Plasma in supersonic shock

Over lunch (2009/Nov/19) some of my colleagues were wondering how fast an object moving through the atmosphere must be going before it causes the air it's hitting to turn into a plasma. I duly contributed to the speculation, but was bereft of necessary numeric data, so looked that up once I got home (and had had some sleep).

An object travelling faster than the speed of sound creates a shock-wave in front of it, as the fact of its imminent arrival can't be communicated through the air (by sound waves) fast enough to catch up with it: thus the air only a short way in front of it is still at rest moments before it gets hit by the object. Molecules that the object directly hits get kicked into motion at speeds comparable with the speed of the object (in the simple case, roughly twice it), so are able to hit air a bit in front of the object; but, in doing so, they cause that air to heat and increase in pressure very rapidly, greatly reducing the distance molecules move before colliding with one another, so that such molecules seldom make it far from the object before being slowed by the air they're hitting, so that the object catches them up. The thin layer of compressed and heated air in front of the object is known as a shock-wave.

So I reasoned that the energy imparted to air molecules in the shock-wave is of order the kinetic energy such a molecule would have at the object's speed. I don't know how realistic this is, but it seems reasonably plausible. If the air shock wave imparts energy comparable with the molecule's ionization energy, we can reasonably suppose it'll ionize some molecules; and ionizing the air sounds like the condition for creating plasma. Taking this with my crude estimate of how much energy is imparted, we can expect ionization to happen if the kinetic energy the molecule would have at the object's speed is of comparable order to the molecule's ionization energies. Conversely, dividing the molecule's ionization energies by its mass and taking the square root will give us a speed: if the object's speed is of the same order of magnitude, we can expect ionization.

Air consists almost entirely of oxygen and nitrogen; argon is next, at almost one percent by volume, with carbon dioxide at only about one part in three thousand. So let's just consider oxygen and nitrogen, which are present as the molecules O₂ and N₂, respectively. The tables (Nuffield Advanced Science Book of Data: Chemistry, Physical Science, Physics; 1972) that I've got handy (at home, not in the canteen at work) give data from which I can work out how much energy it takes to break either molecule into two atoms (free radicals) and how much energy it takes to ionize such atoms. In principle, I'd need to sum these energies to get the energy needed to create ions; but a molecule can be ionized without splitting, or as an incidental side-effect of splitting; and it's not unreasonable to suppose that collisions in the shock-wave will let a molecule first get broken and then get ionized, in two separate collisions, provided the object is going fast enough to do either; so it's reasonable just to consider the maximum of the two energies, rather than their sum.

We can dislodge just one electron to get an ion; or we could dislodge several. Again, if we can produce ions of a given valence (number of electrons stripped), they'll be in the shock-wave and thus potentially subject to collisions, so we may strip yet another electron if collisions have enough energy to do that; so we can consider each step of ionization separately. Let's start with a table giving, for each of nitrogen and oxygen, the successive ionization energies, atomic mass, A, and enthalpy change of atomization, ΔH_a – that last is the energy, per mole of monatomic gas produced, needed to split up the molecules into atoms. Each ionization energy column is labelled by the number of electrons missing after the ionization it describes; it gives the energy needed to remove just one electron, so the entry in column 3 is the energy to knock loose the third electron from an ion with two already missing. Nitrogen has only seven electrons to lose; oxygen has eight.

So, to get orders of magnitude for the object's speed, for each kind of disruption of the air, we just need to divide each kJ/mol quanitty by its corresponding g/mol atomic mass and take the square root:

While we're at it, we may as well consider what happens if we require the collision to split the molecule and ionize it (possibly several steps) all in one go; this time, each kJ/mol quantity in the first table has to be added up with all the ones to its left before we divide by atomic mass and take the square root. The molecule-splitting ΔH_a is first in the sequence, doesn't have anything before it, so is unaffected. Only the energies needed to attain successive ionizations are increased:

To give some sense of scale, satellites in low Earth orbit (just above the atmosphere, at altitudes around 100 km) move at 7.86 km/s; an object leaving Earth's atmosphere at 11.18 km/s straight up will escape Earth's gravity and one doing so at 43.6 km/s will escape the Sun's gravity, if it's going in the right direction. The speed of sound in dry air at 20° centigrade is 343 m/s, so even oxygen's √(ΔH_a/A) of 3.9 km/s is above Mach ten (10 times the speed of sound).

Written by Eddy.