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Edit: This question is no duplicate because here the dropping of the atmospheric density together with the horizontal line, play an important role.
Furthermore none of the answers and question mentioned above adressed the escape velocity.

The question "What would a "Kármán plane look like, a bird, or a plane ?" uses the Kármán line definition to estimate the wing loading of such a plane.

When an airplane would follow the straight line in that definition it can be calculated that within a minute that plane would have to go beyond escape velocity because when going higher the atmospheric density drops dramatically.

To calculate where on the straight line the escape velocity will happen, we will have to know the atmospheric densities at different altitudes.
This question shows us that at 105 km altitude the density has dropped more than half the density at 100 km, and because the speed of the plane is proportional to the square root of the inverse of that density, that speed will increase with $\sqrt{2}$.
So for an orbital velocity at 100 km, this will mean the escape velocity will be reached at 105 km.
To calculate what the distance on the straight line between the 100 km altitude point and the 105 km altitude point is we can apply a right-angled triangle with one side between those points and the two other sides going to the center of the Earth.
The length of those sides are 6478 km and 6483 km respectively so the cosine of the angle at the center of the Earth is thus 6478 divided by 6483 giving an angle there of about 2$^0$.
The sine of that angle gives us that the distance on the straight line between the 100 km altitude point and the 105 km altitude point is about 226 km.

With an orbital velocity of 7 km/sec this means that in about half a minute the Kármán plane will reach the altitude where it has to have escape velocity to maintain a straight line.

As soon as the Kármán plane leaves the altitude of the Kármán line definition then, because of the straight line, it simply cannot maintain the orbital velocity determined by that altitude !

So should not a Kármán plane just follow the curvature of the Earth ?

Cornelis
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  • why does it matter what it does when it is no longer at the karman line travelling at orbital velocity? It's an abstract construct. –  Dec 25 '18 at 11:36
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    How many variants of the “Was Karman wrong?” question is e̶n̶o̶u̶g̶h̶ too many for one user? – uhoh Dec 25 '18 at 12:06
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    The ad hoc and informal way that this was first done used an instantaneous evaluation of forces and involved no propagation of motion into the future. OP continues to play games with questions by pretending the Karman line is something that it's been well established that it isn't. I recommend the OP proposes a new site called orbit golf, similar to code golf in area51 and stop playing games here. – uhoh Dec 25 '18 at 12:13
  • "...it can be rather easily calculated that within a minute that plane would..." consider adding your calculation to the question. "So isn't there a contradiction in that definition..." several answers to several of your previous questions have addressed this for you already. Next time (you can't now, there's already an answer posted) consider leaving out the "Was Karman wrong?" part of the question and just discuss the trajectory without using it to try to prove a point. – uhoh Dec 25 '18 at 13:01
  • @uhoh I try to prove the point that Wikipedia is wrong, not Von Kármán ! On the contrary, to me Von Kárman is right because he din't ignore the centrifugal force ! – Cornelis Dec 25 '18 at 13:28
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    @Conelisinspace Again that is wrong. The calculation was instantaneous. There was no motion, no trajectory. It was evaluated at a single moment in time. Just like I've already stated above – uhoh Dec 25 '18 at 15:56
  • @uhoh And still you are using that calculation there for a very fast moving Kármán plane that does not accelerate towards the centre of the Earth. To me, that looks like a straight trajectory. – Cornelis Dec 25 '18 at 17:12
  • If it was decided to accelerate the vessel to the speed required to maintain lift, then yes. it would need to exceed escape velocity. So what? That's kind of the point of the Karman line - it's ridiculous to try to do anything aerodynamically above it. –  Dec 25 '18 at 19:35
  • So isn't it one way or the other ? Doesn't that mean the Kármán plane has to follow the curvature of the Earth ? That's the point i try to make. – Cornelis Dec 25 '18 at 19:55
  • no, at or below the karman line it does not need to curve towards the Earth. At or above the Karman line it can not curve away from the Earth. –  Dec 25 '18 at 20:29
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    Your calculation is mistaken. You have failed to factor in the centrifugal force of the plane as it accelerates, which would reduce the speed requirement to generate sufficient lift to continue flying in a straight line. –  Dec 25 '18 at 21:30
  • @JCRM I have to admit that you're right ! But i just followed the Wikipedia definition that doesn't account for the centrifugal force. – Cornelis Dec 26 '18 at 10:09
  • Of course it accounts for it @Conelisinspace -- the definition is that the (maximum) lift is the same as the centrifgual force, which would result in it being able to travel in a straight (not earth following) line at and below (but not above) the Karman line. –  Jan 02 '19 at 14:30

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No, it doesn't reach escape velocity.

For simplicity, I'm assuming the hypothetical object has a propulsion system that counteracts the drag - otherwise a steady state is impossible

As it rises, the Coriolis force decelerates the notional vehicle. The vis-viva equation can be used to determine the magnitude of this effect.

As it rises, the atmosphere thins further. Atmospheric models can be used to determine the magnitude of this effect.

Both of these effects diminish the lift. A detailed understanding of the parameters for the coefficient of lift of the imaginary vehicle will be needed to determine the variance.

Eventually, the thought experiment would reach an equilibrium altitude where gravity is equal to the sum of both the lift and the centrifugal force, and would orbit at the higher altitude, for as long as it was able to counteract drag. However, all the time the abstract object is above the Karman line then centrifugal force exceeds the lift. (To clarify: Yes, a Karman plane would follow the curvature of the Earth, however it would do it at a higher altitude than the Karman line. Or at the Karman line, but at a lower velocity. Or at the Karman line, at orbital velocity, but at an AoA that produces no lift)

  • The definition oif the Kármán line includes that the plane travels in a straight line. To stay at that line iit has to travel faster and faster because of the dramatically decreasing atmospheric density. – Cornelis Dec 25 '18 at 12:45
  • @Conelisinspace No it does not. You are just referring to a random sentence in a Wikipedia article that's under development. If you want to argue for or against the sentence, the article has a talk page. – uhoh Dec 25 '18 at 14:57
  • @Conelisinspace, it says "The Kármán line is therefore the highest altitude at which orbital speed provides sufficient aerodynamic lift to fly in a straight line that doesn't follow the curvature of the Earth's surface" Travelling in this straight line, however, moves it above the Karman line so the plane is no longer able, using lift and centrifugal force alone, to travel in a straight line. –  Dec 25 '18 at 17:41
  • Or to put it another way: at the Karman line, it travels in s straight line. Travelling in that straight line moves it above the Karman line, so it no longer travels in a straight line. –  Dec 25 '18 at 17:42
  • Because that highest altitude could be everywhere above the Earth (in theory),isn't the Kármán line a curved line then ? – Cornelis Dec 25 '18 at 18:15
  • Yes, @Conelisinspace, the Karman line is a spherical shell. It's an altitude. –  Dec 25 '18 at 18:17
  • So when the plane travels in a straight line from one point on that shell, then it will move higher and higher above that shell, still travelling in a straight line according to the definition ? – Cornelis Dec 25 '18 at 18:24
  • no, it will be unable to travel in a straight line any more, because the Karman line is the highest altitude at which it was possible to do so. "The Kármán line is therefore the highest altitude* at which orbital speed provides sufficient aerodynamic lift to fly in a straight line that doesn't follow the curvature of the Earth's surface*" –  Dec 25 '18 at 18:25
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    Hang on, @Conelisinspace. You've not been confusing "a straight line" as being the Karman line? –  Dec 25 '18 at 18:32
  • I don't think so but the definition you quoted above says: " to fly in a straight line ". – Cornelis Dec 25 '18 at 19:23
  • yes, it would be able to fly in a straight line at the Karman line. Above the karman line it would not be able to fly in a straight line, but would curve toward the Earth (but not yet to follow the Earth) –  Dec 25 '18 at 19:31
  • When it flies in a straight line at the curved Karman line it will automatically go beyond that curved line. – Cornelis Dec 25 '18 at 20:25
  • If it were to fly in a straight line at the Karman line, it would go beyond the Karman line and no longer be able to fly in a straight line. That's pretty much it @Conelisinspace –  Dec 25 '18 at 20:31
  • that's what a limit is @Conelisinspace, it's the point beyond which something no longer applies -- Would you be happy, instead, if it were "1 millionth of a millimeter below the Karman line is the last point at which the Karaman Aeroplane can continue to travel in a straight line" ? –  Dec 25 '18 at 21:10
  • From your answer: "Or at the Kármán line, but at a lower velocity." That means a velocity lower than the orbital velocity for that altitude. Isn't that a contradiction resulting from the definition ? – Cornelis Dec 26 '18 at 10:20
  • no @Conelisinspace the Karman line (altitude) is defined by orbital velocity, but it is possible to fly along it at a lower velocity precisely because of its definition. –  Jan 02 '19 at 07:47
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    @uhoh and @JCRM. I started a discussion on Meta about why this question was closed by you. https://space.meta.stackexchange.com/questions/1157/why-should-the-question-below-be-closed-when-there-is-no-spacific-answer-to-it – Cornelis Jan 03 '19 at 18:48