Re: neutron stars

From: Allan Adler (ara_at_nestle.csail.mit.edu)
Date: 12/15/04 

Date: 15 Dec 2004 11:33:45 -0500

Bjoern Feuerbacher <feuerbac@thphys.uni-heidelberg.de> writes:

> Yes, I see your point now. But as I have pointed out, the spectra
> are only similar for *very* highly excited states...

Let's see if I can understand why. I think you are saying the following:
(a) The gravitational potential is only like 1/r outside the star, while
    inside at a distance of s from the center it is like the potential
    due to a sphere of radius s (assuming a spherical distribution of mass)
    or maybe something fancier (cf. what I wrote about Chandrasekhar below).
(b) Therefore, one can't simply use the 1/r version of the potential to set
    up the Schroedinger equation.
(c) Different Schroedinger equation, different spectra (most of the time).

> If you want to have better answers for the energy levels, you
> have to assume a mass distribution for the NS (don't know how
> this looks like exactly...), determine the dependence of the
> potential on r from that, and then solve the corresponding
> Schroedinger (or better: Dirac) equation. Good luck...

You mean the Schroedinger or Dirac equation for the electron alone,
not for the electron and some 10^68 neutrons, I hope. Pretending
that the NS is spherical (even though it isn't) isn't bad for a
first approximation and then one reduces to a 1-dimensional Schroedinger
equation, which one can solve numerically, based on the assumed potential
function. That doesn't seem too bad.

Regarding the mass distribution, I read Chandrasekhar's book,
An Introduction To The Study Of Stellar Structore, superficially
a few months ago. Doesn't his theory apply as well to neutron stars
and give some reasonable guesses for the mass distribution?

I should emphasize that just because I think that it isn't too hard
to do doesn't mean I think I can do it.

> > Regarding the electron orbits
>
> I really would prefer if you would stop talking about "orbits".
> That only make sense for highly excited states (Rydberg states).

OK, let me try to understand why. Part of my problem is that there are
a couple of things I think you might mean:
(1) You prefer that I really stick to the formalism of wave functions
    and only talk about the orbital radius as an expectation value?
    (or perhaps as a radius ouside of which the integral of the electron
    density fuunction drops below a certain value)?
(2) On linguistic grounds, you don't want to talk about orbits around a
    NS if the electron is really inside the NS, since orbits around
    something are really understood to be outside the thing being
    orbited? This wouldn't be a problem for me since, e.g., in the study
    of dynamical systems, one routinely talks about orbits even without
    reference to orbiting around something, and in the case of group
    actions one also speaks of orbits in a similar spirit.

> You also have to consider that the electrons themselves interact
> with individual neutrons - so it is not clear at all if there
> are stable states possible!

That's true. But if the mean free path of the electron (if I can speak
of paths) is long enough, e.g., it could take a long time for that to
happen, e.g. longer than the age of the universe. Also, even if one
electron collides with one neutron (by the way, what kinds of reaction
products can result from such a collision?), there might be others
available to take its place. For example, the light emitted by neutron
stars might, at deeper layers of the star, spawn electron-positron pairs
in abundance which would then move under the influence of gravity inside
the star.

Even quantum mechanically, if the electron density function is concentrated
in a sufficiently small region, it might really be improbably that it would
actually be scattered by an individual neutron.

> > What an odd state of affairs: with ordinary atoms, the electron goes around
> > the nucleus. With the electron in its ground state about the neutron star,
> > the nucleus literally goes around the electron.
>
> Huh? What's moving is still (mainly) the electron, not the NS.

I didn't mean that as one big neutron star in orbit around a tiny electron
far away from it. I just meant that the electron (conceived of classically)
is located very near the middle of the star or, perhaps more correctly,
that the electron's wave function is concentrated in a little sphere
at the center of the star, with the neutrons in a big spherical shell
around it. That's kind of the reverse of the picture one has of a
nucleus, where the neutrons are in the middle (with the protons) and the
electrons form a kind of shell around it.

> > So, the electron is like
> > the real nucleus. In imitation of cytology, let's call it the nucleolus.
> > There could be a lot of electrons in the nucleolus, occupying low energy
> > levels. For that confined region, the electrons do interact and we do get
> > something like electron shells, aufbau, isotope effect and all the stuff
> > we formerly ruled out.
>
> Yes, but nevertheless the spectra would be totally different from
> atomic spectra - see above.

I think I see your point, but I'm not entirely convinced. Let's suppose
that the electron's wave function is concentrated in a little sphere
several orders of magnitude smaller than the internucleon distance
(even if we disown your original estimate of 10^-28 m on the grounds
that the potential doesn't really grow like 1/r). That could make it
unlikely for the electron to collide with any individual neutrons.
What is the range of the forces that would contribute to an interaction
between an electron and an individual neutron?

Let me try to guess what could go wrong. Closer to the center of the
NS, the gravity is less, so there is less reason for the expectation
value of r to be small, so the electron isn't confined to the little
sphere I want to keep it in, so it moves out further and bumps into
neutrons.

On the other hand, maybe when it bumps into the neutrons, trying to get
out of the little sphere, it bounces back in...

I guess all these intuitions won't be worth anything without some wave
functions and some computations supporting them. I'm probably not up
to that, even if it is possible.

Well, there are similar things that people have thought about. For example,
in beta decay of a nucleus, the electron does have some way of getting out
of the nucleus, so maybe it isn't so difficult. But maybe the half life
for beta decay is a measure of how long the electron keeps bouncing around
until it finally gets out. I guess I'm assuming there is an electron bouncing
around inside, hence that some neutron decayed. I think I've seen it said
that the cross section for neutron decay inside a nucleus is different from
what it is for a few neutron. But maybe another way to look at it is that
the cross section for neutron decay is the same as for a free neutron
but the electron produced doesn't normally get out of the nucleus and
eventually recombines with a proton. Maybe the ratio between the cross
section for decay of a free neutron and the cross section for beta decay
is a measure of the mean free path of an electron bouncing around in a
nucleus.

I'll be the first to admit that I don't know what I'm talking about.
It would be nice to know whether people actually do try to figure out
stuff like the mean free path of an electron inside a nucleus.

Maybe instead of free associating like this, I should go back to reading
Preston's book, Physics of the Nucleus, which I had put aside because I
was so busy. If there are books about models of the structure of neutron
stars, I'd be glad to know about them too.

In view of what I write below about nuclei, I guess the same questions,
mutatis mutandis, apply to the mean free path of a photon in the nucleus
(e.g. decay of an excited nucleus to the ground state and emission of
a gamma ray).

> > There could also be positively charged particles in the nucleolus, say
> > positrons, also in orbits. That being the case, we might also have
> > positronium in the nucleolus. Then there is all the pressure from
> > the neutron star, resulting in liquid or crystalline positronium.
> > (Sorry, I couldn't resist.)
>
> Ouch.

One reason I started thinking about creation of electron-positron pairs
inside the NS was to try to find a source of positronium. They might form
spontaneously at a certain rate, but the conditions inside the NS might
make it harder for them to decay.

Anyway, it occurred to me that one might also want to consider the paths
that light could follow, particularly since it is so closely related to
the electron-positron pairs and since neutron stars do radiate. A photon
path can be bent by gravity. Is it possible that some photons could be
traveling in circular orbits under the gravitational influence of the NS?
Wouldn't that be different from the mechanism preventing photons from
escaping from a black hole, even though it would still be due to the
neutron star's gravity?

It has been pretty stimulating to follow up on the OP's original question
about whether a neutron star can be regarded as a nucleus. One can undoubtedly
take it a lot further, but now I think I have to stop doing this. I'll be
interested in the answers to my naive questions, but I have to stop posting
on this topic. It's been fun, though. I don't often get a chance to play
with this kind of stuff. 

-- 
Ignorantly,
Allan Adler <ara@zurich.csail.mit.edu>
* Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions and
* comments do not reflect in any way on MIT. Also, I am nowhere near Boston.

Relevant Pages

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