My comments in bold are included.   --  Gene

________________________________ From: Warren Smith <warren.wds@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, December 23, 2012 12:27 PM Subject: [math-fun] Sci-fi myths... "Neutronium"

In various sci-fi stories, the wonderful substance "neutronium" plays a role. This is the superdense stuff neutron stars are made out of, except some amazing advanced aliens find a way to make or get hold of chunks of it.

For example in one fine Larry Niven story, there is a chunk of it about the mass of Earth's moon orbiting some planet, and foolish trinocs try to get it due to wrongly thinking it is a valuable slaver stasis box, thereby meeting their doom from its immense surface gravity.

So, let's think a bit about the actual properties of neutronium. Do they live up to the hopes of sci-fi writers?

They are correct it is very dense. The estimated density of the nuclei of atoms averages about dens=4*10^17 kg/m^3, which is 4*10^14 times denser than water.  Thus the mass of the Moon would fit in a sphere of radius 35 meters, which I thunk is rather larger than Niven had in mind.

The problem is, neutrons are unstable. Neutron has a mean life of 15 minutes for decay into proton+electron+antineutrino+energy. The energy release for decay of a free neutron is 782 KeV (which is about 1/1201 fraction of its rest-mass energy) but for neutrons in different atomic nuclei it can differ, sometimes being as large as 1.5MeV (?).

Neutrons bound in nuclei can be stable.  The set of isobaric (equal mass number) nuclei has a minimum energy for a certain number of protons and neutrons.  Neutron rich members decay by turning a neutron into a proton and emitting an electron + antineutrino.  Proton rich members decay by turning a proton into a neutron by capturing an electron or emitting a positron, and emitting a neutrino.  A complication on this simple situation is that for even mass isobars even-even nuclei are more stable than odd-odd.

Anyhow, for a ball of neutronium of radius=r and mass=M, we expect it should be unstable to neutron decay if the gravitational escape energy for a neutron-mass is less than 782 KeV. That is,     Mcrit = dens * rcrit^3 * (4*Pi/3) and     mneutron * G * Mcrit / rcrit = (782 KeV)*c^2 = 1.25290199 × 10^(-13) joules Solving, I find     Mcrit = 9.2 * 10^27 kg = about 4.8 jupiter masses     rcrit = 818 meter

conclusion: A chunk of neutronium smaller than about 5 jupiter masses or which fits in a ball of radius<=818 meters cannot exist; if you had one, it would explode.

Actually, even larger balls should also be unstable since we do not need the decay products to escape all the way to infinity -- our argument really only provides very weak lower bounds on Mcrit and rcrit.

A low mass neutron star would decay as usual, the antineutrino escaping while the protons and electrons remain forming a white dwarf.  The factor deciding which configuration is more stable is the energy at the top of the Fermi surface.  For a given Fermi energy, neutrons, being more massive, can be packed to a greater number density than electrons.  A famous calculation by Chandrasekhar is that a white dwarf becomes unstable to collapse above 1.4 solar masses.

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