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On Earth the Karman Line is used as the boundary of space, and I believe it is defined as the height where you would have to go faster than orbital speed in order to obtain aerodynamic lift. Therefore planets like Mars and Venus also have Karman Lines.

However, places like Mercury or the Moon have no atmosphere. Yet surely nobody would classify Neil Armstrong's lunar bunny-hops as suborbital spaceflights. How would you define the space boundary on these planets? Should it be defined as the minimum height where it is possible to complete one stable orbit?

Yet surely nobody would classify Neil Armstrong's lunar bunny-hops as suborbital spaceflights.

Why not? There's no essential difference between a high-eccentricity trajectory with an apolune of 1 meter and one with an apolune of 100 kilometers.

How would you define the space boundary on these planets?

If the body doesn't have an atmosphere dense enough for its Kármán line to be above the surface, space begins at the surface. What other sensible definition is there?

Ultimately, "the edge of space" is an agreed upon convention. In other words it is essentially arbitrary, something which only humans even care about (well maybe space aliens too <grin>). Yes, there are various physical properties which can be used to define this boundary and the measurements themselves are not arbitrary, none-the-less, the choice of WHICH system of measurement to use is arbitrary.

Back in the 60's and 70's, in America, "the edge of space" was deemed to be 100 miles. It was defined this way because there was essentially no atmosphere at that altitude and because it was a nice round number to work with. This is what I was taught in school.

Now that the metric system has become prevalent, the definition has changed from 100 miles to be 100 kilometers, because the atmosphere at that height is still thin enough to be thought of as essentially a vacuum, and because under the metric system it is a nice round number.

The choice of various physical properties to justify these numbers are secondary to the fact that human brains prefer nice round numbers. We could have just as easily defined space as being above the maximum height that is achievable by a high altitude balloon, or by a bird. Or some other measure that resulted in a nice round number.

Given our earth centric view, we have always considered the presence of an atmosphere to define the absence of space, and we consider by various measures that space is roughly equivalent to a vacuum -- the only point of disagreement being the specific density at which we define the atmosphere to be at a vacuous state, and that choice is fairly arbitrary.

From that perspective I think it is reasonable to say that space extends to the surface of any planet which lacks an atmosphere.

The lowest achievable orbit is a separate matter and does not itself define where space starts. However since it is impractical to sustain an orbit inside of an atmosphere, we can use that as a way to define what is not space. In other words if you can't orbit at that altitude due to the presence of an atmosphere then you are not in space.

On a planet without an atmosphere the lowest achievable orbit has three constraints. The orbit must be high enough to clear any physical obstacles, such as mountains. And it must be slow enough to avoid achieving escape velocity for that specific planet's gravity, while also being fast enough to avoid colliding with the planet (i.e. Free-fall).

None of which is useful for defining the beginning of space itself, except in an arbitrary way. For instance on a planet with very tall mountains, the lowest achievable orbit is much higher than on a planet with a smooth surface.

The Kármán line does not easily apply to planets without an atmosphere because it is really just another way to define the point at which the atmosphere becomes so thin that it can effectively be considered to be a vacuum. Basically it is a way of saying that if the atmosphere is thick enough support an airplane then you are not in space. We could apply this to a planet without an atmosphere but that feels like a misapplication of it's intent.

More intuitively, think of an astronaut taking a Space-Walk. If the ship is in-between planets, then the astronaut experiences space in a particular way, specifically as a vacuum with extremes of light and temperature. Clearly this is Space. Now consider that same astronaut going for a walk on the moon. Beyond the addition of some rocks and some gravity, their experience of Space is identical to the one that they had while in-between planets. In both instances it is reasonable to say that the astronaut is In Space.

From which we can conclude that Space, as defined by humans, extends to the surface of the moon and any other planets (meteors etc) which lack a significant atmosphere.

Where does space begin on planets without atmospheres?

Where the ratio κ=1 for the ion cyclotron and ion collision frequencies - for Earth that is 118 ± 0.3 km.

The lunar surface is surrounded not by an atmosphere but by a "surface boundary exosphere". Wikipedia says little but NASA's article: "NASA Mission to Study the Moon's Fragile Atmosphe" explains:

At night you could jump out of the exosphere.

"During the lunar night, the Moon's exosphere mostly falls to the ground. (Just imagine if our atmosphere fell to the ground at night!) When sunlight returns, the solar wind kicks up new particles to replenish the exosphere.

... intense ultraviolet sunlight kicks electrons off particles in the lunar soil, giving those particles an electric charge that can cause them to levitate. Ambient electric fields lift these charged dust particles as high as kilometers above the surface, forming an important part of the exosphere.".

Source: "Rocket‐based measurements of ion velocity, neutral wind, and electric field in the collisional transition region of the auroral ionosphere", (April 7 2009), by L. Sangalli, D. J. Knudsen, M. F. Larsen, T. Zhan, R. F. Pfaff and D. Rowland.

See Wikipedia's webpage "Outer Space - Boundary":

"There is no clear boundary between Earth's atmosphere and space, as the density of the atmosphere gradually decreases as the altitude increases. There are several standard boundary designations, ...".

Where there is solid rock there is no outer space.

Where there is the vacuum of space, there is space.

If you had an object a hundred meters in diameter with a large cave you would have space in the cave, you certainly wouldn't have atmosphere, such a small object doesn't have enough gravity to hold enough hydrogen to constitute an atmosphere.

On Earth the Kármán Line is used as the boundary of space, and I believe it is defined as the height where you would have to go faster than orbital speed in order to obtain aerodynamic lift. Therefore planets like Mars and Venus also have Kármán Lines.

Source Wikipedia: "The Kármán line lies at at an altitude of 100 km (62 mi; 330,000 ft) above Earth's sea level and commonly represents the boundary between Earth's atmosphere and outer space. This definition is accepted by the Fédération aéronautique internationale (FAI), which is an international standard-setting and record-keeping body for aeronautics and astronautics.

The line is named after Theodore von Kármán (1881–1963), a Hungarian American engineer and physicist, who was active primarily in aeronautics and astronautics. He was the first person to calculate that the atmosphere around this altitude becomes too thin to support aeronautical flight, since a vehicle at this altitude would have to travel faster than orbital velocity to derive sufficient aerodynamic lift to support itself.".

The line defining the 'edge of space' for Earth at 100 km is convenient, just like international waters are 200 nautical miles from the baseline. Not everyone agrees on it, and it is not scientifically correct; it's agreed upon by treaty.

The line defining the 'edge of space' was determined to be 118 ± 0.3 km.

"Measured ion drifts in the 150–198 km and 92–105 km altitude ranges are consistent with $scriptsizeoverrightarrowE$ × $scriptsizeoverrightarrowB$ motion to within 16 m s$^−1$ rms and with neutral wind velocity to within 20 m s$^−1$, respectively. From these measurements we have calculated the ratio κ of the ion cyclotron and ion collision frequencies, finding κ = 1 at an altitude of 118 ± 0.3 km.

The transition between the two regimes is controlled by the ratio of ion cyclotron to ion collision frequencies, $κ_j = Ω_j/v_j$, where $j$ is a species index. Consequently, collision frequencies can be derived from independent measurements of electric and magnetic fields $scriptsizeoverrightarrowE$ and $scriptsizeoverrightarrowB$, bulk flow velocity $scriptsizeoverrightarrowE$, and neutral winds $scriptsizeoverrightarrowv_j$ [e.g., Egeland et al., 1973].

However, places like Mercury or the Moon have no atmosphere. Yet surely nobody would classify Neil Armstrong's lunar bunny-hops as suborbital spaceflights. How would you define the space boundary on these planets? Should it be defined as the minimum height where it is possible to complete one stable orbit?

This is answered above.