Let’s pick this up right where the airspeed corrections leave off — Page 109, Lesson 104

Let’s pick this up right where the airspeed corrections leave off. We’ve already dealt with instrument error and pressure error to get from Indicated Airspeed to Calibrated Airspeed. Now we’re at the point where we have CAS, and we need to turn it into True Airspeed. And this is where the density of the air comes in. First, a quick note on compressibility, because it frames everything we do next. The ASI is already calibrated to allow for compressibility at ISA at Mean Sea Level. Under those exact conditions, no compressibility correction is necessary. And here’s a handy rule: compressibility correction is small at True Airspeeds lower than 300 knots, so no correction is considered necessary below that. So the sequence is always the same. You calculate the Density Error correction first to get TAS. If that TAS is 300 knots or less, you’re done, no further correction. If it’s greater than 300 knots, then you apply the Compressibility Error correction. Let me show you the full chain of corrections, because it’s a neat summary. We start with Indicated Airspeed. We apply Instrument Error, then Pressure Error, which is also called Position Error, and that gives us Calibrated Airspeed, CAS. Now, from CAS, we split into two paths. At low speed, 300 knots or less, we apply Density Error and go straight to True Airspeed. At high speed, greater than 300 knots, we apply Density Error to get Equivalent Airspeed, EAS, then we apply Compressibility, and that gives us True Airspeed. So the density correction always comes first, and compressibility only kicks in above 300 knots. Now, how do we actually correct CAS to TAS for density? We divide by the square root of the relative density. You’ll recall this formula from Phase 1: CAS equals TAS times the square root of relative density. So to solve for TAS, we divide CAS by the square root of relative density. Relative to what? That’s the key question. It’s relative to ISA at Mean Sea Level, which is 1225 grams per cubic metre. Let me give you the standard ISA atmosphere table so you can see the values. At Flight Level 00, ISA temperature is 15 degrees Celsius, pressure is 1013.25 hectopascals, density is 1225 grams per cubic metre, and there are 27 feet to a hectopascal at that level. At FL50, temperature is 5 degrees, pressure 843 hectopascals, density 1056 grams per cubic metre, 32 feet per hectopascal. At FL100, minus 5 degrees, 697 hectopascals, 905 grams per cubic metre, 37 feet per hectopascal. At FL150, minus 15 degrees, 572 hectopascals, 771 grams per cubic metre, 43 feet per hectopascal. At FL200, minus 25 degrees, 466 hectopascals, 653 grams per cubic metre, 51 feet per hectopascal. And at FL250, minus 35 degrees, 376 hectopascals, 549 grams per cubic metre, 61 feet per hectopascal. Let’s work an example. Imagine we’re flying at 100 knots CAS at FL200 in an ISA atmosphere. The density at that level is 653 grams per cubic metre. To find relative density, we divide the density at our level by the density at MSL. So 653 divided by 1225 equals 0.5331. Then we take the square root of that, which is 0.7301. So with a CAS of 100 knots, TAS equals 100 divided by 0.7301, which is 137 knots. Now, you might think, well, if we had an instrument that measured air density directly, we could just do this calculation. But there is no flight deck instrument that gives such a reading. So how do we get density? We use the gas laws. From your Principles of Flight lectures, you’ll remember that P over T times r is a constant, or r is proportional to P over T, where P is pressure, T is temperature, and r is density. So if we know any two of pressure, density, and temperature, we can find the third. And from your Piston Engine lectures, you’ll remember the Combined Gas Law, which combines Boyle’s Law and Charles’ Law. It relates volume, pressure, and temperature. It can be shown as P times V over T equals K, or alternatively P times V equals K times T. And for two states, P1 times V1 over T1 equals P2 times V2 over T2. So that’s the theory behind it. We can’t measure density directly, but we can measure pressure and temperature, and from those we can derive density. That’s the foundation for the density error correction. And once we have TAS, we check if it’s above 300 knots to decide whether compressibility correction is needed.