Chemical calculations

An offhand comment led to a set of hybrid calculations.

Some years ago, when our astronomer was working at Cerro Tololo Interamerican Observatory in Chile, he attended a briefing on orbital debris. Among the tens of thousands of satellites in orbit around the Earth, most are defunct; and there are orders of magnitude more pieces, from large fragments to paint chips.  The subject was not something he ever wound up dealing with directly, but one comment by the presenter caught his attention: that some of the debris took the form of droplets of liquid sodium, leakage from the reactor coolant of a Soviet surveillance satellite.

Now, there are good engineering reasons to use liquid sodium as a coolant in some situations.  Sodium metal melts at just under the boiling point of water, so it’s convenient for machines that are hot but not extremely so; it’s not corrosive, as water is; and it has a reasonable heat capacity.  It is not in wide use on Earth because it bursts into flame on contact with water, leaving a highly caustic solution.  In space that would not be a problem.

What caused our astronomer to prick up his ears was the word “liquid.”  We are familiar with liquids on Earth, and our tutor’s Chemistry students are now learning the basic trinity of states of matter: solid, liquid, gas. But (as they should learn later) liquids only form above certain pressures, varying with the material.  At atmospheric pressure carbon dioxide is never found in liquid form; if you warm up “dry ice,” it goes directly to gaseous form.  Under the thinner atmosphere of Mars, water is never liquid.  In the pressureless vacuum of space, there should be no liquids.  The astronomer’s trinity of states is gas, dust, and plasma.

Well, the presenter seemed credible, and our astronomer was left wondering.  Much later he picked up the idea again, and pulled out his ancient, revered Physical Chemistry textbook.  The key word this time, he found, was “droplet.”  A liquid falling in air, or floating in space, forms separate drops, bound by surface tension.  He did some calculations and found that the surface tension of a liquid could act as a sort of pressure.  Further calculations concerning heat absorbed from the Sun and radiated by the droplet showed that it could stay warm enough for sodium to remain liquid, and the pressure of surface tension could keep it in that state.

But for how long?  Liquids evaporate.  Now he was into territory not covered in his PC class.  A theory of evaporation has no doubt been worked out long ago, and imparted to those graduate students in need of it; but none of us had seen it.  Our astronomer proceeded to develop one.  With some reasonable assumptions and the numerical calculation of exponential integrals, he had something that fit the data at hand for a number of substances.  Not exactly, but well enough.  Applied to sodium in space, he calculated the lifetime of a droplet with a nominal radius of 10cm under different assumptions as 50 days, 72 days, 666 years and 1120 years.  The enormous range of answers comes in part from the fact that he didn’t have some numbers for sodium itself, and had to assume several plausible ones; and that an exponential integral is very sensitive to small differences in the numbers you input.

So sodium droplets are credible as orbital debris.  The point we notice, however, is that to answer an essentially astronomical question most of the help came from a Chemistry textbook.  Dividing up knowledge into separate subjects is sometimes convenient, but not always.