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Author Topic: Do planes have to adjust their height due to the curvature of the Earth?  (Read 3409 times)

Offline chris

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Does a plane need to compensate to maintain a steady altitude while flying, or will it naturally follow the curvature of the Earth?


 
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Offline alancalverd

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Since altitude means distance above sea level, and the distance is measured radially from the centre of the earth (i.e. "straight up"), according to the physics textbooks a perfectly trimmed aircraft in a stable, homogenous atmosphere will maintain constant altitude without further intervention.

BUT (and it is a heck of a big but!)

none of the required conditions ever applies in practice!

The air is moving in 3 dimensions all the time, air density varies from place to place, gliders and balloons cannot fly "straight and level", and any other aircraft gets lighter and changes its angle of incidence as it burns fuel.

Additionally whilst we can in principle use GPS to maintain a true constant altitude (precision GPS systems  actually take account of the local nonsphericity of the planet) the mandatory reference  for all aircraft in flight is "pressure altitude". This varies at low level according to "QNH" - the promulgated lowest forecast pressure (corrected to sea level)  in the region you are flying in, and above a "transition altitude" (which varies from place to place: can be a low as 3000 ft or as high as 12,000 ft) all aircraft fly at "flight levels", 100 foot increments based on a notional standardised sea level pressure of 101.32 hPa and a standard barometric altimeter.   

Given all these complications, the business of maintaining vertical position (according to the direction of flight) and avoiding lightning, turbulence, icing, fog, volcanic ash.....involves so much short-term tweaking of controls that nobody other than astronauts notices that we are actually following the curvature of the earth.

It's interesting to travel in the back of a big plane like a 380. The autopilot maintains itself at 40,000 ft (actually FL400) within 1 foot, but in order to do so, the tailplane has to thrash up and down like a porpoise. 

And just to make it complicated, "height" means vertical distance from the actual surface, which inconveniently can be anywhere from 1000 ft below to 29,000 ft above mean sea level.
 
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Offline jeffreyH

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So it's quite a simple procedure then.
 

Offline agyejy

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Does a plane need to compensate to maintain a steady altitude while flying, or will it naturally follow the curvature of the Earth?

There is actually a relatively simple way to think about this. If we take "a steady altitude" to mean that the plane is not accelerating in the up and down direction that means the force of gravity on the plane must be balanced by the lift (and a little bit by a centrifugal correction). Now forces are vector quantities so they must match in magnitude and direction in order to properly cancel. The force due to gravity always acts radially toward the center of the Earth (barring minor corrections for topology and density) and so the lift of the plane must always act radially away from the center of the Earth in order to maintain the "steady altitude" condition as we've described it. The only way for the plane to make sure that its lift force always points in the right direction is by following the curve of the Earth. So in a sense you could say that yes a plane will naturally follow the curvature of the Earth as long as the pilots want to keep the instantaneous vertical acceleration at or near zero.
 

Offline chris

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Thanks for these perspectives.

I would anticipate that if an aircraft flies far enough in a straight line, owing to the curvature of the Earth, it will gain altitude. The ultimate gain in altitude will be limited by the density of the air which will restrict the rate of lift and also the ability of the engines to function.

 
 

Offline alancalverd

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So it's quite a simple procedure then.

Fortunately, yes. I've just demonstrated it to a prospective pilot. The trick was "if the number on the altimeter increases, lower the nose. If it decreases, raise the nose. Try to keep it at 3,000 ft." And to nobody's  surprise, we followed the curvature of the earth for 200 miles. Which is just as well as otherwise we would have landed an embarrassing 133 feet above the runway (curvature = 8 inches per mile - you can see it on a long runway!).
 

Offline chris

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Someone sent me this answer by email:

"This is to do with centripetal force, and gravity. It's same as if you have a tennis ball on a string and then swing the ball, if it doesn't have thrust, eventually it will stop swinging. However if you keep spinning it, it will swing the length of the string. It's moving in a circle, but doesn't fly off because the string is holding it on this arc. A plane is tethered to the earth by gravity, to resist it requires huge force to propel it so that the wings create enough pressure to lift the plane, as you know. However once it reaches cruising altitude it throttles back and that length of string has been determined. To lengthen the string would then require more thrust, which doesn't happen, therefore the plane stays at the same height, regardless of earth's curvature."

Discuss...
 

Offline alancalverd

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therefore the plane stays at the same height, regardless of earth's curvature.


A tautology, or a nonsense, depending on what you mean by "regardless"! Since height is the distance above the surface, and the surface is curved, the plane must follow a curved path in order to remain at the same height. But what do we mean by "curved"? In terms of the gravitational potential, the plane is maintaining a constant contour, and having no reference other than the surface of the earth, it has no concept of curvature.
 

Offline Colin2B

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Someone sent me this answer by email:

"This is to do with centripetal force, and gravity.
Discuss...
Alan will correct me if I'm wrong, but I don't see this as a balance between centripetal force and gravity. If the plane flies faster it does not fly higher except by additional lift over the wings. As has been said, centripetal effect would be a small correction only.
 

Offline alancalverd

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Pedantically, as the lift vector is radial to the center of the earth, it is a centrifugal force which precisely balances the centripetal force of gravity if we are flying at constant altitude.

In trimmed flight you have a 4-vector balance between thrust, drag, lift and weight. Now if the atmosphere were infinite and of constant density and you suddenly switched gravitation off and reduced the thrust to compensate for drag only, the plane would obviously continue on a tangential course. 
 

Offline Colin2B

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good point, I hadn't viewed the lift vector as a centrifugal force
 

Offline chris

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So can we nail this down to a clear, succinct answer...

Assuming constant weather conditions, and without intervention from a pilot, a plane flying at constant speed (thrust unchanging) will naturally follow the curvature of the Earth; true or false?
 

Offline alancalverd

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Offline jeffreyH

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Is this thread worth pinning?
 

Offline Colin2B

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Is this thread worth pinning?
I would still be concerned about the ball on a string analogy. The string at extension is rigid and would keep the ball at constant height no matter how Fast it is travelling. Also the string only has to balance the centrifugal force whereas for a real plane you need to account for the variation of gravity with height.
I prefer Alan's answers.
 

Offline chris

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variation of gravity with height

Does that really make a difference? Relative to the radius of the Earth, the altitude range over which a plane flies means that gravitational variation is negligible isn't it?
 

Offline jeffreyH

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Well the cord is inflexible and once you reach a certain speed of rotation any increase does not change the 'altitude'. If this was instead a perfectly elastic cord then change would then be a more realistic model of the flight of the plane. Even modelling a stall when the rotation slows too much.
 

Offline Colin2B

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Does that really make a difference? Relative to the radius of the Earth, the altitude range over which a plane flies means that gravitational variation is negligible isn't it?
Yes, but the centrifugal force is also negligible. To get the effect of the ball on a string (rather than aerodynamic lift) the plane would need to travel about 18,000 mph if my calc is right.

Well the cord is inflexible and once you reach a certain speed of rotation any increase does not change the 'altitude'. If this was instead a perfectly elastic cord then change would then be a more realistic model of the flight of the plane. Even modelling a stall when the rotation slows too much.
Yes, I did think about that and it is a better model. My concern was that someone would misunderstand the analogy and think it was more like a satellite than an aerodynamics problem.

However, we have discussed the issue so no problem if this is made sticky as folks can understand the limits of the analogy.

Overall it is a really interesting question, thanks for raising it.
 

Offline alancalverd

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variation of gravity with height

Does that really make a difference? Relative to the radius of the Earth, the altitude range over which a plane flies means that gravitational variation is negligible isn't it?

If you think 0.3% is negligible, yes.  I'd hesitate to load a plane with such a small safety margin, but the variation is easily measurable.
 

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