight, your airplane needs, say, 6500 feet of runway to get airborne. If the airport you are parked at only has 4000 feet of runway, you're hosed. You either get rid of some weight, or wait until evening when the air cools.
What this means for air tanker pilots is that tankers working wildfires, e.g. in California or the southwest, may be forced to carry lower loads of retardant in order to compensate for higher density altitudes.
In my final article on density altitude, posting this Sunday, I will write a short piece on how density altitude affects helicopters.
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