I am interested in how the original value of ~100 km was calculated for the Kármán line of earth. What equations and constants would need to be used to reproduce this value?

Note: By constants, I'm not referring to things like the gravitational constant but am referring for instance to the wing area in the equation for lift coefficient.

  • $\begingroup$ It sounds as if you're really asking about how to calculate the lift generated by some arbitrary wing. Maybe your question would be better phrased in these terms. $\endgroup$ – John Rennie Jul 8 '12 at 7:29
  • $\begingroup$ I'm really trying to calculate the karman line for planets other than earth. $\endgroup$ – Error 454 Jul 8 '12 at 8:07

In principle calculating the latitude of the Kármán line for any planet is simple. For some planet of mass $M$ the orbital speed $v_o(r)$ at a distance $r$ from the centre of the planet is just:

$$ v_o(r) = \sqrt{\frac{GM}{r}} $$

where $G$ is the gravitational constant. So far so good, though from here it gets a bit messier. If you're flying a plane with some mass $m$, then the gravitational force pulling you down is (this is just Newton's equation):

$$ F_g = \frac{GMm}{r^2} $$

and using the expression for the lift from the Wikipedia article you mention, the lift opposing the gravitation force is:

$$ F_{lift} = \frac{1}{2} \rho v^2 A C_L$$

To work out what speed you need to fly you just set $ F_g = F_{lift} $ and you get (after a bit of rearrangement):

$$ v_{fly}(r) = \sqrt{\frac{2GMm}{\rho A C_Lr^2}} $$

The Kármán line is the height at which $v_o(r) = v_{fly}(r)$. Setting these equal and removing the common factors of $G$, $M$ and $r$ gives:

$$ \frac{2m}{\rho(r) A C_Lr} = 1 $$

This is a simple equation and solving for $r$ gives you the height of the Kármán line. The problem is that the atmosphere density $\rho(r)$ is a function of height, and it varies in a non-trivial way because it depends on the temperature and the temperature varies in a non-trivial way with height. To work out the Kármán line on an extraterrestrial planet I suspect you'd need to measure the temperature as I doubt it would be easy to predict it.

I note that the mass of the plane $m$ appears in the formula. I'd guess Kármán took some representative value of the ratio of the plane's mass to the wing area, $m/A$, and used that in his calculation.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.