# Could a planet's Karman line hypothetically occur under a liquid surface?

The Karman Line is one of the most commonly-used definitions of the "edge of space". As an airplane flies higher in the atmosphere, the air gets thinner and thus the lift decreases. This can be compensated by flying at a faster speed. The Karman line is the altitude at which you would need as much speed as the orbital velocity. You are no longer flying; you are in orbit.

Earth has a gaseous atmosphere, and a Karman line that calculates to about 100 km.

This Space.SE question examines the Karman line of a planet without an atmosphere (i.e. a solid surface). The general consensus of the answers is that the solid surface itself is the "edge of space". The moon is such a body.

So we have...

• the edge of space with a solid surface (moon).
• the edge of space with a gaseous atmosphere (Earth).

What about a planet (or moon) with a liquid surface -- namely, could there be any contrived, theoretical scenario where the Karman line occurs below sea level?

The oceans do not necessarily have to be water (e.g. ammonia, mercury, or hydrocarbons are fine). You may adjust temperature, pressure, and gravity to any plausible values that support liquid oceans. Presumably, to keep the oceans from boiling away, there would need to be a solid crust above the ocean, or some atmosphere inadequate for flight (your choice).

Interestingly, such a possibility would mean that no creature or vehicle could "swim" to the surface of their ocean.

Obviously, Earth itself proves you can have a Karman line above a liquid sea level.

This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

• Don't accept right off the bat! It is good practice to leave questions open for at least 24 hours to see if other answers appear. – kingledion Sep 11 '18 at 20:02
• Just an FYI: if you want the hard-science notice to go along with the tag, flag it for mod attention and they'll add the post notice. Just flagged accordingly (further info on that can be found in the hard-science tag wiki.) – FoxElemental Sep 11 '18 at 21:06
• Water will sublimate in a vacuum. I followed this reddit link and found this wikipedia page about Ionic Liquids. They are salts in liquid form and do not evaporate/sublimate in a vacuum like water does. – John Locke Sep 11 '18 at 21:44
• @JohnLocke: Water ice will sublimate in air at normal pressure, not only in a vacuum. In vacuum liquid water will boil . . . – AlexP Sep 12 '18 at 0:21
• @AlexP The karman line is where the medium is so thin that orbital mechanics is holding you up more than aero/hydrodynamics. Therefore, the karman line is at least as tall as the highest point where matter is found in a relevant density, meaning if the planet has an atmosphere, the karman line will be at the top of the atmosphere, not, as the OP asked, under water, nor anywhere near the surface, it will end way above water. So if you have no atmosphere, you need something that will not sublimate. – John Locke Sep 12 '18 at 0:45

# No

As long as you are in a liquid, the density will be high enough that an airfoil shape will be able to give you lift. In that case, you can always get to the surface of the liquid, with a sub-orbital velocity. Therefore, the Karman line can't be below the surface of the liquid.

### Definition of the Karman line

The Karman line's mathematical definition is

$$\frac{1}{2}\rho v_0^2SC_L = mg$$

where $v_0$ is the orbital velocity; $m$ and $S$ are mass and wing area; and $C_L$ is coefficient of lift. The wing loading of an airplane is $m/S$ and is around 600 kg/m$^2$ for a commercial airplane. Via Aviation.SE we can get lift coefficients. Lift varies based on angle of attack, but $C_L=1$ is a good enough approximation. We can plug this into the equation to get:

$$\frac{1}{1200}\rho v_0 = g.$$

So now we have a relationship between orbital velocity, gravity, and fluid density. Given a fluid density of water at 1000 kg/m$^3$, the surface gravity must be 0.83 times the orbital velocity, in units of m/s$^2$ and m/s, respectively, for a the Karman line to be below the liquid level.

### Relationship between escape velocity and surface gravity

Now, escape velocity is not the same as orbital velocity, but it can give us an approximation of what orbital velocity is. LEO on Earth is ~7 km/s while escape velocity is 11.2 km/s. This will be close enough an approximation, as we will see.

Escape velocity can be expressed as a product of surface gravity by

$$v_0 = \sqrt{2gr}$$. We will use escape velocity as a stand-in for orbital velocity. Combine this equation with $g = 0.83 v_0$ and we get $v_0 = 1.7 r$, with units of m/s and m.

If escape velocity is the same as Earth, then the planet needs to be 19,000 km in radius (Earth is 6370) with surface gravity of 9260 $g$. If the planet is to have surface gravity of 9.8 m/s$^2$, then the escape velocity of this 'planet' is 12 m/s and its radius is 20 meters.

### Calculation of required mass

So here you can see the impossibility forming. Escape velocity is

$$v_e = \sqrt{\frac{2GM}{r}}.$$ If we plug in 12 m/s and a radius of 20 meters, we get a mass of $2.2\times10^{13}$ kg; this is a density of $4.4\times10^{8}$ kg/m$^3$ which is electron-degenerate matter.

# Conclusion

The only way to make this happen is to put a liquid ocean over a small asteroid's worth of electron degenerate matter. So, no, this cannot happen.

• And I don't dare imagining what happens upon reaching orbital velocity into a liquid... – L.Dutch Sep 11 '18 at 19:27
• It would be brief, but exciting. – Keith Morrison Sep 11 '18 at 19:33
• @DrSheldon, if you are in a liquid, you are, by definition, in a material of less density than if that same liquid was a gas. Thus, you don't even need an airfoil to reach the surface. You can theoretically use that same liquid in gaseous form to float to the surface, ie needing a horizontal speed of 0. – Keith Morrison Sep 11 '18 at 19:40
• @KeithMorrison A rocket doesn't need a horizontal speed to get to the top of the atmosphere either, and it isn't in a liquid. – John Locke Sep 12 '18 at 0:50
• @JohnLocke Yes but it also will have a vertical speed of close to zero as well, and rockets can't do that. – fyrepenguin Sep 12 '18 at 2:27