At constant CAS, high altitude means lower air density and therefore higher TAS: this decreases the aerodynamic damping during a rotation.

For this reason, I cannot understand why directional stability should decrease with altitude, as stated in this chapter:

From Principles of Flight CAE books

Shouldn't the restoring (yawing) moment resulting from a sideslip be bigger when the aerodynamic damping is less effective?


Guys, thank you so much for your answers. I should have provided some more background about this subject. I am an ATPL student, and I came across this question while preparing for my Principles of flight exam:

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The only informations I could find on the book are the chapter I posted above and this:

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I could not find any reference on the book -and on EASA learning objectives for the ATPL theory- about the Mach-related effects on directional stability.

The chapter above, as you said, is quite confusing. The Cn-B graph shows how strong the initial tendency of the nose to yaw towards the relative wind is, "closing" the angle of sideslip: the steeper the slope of the curve, the quicker the tendency of the A/C to re-align with the airflow. If I understand correctly, the graph doesn't tell anything about the dynamic behaviour of the A/C, it's all about static (directional) stability.

So, I cannot really understand why aerodynamic damping (or some other effect related to an increase of altitude, other than compressibility) should flatten the slope as the book -and the question- seem to suggest.

This is a super fascinating topic -as everything related to Principles of flight- so I would love to have a deeper understanding of it, not just because I have to pass the exams.

By the way, if you have any book/source to suggest it would be amazing.

  • $\begingroup$ The aerodynamic damping link is dependent on the aerodynamic forces. What happens to the magnitude of aerodynamic forces with increasing altitude? $\endgroup$
    – ROIMaison
    Commented Feb 15, 2022 at 14:29
  • $\begingroup$ @ROIMaison The aerodynamic damping is dependent on the rotation (tangential) velocity about an axis and inversely proportional to the True Airspeed of the aircraft. At constant CAS (constant dynamic pressure), the magnitude of aerodynamic forces doesn't change for a given AoA, am I right? $\endgroup$ Commented Feb 15, 2022 at 14:36
  • $\begingroup$ The way I read it, the indicated section talks about high altitude (= low-density) flight, and notes that aerodynamic damping is low because the aerodynamic forces on the rudder are low. The text doesn't mention constant CAS. $\endgroup$
    – ROIMaison
    Commented Feb 15, 2022 at 16:20
  • 2
    $\begingroup$ Aren't we talking about dynamic stability here? Static margin is unchanged, as the vertical aerodynamic center is at the same spot relative to the CG, so static stability should be the same. The effect is on dynamic behaviour. The reduced damping is mostly because restorative forces decline with air density while inertial forces remain the same. $\endgroup$
    – John K
    Commented Feb 15, 2022 at 16:58
  • $\begingroup$ @JohnK Uuuhh - that (restorative forces decline with air density) from you! What declines are the motion-induced forces because TAS grows with altitude. Restoring forces stay constant because (as you correctly say) the vertical aerodynamic center is at the same spot relative to the CG and dynamic pressure stays constant. $\endgroup$ Commented Feb 17, 2022 at 7:20

4 Answers 4


I don't think the person who wrote the question in attachment IC-081-050 understands the topic of lateral stability well enough to be qualified to judge others on it. Also, the text you quote jumps between static and dynamic stability without making clear each time what is meant with the term "stability".

Definitions first: Static stability concerns the forces and moments that oppose a change of state. Dynamic stability concerns the forces and moments that oppose a movement. In both cases higher stability means higher forces and moments. The difference is in their cause.

From now on, I focus on lateral stability only. Static stability first: A typical change of state is flight into a lateral gust: Suddenly, all of the aircraft flies at a changed sideslip angle. If we neglect niceties like that the tip of the fuselage encounters the gust before the tail does, the effects are:

  • Destabilizing c$_{n\beta}$ contribution of the fuselage which is independent of Mach or angle of attack since the center of pressure of the fuselage is well ahead of the center of gravity.
  • Stabilizing c$_{n\beta}$ contribution of the wing which is proportional with aspect ratio, sweep angle and dihedral and grows with the square of the lift coefficient.
  • Strongly stabilizing c$_{n\beta}$ contribution of the vertical tail which initially is constant but deteriorates with higher sideslip when the tail reaches and exceeds its maximum side force and stalls like a wing. This contribution grows slightly with an increase in the subsonic Mach number but drops significantly with higher dynamic pressure due to elastic deformation of the tail and the fuselage it is attached to.

Next, what is meant by higher altitude? Your first sentence tells us it is flight at higher TAS but the same CAS and, hence, the same dynamic pressure and lift coefficient but a higher Mach number. c$_{n\beta}$ should increase ever so slightly from the higher side force slope on the tail but due to the higher TAS, any excursions from a straight flight path should grow proportionally to it. I guess that those very noticeable higher excursions are the reason for the author of attachment IC-081-050 to think that c$_{n\beta}$ shrinks with altitude, but that is wrong.

Flying faster at the same altitude will significantly decrease c$_{n\beta}$ due to elastic deformation, but if CAS is kept constant, this effect will not be present. It could also be that the author of attachment IC-081-050 confuses the effect of higher dynamic pressure on c$_{n\beta}$ with that of higher TAS at higher altitude, but that is equally wrong.

A third reason for the wrong impression that c$_{n\beta}$ changes with altitude is that due to the higher flight speed the same gust will cause a smaller change in sideslip angle at higher altitude. But the coefficient is not c$_{n\;gust\:speed}$, but c$_{n\beta}$, so for being different, a different yawing moment change must be caused by the same change in sideslip angle at different altitudes, which is wrong.

Now on to dynamic stability: A typical movement is expressed as the yaw rate $r$. Dynamic yaw stability is determined by yaw damping. As you correctly say it decreases with higher TAS and in many jets this requires the use of a yaw damper for flights at higher altitude. Because dynamic stability has its own coefficient (c$_{nr}$), this has nothing to do with the slope of c$_{n}$ over $\beta$. For dynamic stability the yawing moment coefficient must be plotted over yaw rate which is clearly not the case in attachment IC-081-050. Of course, it is also possible that the author has confused static and dynamic stability and attributed the altitude dependency of the latter to the former. But this would be a really bad mistake to make.

Also, the paragraph highlighted in yellow from your text is, while not outright wrong, at least misleading. Only if you interpret the slope the author talks of here as the slope of c$_{n}$ over $r$ will I agree with all he or she says. The context, however, implies he or she means the slope of c$_{n}$ over $\beta$ which would be wrong.

  • $\begingroup$ Does the X-15 wedge tail have anything to do with this? In very thin air at very high speeds, does the shock wave actually blank the stabilizer behind it? Will the effect (on damping) be noticeable at high subsonic. (Is choice 3 the right answer?) $\endgroup$ Commented Feb 16, 2022 at 11:46
  • $\begingroup$ @RobertDiGiovanni The wedge was chosen because it would create more drag to fair the blunt trailing "edge" than to cut off the tail at the thickest point. But that is really only helpful in hypersonic flight and makes very little difference at subsonic speed. The lever arm of the drag force is small (tail length times sin($\beta$)). It should not be the reason for a change of c$_{n\beta}$ with density. $\endgroup$ Commented Feb 16, 2022 at 15:23
  • $\begingroup$ @PeterKämpf thank you very much for the detailed answer. I believe that the author of the question simply wants us student pilots to state that dynamic stability decreases with higher TAS, but posted the wrong graph. We are required to comply with EASA so called "learning objectives", and one of them (posted in the picture above) is about the effects of altitude on dynamic stability. I don't know why they provided the cnβ graph attached to the question when they are clearly asking about dynamic stability. Thank you again for your help! $\endgroup$ Commented Feb 17, 2022 at 13:08
  • 1
    $\begingroup$ @Andrea, I had great difficulty answering my pilot theory exam (while having PhD on the subject) exactly because of such issues. Every second question (or offered answer) was wrong in one way or another. One needs to translate the question to 'what they actually wanted to say'. And what they (pilots/instructors) know is that 'flying is harder at high altitudes'. Why? Obviously, 'stability' is lower! Now let's find a graph about stability... $\endgroup$
    – Zeus
    Commented Feb 18, 2022 at 0:57
  • $\begingroup$ @AndreaAppiani I have long lost all hope for adequate understanding in officialdom of the topics they administer. I work on an IoT thingy with a small Li-Ion battery and am required to lie to my customers, because I must state that tampering with the battery will cause fire (it will not, not enough energy) and is prohibited. At the same time I must give detailed instructions how the battery can be removed for safe disposal. Go figure! – $\endgroup$ Commented Feb 18, 2022 at 5:47

We've got a rather complex "can of worms" here. Some of the quoted material seems to confuse cause with effect, or to confuse static effects with dynamic effects.

Shouldn't the restoring (yawing) moment resulting from a sideslip be bigger when the aerodynamic damping is less effective?

It seems like you are imagining that with less damping acting in opposition to yaw rotation, the restoring moment from sideslip will somehow be able to act more quickly or more effectively to introduce a yaw rotation to end the sideslip.

That's not a correct way to view the situation.

To understand why, you have to realize that the root cause of aerodynamic damping of yaw rotation is that the yaw rotation automatically creates a difference in sideslip angle between nose and tail. The whole fuselage can't all be experiencing zero sideslip. If the tail (vertical fin) "weathervanes" into alignment with the local airflow, the nose will be experiencing a sideways flow that generates a yaw torque that opposes the yaw rotation. Similarly, if the rudder is used to keep the nose aligned with the local airflow, the vertical fin will be experiencing a sideways flow that generates a yaw torque that opposes the yaw rotation. And note that the steeper the positive slope of the curve shown in the textbook, the stronger this opposing yaw torque will be.

At high altitude and high TAS the aerodynamic damping on the rudder is less effective, so directional stability decreases shown by a reduced positive slope.

That's not really a correct way to view the situation either. A reduction in directional stability ("reduced positive slope") really can't be explained as a result of "damping". A reduction in directional stability ("reduced positive slope") could be the result of effects related to high Mach, but it's not completely clear that this is in fact what the quoted text is trying to describe.

The truth is that whenever we talk about "damping", we need to specify exactly what mean to say is being "damped". Generally we are talking about a dynamic effect--something related to the rate of something else. So is the quoted passage trying to address aerodynamic damping of the rotation rate in yaw, pitch, or roll? Or is it trying to address damping of an oscillation, such as the pitch phugoid, or such as the "Dutch roll" oscillation? Or are we actually talking about a reduction in the "weathervane" yaw torque per degree of sideslip (i.e. the static directional stability) for a given IAS-- which would not be a conventional usage of the term "damping"? The quoted passage fails to clarify this.

It's true that some oscillation tendencies tend to be most pronounced at high altitude, for a given CAS or IAS, but the quoted passage hasn't adequately explained why this should be so.

Increased static directional stability would be expected to cause increased damping of yaw rotation rate. It's confusing to suggest that increased damping of yaw rotation rate causes increased static directional stability, and decreased damping of yaw rotation rate causes decreased static directional stability, as the quoted passage seems to do.

There are at least two different reasons why an aircraft tends to be more prone to various dynamic oscillations at high altitude (and therefore high TAS), for a given IAS--

One, strong damping of yaw or pitch rotations tends to reduce "Dutch roll" or pitch "phugoid" oscillations, respectively. And for a given IAS and G-load, a higher TAS will be associated with a larger radius (smaller rate) of curvature of the flight path, so the stabilizing effect of yaw or pitch damping will play less of a role in the aircraft's dynamics.

And two, at high Mach numbers the slope of the static directional (yaw) stability curve is indeed reduced.

  • $\begingroup$ Thank you very much, in the meantime I edited the question to add some more background. $\endgroup$ Commented Feb 15, 2022 at 20:46
  • $\begingroup$ at high Mach numbers the slope of the static directional stability curve is indeed reduced - proof, please! With Mach the sideforce-over-sideslip curve becomes a bit steeper, making the tail more effective. What leads to the reduction at higher Mach is elastic deformation from higher dynamic pressure. Please don't confuse the two! $\endgroup$ Commented Feb 17, 2022 at 7:24
  • $\begingroup$ @PeterKampf -- looking into it. So, you are essentially saying that the only reason supersonic fighters tend to feature much larger vertical tails than their 1940's piston-driven counterparts, is because of elastic deformation? $\endgroup$ Commented Feb 18, 2022 at 15:53
  • $\begingroup$ @quietflyer No, the question is about subsonic flight and the supersonic airplanes need the larger tails because of thinner air on the upper side and the reduced supersonic lift curve slope - all purely supersonic effects. Elastic deformation has its share but is not alone. $\endgroup$ Commented Feb 19, 2022 at 6:25

After much thought and credit to other responders ...

why does static directional stability decrease with altitude?

It will not, if aircraft stays below critical Mach number, although increased stagnation pressure on the forward fuselage relative to other surfaces should be considered (even 300 knots is pretty fast!)

damping on the rudder is less effective, so directional stability decreases

Yes, but not directly. Damping has nothing to do with directional stability. In the simplest sense, imagine a gold pendulum under water, at sea level, and on top of Mt. Everest. Viscous drag limits the overshoot of each swing by limiting the velocity, therefor angular momentum of each swing. All three pendulums eventually stop.

The correcting force (gravity) is the same in each case if each pendulum starts from the same place.

With swept wing large airplanes, like airliners, the lack of damping manifests into instability because, at a certain degree of Velocity and deflection, overshoot becomes self-reinforcing and the yaw-roll cycle gets worse and worse.

At altitude, lift requirement is the same, therefor the lift differential (of the swept wing) for a certain degree of yaw is the same.

Because of lower damping, the velocity of the roll is faster. This couples with the yaw created by drag differential (of the wings).

The sideslip adds fuel to the fire with each yaw-roll cycle.

  • $\begingroup$ @Andrea Appiani perhaps the highlighted portion is only saying that stability is lost because with less damping (drag against motion), the aircraft will yaw more for a given input. The key words may be sideslipping to the right, which indicates greater spiral instability. I have seen much confusion here because (especially with swept wings) of dihedral effect, a yaw to the right produces a skid to the left, which rolls the plane to the right. So we see a loss of damping at higher altitudes in roll as well. Appreciate (as an ATP) you are trying to understand all this a bit better. $\endgroup$ Commented Feb 18, 2022 at 7:05

Shouldn't the restoring (yawing) moment resulting from a sideslip be bigger when the aerodynamic damping is less effective?

No - damping is a force proportional to velocity, in this case yaw rate $\dot{\beta}$. The restoring moment is a function of yaw angle $\beta$, which will be smaller at lower air density.

Once the aeroplane starts responding to the restoring moment by moving back towards neutral, the damping starts playing a roll, with lower damping generating faster return to neutral and more overshoots.

At high altitude and high TAS the aerodynamic damping on the rudder is less effective, so directional stability decreases shown by a reduced positive slope. This can lead to an increased tendency to Dutch roll with increasing altitude.

Reduced damping:

  • cannot cause a reduced positive slope, since one is a function of yaw rate, the other a function of yaw angle.
  • does cause increased tendency to Dutch roll with increasing altitude.

The dynamic pressure is eliminated from the moment coefficient $C_{n \beta}$ and the argument that reduced dynamic pressure causes reduction of the coefficient slope does not make sense.

  • $\begingroup$ The coefficient is c$_{n\beta}$, so the sideslip angle is the basis, not gust strength! $\endgroup$ Commented Feb 16, 2022 at 15:28
  • $\begingroup$ @PeterKämpf With a constant $C_{n \beta}$, dynamic pressure reduces at increasing altitude. All aerodynamic entities change, mass does not, yet you state that dynamic pressure is of no interest, only the dimensionless coefficient? $\endgroup$
    – Koyovis
    Commented Feb 17, 2022 at 2:24
  • $\begingroup$ Reduced density! Dynamic pressure stays the same, hence same IAS but higher TAS at altitude. $\endgroup$ Commented Feb 18, 2022 at 11:21

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