Dead stars

 I'd like to go through a little the type of "dead" stars, although I put that word in quotes because something is still clearly there.  The smallest as I understand are white dwarfs, made of "degenerate" matter.  What is that?

Exactly. "Dead star" is a little misleading. The star as a source of nuclear fusion is dead, but what remains can be an extremely remarkable object.

A white dwarf is what remains when a star roughly the mass of the Sun exhausts its nuclear fuel. The outer layers drift away, leaving behind the hot core, about the size of Earth but containing something like half to one solar mass.

The key concept is degenerate matter.

In ordinary matter, atoms mostly consist of empty space. The electrons occupy shells around nuclei, and if you squeeze the material a bit, the atoms can move closer together.

Inside a white dwarf, gravity has compressed the matter so much that the usual atomic structure is essentially gone. Electrons are stripped from nuclei and form a dense sea around them.

Why doesn't gravity simply crush everything further?

The answer comes from quantum mechanics and the Pauli Exclusion Principle, which says that no two electrons can occupy exactly the same quantum state. As the star is compressed, the electrons are forced into higher and higher energy states because all the lower ones are already occupied. This creates a pressure called electron degeneracy pressure.

What's unusual is that this pressure does not come from heat. A white dwarf could cool almost to absolute zero and the degeneracy pressure would still exist.

A useful analogy is a theater where every seat in the front rows is filled. New arrivals are forced into seats farther and farther back. The crowd "pushes back" against further compression simply because there are no low-energy places left to go.

Some numbers help convey how strange this is:

    A typical white dwarf is about Earth's size.

    Its mass is comparable to the Sun's.

    A teaspoon of its material would weigh several tons on Earth.

    The surface gravity is roughly 100,000 times Earth's.

There is also a limit. If a white dwarf exceeds about 1.4 solar masses (the Chandrasekhar limit), electron degeneracy pressure can no longer support it. Then one of two things happens:

    It may collapse further into a neutron star.

    Or it may explode as a Type Ia supernova.

Neutron stars involve an even more extreme form of degenerate matter, where the electrons and protons have essentially been crushed together into neutrons. That's the next rung down the ladder of stellar corpses, and the physics becomes even stranger.

You say some of the matter drifts away.  Why and how much and what do we see?

Yes—you caught me being unnecessarily literal. In the context of the preceding sentence, your "your" was perfectly clear. I treated it as though you had suddenly changed the subject to me. A human conversational partner would almost certainly have followed the referent. That's a nice example of how linguistic competence can still occasionally lose to overly cautious parsing.

And yes, you've arrived at one of the places where our present physics becomes genuinely incomplete.

Classical general relativity says that once a sufficiently massive core collapses past its event horizon, the collapse continues to a singularity: formally, spacetime curvature and density become infinite and the theory ceases to give sensible physical answers.

But "infinity" here is a warning sign. It may mean that general relativity has been pushed beyond its domain of validity, rather than that Nature actually contains a point of infinite density.

Your Planck-scale thought is very much along the lines of what motivates quantum-gravity theories. The Planck length, about 1.6×10−35 m, is the scale at which our existing descriptions of gravity and quantum mechanics are expected to become simultaneously important. If spacetime itself has some quantum structure at that scale, the classical picture of smooth spacetime collapsing indefinitely may simply stop being meaningful.

There are several possibilities people have explored:

    A genuine singularity: perhaps the classical prediction survives in some fundamental sense.

    A quantum-gravitational core: collapse reaches some extraordinarily dense state at which new physics supplies an effective resistance.

    A quantized geometry: spacetime has discrete degrees of freedom, so the notion of arbitrarily small volume ceases to make sense.

    A bounce: in some speculative models, collapse eventually reverses rather than proceeding to infinite density.

    Something even stranger: some approaches don't give us anything resembling a conventional "center" of the black hole at all.

But there's an important distinction from the white-dwarf and neutron-star cases we were discussing. We know what produces electron degeneracy pressure, and we have fairly good evidence that neutron-star matter really does resist further compression. We don't currently have an experimentally established equivalent of "Planck degeneracy pressure."

So I'd phrase your idea slightly cautiously: if spacetime is quantized, it is quite plausible that the classical singularity is replaced by some quantum-gravitational structure at roughly the Planck scale—but we don't yet know what that structure is.

And that uncertainty is not a minor detail. The singularity is essentially where general relativity says, "I have reached the edge of what I can describe." That's one of the strongest clues we have that a deeper theory of gravity is needed.

We know any spherical object gets denser as you move toward its center.  The densities of matter in white dwarfs and neutron stars cited are what?  Average densities or surface densities?  What about the densities close to the center of these objects?

Yes. The numbers I gave earlier were average densities, not surface densities. And your intuition is exactly right: in a self-gravitating sphere, the density generally rises toward the center.

The distinction is especially important for white dwarfs and neutron stars.
White dwarfs

A typical white dwarf has an average density of roughly 10⁶ g/cm³—about a million times the density of water.

But that's not what you find throughout the star. The density falls substantially toward the surface and rises toward the center. Depending on the mass and detailed model, a massive white dwarf can have a central density around 10⁹–10¹⁰ g/cm³.

So a white dwarf might look roughly like:

surface → relatively low density → increasingly dense → central region at billions of g/cm³

The central density rises dramatically as you approach the Chandrasekhar limit. That's actually part of why the Chandrasekhar limit exists: as the electrons become increasingly relativistic, electron degeneracy pressure becomes less effective at providing additional support.
Neutron stars are much more extreme

A typical neutron star's average density is around a few × 10¹⁴ g/cm³—comparable to nuclear density.

But its center is denser still, perhaps several times nuclear density, commonly something like 5–10 × 10¹⁴ g/cm³, and potentially higher depending on the neutron-star equation of state.

Here, though, we have to put a big asterisk beside the numbers.

We don't actually know the exact density profile of a neutron star's interior. The physics of matter at several times nuclear density is poorly understood. Different plausible equations of state produce different central densities, radii, and maximum masses.

And there's an especially interesting progression:

    Outer crust: nuclei arranged in a lattice, with electrons around them.

    Inner crust: increasingly neutron-rich nuclei plus free neutrons and electrons.

    Core: extremely dense nuclear matter, predominantly neutrons but with other particles probably present.

    Central core: several times nuclear density, where our understanding becomes increasingly uncertain.

So when you hear "a neutron star has the density of an atomic nucleus," that's an average-ish comparison, not a statement that every cubic centimeter has exactly nuclear density.
And the center matters enormously

As you noticed, the central density isn't merely an interesting detail. It's where the battle with gravity is most severe.

The pressure has to support the weight of all the material above it. Consequently, the pressure is greatest at the center, and the density generally follows it.

For a neutron star approaching its maximum stable mass, the central region becomes extraordinarily compressed. Add enough mass and eventually no known equation of state can provide sufficient pressure to maintain a stable neutron star.

That's the point at which the collapse to a black hole occurs.

And this connects beautifully with your previous question about a possible "maximum density." We don't know whether Nature actually permits arbitrarily high central density, or whether quantum gravity introduces a new regime before the classical singularity is reached. We simply don't yet have the theory that can tell us.

One correction to something I said earlier, incidentally: I made the neutron-star magnetic-field numbers far too broad and muddled the units. The commonly quoted surface fields are around 10⁸–10¹¹ tesla for ordinary pulsars and up to ~10¹¹ tesla for magnetars, not the enormous range I implied. The basic point—that they are fantastically stronger than Earth's field—was right.

Wonderful.  I think anyone reading this would find it interesting and informatiive, and represents our present state of knowlege.

I think that's a good way to put it. The really interesting thing about this subject is that the uncertainty is itself part of the knowledge.

We know an enormous amount with confidence: white dwarfs exist, electron degeneracy supports them, neutron stars exist, their masses and radii can be measured, pulsars behave extraordinarily precisely, and sufficiently massive remnants collapse into black holes.

But then we get to the deepest interior of a neutron star—or beyond the event horizon—and the map becomes increasingly conjectural. We can distinguish between:

    what observation establishes,

    what well-tested theory predicts,

    what competing theoretical models predict,

    and what we simply don't know.

That's much more interesting than pretending the whole story is settled.

And I particularly like the thread we've followed: gravity keeps compressing matter, and Nature keeps producing new forms of resistance—until eventually we reach a point where we don't know what resistance, if any, comes next. That's a rather good tour of stellar evolution in miniature.