Mach and TAS
We checked on with Seattle Center doing .74 at FL340. Above us at FL400 was another aircraft that we were gradually reeling in. When they checked on, Center asked “Airline 1456, say Mach number?” “Airline 456, we’re doing .75.” Perplexed at first, I scribbled some napkin math on the back of an old ATIS and found an answer for why we were faster despite the lower Mach number: it all had to do with temperature. Let’s go through the math of it to form a solid understanding, then come up with some simple rules for planning purposes.
The relationship between TAS and Mach is a function of temperature:
TAS=√(T/T0)*M*661, where TAS is in knots, T is temperature in Kelvin, and T0 is ISA standard temperature, 15° C, or 288.15K.
As we climb into colder air, the standard lapse rate is 2 degrees per thousand feet up to the tropopause. That means that T, the numerator in the fraction, keeps getting smaller, so TAS will get slower at a constant Mach number as we climb. To form some helpful rules, let’s start with a base scenario and look at how things change. Assume we’re doing .75 at FL350:
TAS=√(218.15/288.15) * .75 * 661 = 431.2 KTAS
Now if we go up 1000 feet, T will drop 2 degrees, and we end up with 429.2 KTAS, for a 2-knot change, or about half a percent change to your TAS. This relationship holds throughout most of the 30s until you hit the tropopause.
Approximation: you lose 2 knots or .5% TAS for every thousand feet with ISA lapse rates at constant Mach.
Approximation: you gain 1 knot for every degree warmer (memory aid: “Heat up” means “speed up”) at constant Mach.
Plotting a set of values for .73, .74, and .75 at various altitudes in the 30s shows more overlap than difference:
The relationship between TAS and Mach is a function of temperature:
TAS=√(T/T0)*M*661, where TAS is in knots, T is temperature in Kelvin, and T0 is ISA standard temperature, 15° C, or 288.15K.
As we climb into colder air, the standard lapse rate is 2 degrees per thousand feet up to the tropopause. That means that T, the numerator in the fraction, keeps getting smaller, so TAS will get slower at a constant Mach number as we climb. To form some helpful rules, let’s start with a base scenario and look at how things change. Assume we’re doing .75 at FL350:
TAS=√(218.15/288.15) * .75 * 661 = 431.2 KTAS
Now if we go up 1000 feet, T will drop 2 degrees, and we end up with 429.2 KTAS, for a 2-knot change, or about half a percent change to your TAS. This relationship holds throughout most of the 30s until you hit the tropopause.
Approximation: you lose 2 knots or .5% TAS for every thousand feet with ISA lapse rates at constant Mach.
Approximation: you gain 1 knot for every degree warmer (memory aid: “Heat up” means “speed up”) at constant Mach.
Plotting a set of values for .73, .74, and .75 at various altitudes in the 30s shows more overlap than difference:
Now if we want to hold TAS constant and adjust Mach, we can plug these approximations in to make changes. If we’re doing .75 and making .01 adjustments, then each point gives .01/.75=1.3% change, or 5.8 KTAS (let’s round up and call it 6 KTAS/.01M). Highlighting the two points on the above chart, or picking any other two, shows us:
Approximation: for the same TAS, adjust Mach .01 every 3,000 feet at cruise.
Going back to our .74 at FL340 scenario, we were going 3 knots faster than the plane doing .75 at FL400. In the end, we both got the Seattle Shuffle, but they were put in line ahead of us. So much for reeling them in that day.
Approximation: for the same TAS, adjust Mach .01 every 3,000 feet at cruise.
Going back to our .74 at FL340 scenario, we were going 3 knots faster than the plane doing .75 at FL400. In the end, we both got the Seattle Shuffle, but they were put in line ahead of us. So much for reeling them in that day.