# What's the *calculation* for the altitude near the Kármán line where the lifting force equals the centrifugal force? [duplicate]

Edit:
This question is not a duplicate of the associated question and has no answers to it that were posted for that associated question. This question asks specifically for a calculation, the associated question doesn't ask for that, and no answers with calculations were given there.

According to Wikipedia about the Kármán line:

In the final chapter of his autobiography Kármán adresses the issue of the edge of outer space:

...or 24 miles up. At this altitude and speed, aerodynamic lift still carries 98 per cent of the weight of the plane, and only two per cent is carried by centrifugal force, or Kepler Force, as space scientists call it. But at 300,000 feet (91,440 m) or 57 miles up, this relationship is reversed because there is no longer any air to contribute lift: only centrifugal force prevails. This is certainly a physical boundary where aerodynamics stops and astronautics begins, and so i thought why should it not also be a jurisdictional boundary ? Haley has kindly called it the Kármán Jurisdictional Line. Below this line space belongs to each country. Above this level there would be free space.

If von Kármán is right with his description of the Kármán line, then isn't there a region near that line where the lifting force equals the centrifugal force ?

Question: Is it possible to calculate the altitude where the lifting force would equal the centrifugal force near the Kármán line ?

Afterword: From the accepted answer below can be concluded:

.........60 km........... for the North American X-15 hypersonic aircraft

.........70 km........... for the Fokker F27 turboprop airliner

.........80 km........... for the Schleicher ASW 22 sailplane

.......100 km............for the Kármán bird

• Kármán’s “no longer any air” here is imprecise; the passage is describing exactly the thing you’re asking for. Note that the abstractly defined Kármán line can’t map to a precise altitude, because it’s dependent on wing loading, which is in turn dependent on the particulars of the aircraft design. Oct 31, 2018 at 21:47
• What's this obsession with the Karman line recently... Oct 31, 2018 at 21:57
• The Wind and Beyond: Theodore Von Kármán, Pioneer in Aviation and Pathfinder in Space Tódor Kármán, Theodore Von Kármán, Lee Edson Little, Brown, 1967 has two other authors and was published four years after his death, so not exactly a pure autobiography. Quotes like "This is certainly a physical boundary where aerodynamics stops and astronautics begins" though using first person, was written by non-scientists. I believe this was pointed out in previous comments on one of the previous half-dozen Karman questions.
– uhoh
Oct 31, 2018 at 23:04
• That paragraph includes "Seven major academic journals then followed with book reviews by noted authors: As the book was non-technical, written for the general reader, Thomas P. Hughes cited that as problematic given the technical context of Kármán's work."
– uhoh
Oct 31, 2018 at 23:17
• I too would love to know the motivation behind the recent torrent of Karman line questions. Nov 1, 2018 at 1:26

Ignoring the history, let's try and make this a straight physics question. Consider an aircraft of mass $$m$$ and some fixed shape, following the curve of the Earth around at an altitude of $$h \,m$$ above the Earth's surface and a velocity of $$v \,m/s$$. We'll assume that $$h$$ is small compared to the radius $$R$$ of the Earth.

Now the aircraft is accelerating (towards the centre of the Earth) since its path is not straight. This acceleration is roughly $$v^2/R \,ms^{-2}$$. It is doing so as a result of two forces, Earth's gravity exerting a force $$mg$$ downwards and a lift force of $$A\rho v^2$$ upwards, where $$\rho$$ is the density of the air and $$A$$ is a constant depending on the design of the plane. So we find that

$$mg - A\rho v^2 = mv^2/R$$

Now if I understand the question you want to know when $$A\rho v^2 = mv^2/R$$ This will occur at an altitude when $$\rho = m/RA$$

To actually put a number on this one would need to know what is realistic for $$m/A$$ for a plane able to fly at hypersonic speeds, and how $$\rho$$ decreases with altitude.

Thanks to @Hobbes in a comment and another answer We get that $$m/A$$ for the X-15 was in the general ballpark of 1600 (SI units) -- that's twice the wing loading divided by the lift coefficient. So we want $$\rho = 1600/(6\times 10^6) = 2.6 \times 10^{-4} kg/m^3$$ which seems to be at roughly 60 000m.

This should definitely be taken with a large pinch of salt as there are many approximations made.

• Sure. It follows from the equations above -- $v^2 = 1/2 gR$. We could put numbers on it, but in fact it's enough to notice that for orbit $v^2 = gR$, so for your question velocity is $1/\sqrt{2}$ of orbital velocity. This is true independently of the altitude (provided it's small compared to $R$) and of the aerodyanmics. Nov 1, 2018 at 11:50
• @uhoh Please keep a courteous tone in your criticisms. Nov 1, 2018 at 17:04
• @uhoh. No worries. Having glanced at bits of the various Karman line questions, I was intrigued by the idea that there was, even vaguely, a logically based notion of where the transition from air to space flight happens. Also I think I understand this question of "centrifugal force" fairly well and I wanted to try and untangle it here. Nov 1, 2018 at 18:01
• @Conelisinspace That force only exists if you are applying the laws of motion of a rotating frame of reference (rotating about the centre of the Earth at angular velocity $v/R$). In that frame the object is stationary, and gravity is balanced by what would be called "centrifugal force". In an inertial frame the force is directed inwards so as to maintain the acceleration, also inwards, of the object, and must be provided by something for the object to move in a circle. In this case by the difference between gravity and lift. Nov 18, 2018 at 22:37

The question uses false premises and misrepresentations as follows, and tries to prove a point rather than actually ask a question in good faith.

Question: If von Kármán is right with his description of the Kármán boundary, then isn't there a region within that boundary where the lifting force equals the centrifugal force ?

• The figure, which is now rounded to 100 km, is an altitude, not a "boundary". This has been addressed for the OP already; see this answer.

• By using the term "boundary", the question introduces a false premise, and this is not the first time in the OP's recent series of von Kármán questions.

• The circa 100 km figure is named after von Kármán, but where is the source showing exactly what von Kármán said? Let's look at the quote next.

Wikipedia says:

In the final chapter of his autobiography Kármán adresses the issue of the edge of outer space:

that von Kármán says:

This is certainly a physical boundary where aerodynamics stops and astronautics begins, and so i thought why should it not also be a jurisdictional boundary?

However:

The Wind and Beyond: Theodore Von Kármán, Pioneer in Aviation and Pathfinder in Space Tódor Kármán, Theodore Von Kármán, Lee Edson Little, Brown, 1967 has two other authors and was published four years after his death, and so it is not an autobiography.

Quotes like "This is certainly a physical boundary where aerodynamics stops and astronautics begins" though using first person, was written by non-scientists. I believe this was pointed out in previous comments on one of the previous half-dozen von Kármán questions.

That paragraph includes "Seven major academic journals then followed with book reviews by noted authors: As the book was non-technical, written for the general reader, Thomas P. Hughes cited that as problematic given the technical context of Kármán's work."

Further:

Haley has kindly called it the Kármán Jurisdictional Line. Below this line space belongs to each country. Above this level there would be free space.

Andrew G. Haley was an attorney, suggesting there could be several things at play; Haley "...has been described as the world’s first practitioner of space law" and may have been in need for some kind of legal altitude for the "beginning of space".

Question: If von Kármán is right with his description of the Kármán boundary, then isn't there a region within that boundary where the lifting force equals the centrifugal force ?

• OP has not shown what von Kármán has actually said, what's been non-technically paraphrased, and what has been simply loosely attributed without evidence. Therefore questioning "If von Kármán is right with his description..." is moot.

• This is yet another false premise, and this is not the first time even the second time in the OP's recent series of von Kármán questions.

• Von Kármán himself called it a boundary. Nov 1, 2018 at 7:43
• apparently I can indeed
– uhoh
Nov 1, 2018 at 8:29
• So you're asserting that the quotes of von Kármán in the Wikipedia article about the Kármán line are false ? Nov 1, 2018 at 9:13