# I Low mass limit of a neutron star

#### tzimie

Note: this is QM question, not about stellar science. I am not asking what are the lightest neutron stars found in the Universe.

The same star (say, 1 sun mass) can exist both in a form of a white dwarf and a neutron star. Both states are stable.

However, lets say I start to stripe outermost levels of a neutron star one by one (*), making it lighter and lighter. I expect that when neutron star becomes really light, the pressure would push matter outwards, creating an explosion or just increase the volume dramatically, creating a white dwarf back.

So what is that low mass limit?

(*) This is just a thought experiment, but you can actually do it by dropping small amounts of antimatter to a neutron star, so the outermost levels would burn to light, which escapes. Of course, you have to do it slowly leaving time for a star to cool down.

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Staff Emeritus
The same star (say, 1 sun mass) can exist both in a form of a white dwarf and a neutron star.
I don't believe you can have a neutron star that light.

Before discussing why something is true, we first need to understand if it is true.

#### tzimie

I don't believe you can have a neutron star that light.

Before discussing why something is true, we first need to understand if it is true.

https://en.wikipedia.org/wiki/GW170817

The masses of the component stars have greater uncertainty. The larger (m1) has a 90% chance of being between 1.36 and 2.26 M☉, and the smaller (m2) has a 90% chance of being between 0.86 and 1.36 M

#### PeterDonis

Mentor
I don't believe you can have a neutron star that light.
AFAIK our current understanding of the equation of state of neutron star matter does not rule this out; I believe it allows neutron stars as light as about 1/10 of a solar mass to be stable. But our current understanding of the equation of state of neutron star matter is not very good.

what is that low mass limit?
See above.

Staff Emeritus
J0453+1559 has what might be a NS at M = 1.174 (with a 0.4% uncertainty). It could be a WD, although the orbit better fits a NS. Apart from that, the distribution of neutron stars is in the 1.1-2.0 solar mass range.

Using a measurement of 1.11 solar masses with a 23% uncertainty as evidence for M < 1 is just silly.

Peter is right, many EOS's allow for stable neutron star matter at M ~ 0.1. However, these require cold neutron star matter. For hot neutron stars - that is to say, all of them - the limit is close to M = 1. (Indeed, this is one of the reasons we don't know the EOS well. All EOS's under consideration have similar predictions for hot NS matter). So even if a smaller object would be stable, there is no known way to form one, nor do there appear to be any observed.

#### PeterDonis

Mentor
For hot neutron stars - that is to say, all of them - the limit is close to M = 1.
More precisely, all of them that we have observed. That might be because all of the neutron stars we have observed are fairly young ones--that is, the supernovas that formed them were fairly recent, on cosmological time scales. (The Crab Nebula pulsar, for example, is about 1000 years old.) As these objects age and grow colder, they will get harder to observe, at least with our current technology.

However, it's also true that the main process we are aware of for forming neutron stars--a supernova--requires a star significantly more massive than the Sun (somewhere about 2 to 3 solar masses). A fair fraction of that mass gets ejected during the explosion, but that still leaves a remnant of about 1 solar mass or so. So even a cold neutron star that is a remnant from a supernova that happened billions of years ago would not, on cosmological grounds, be expected to be much ligher than 1 solar mass. I don't know if other mechanisms for forming neutron stars have been proposed that could make cold neutron stars significantly lighter than 1 solar mass.

#### tzimie

However, these require cold neutron star matter. For hot neutron stars - that is to say, all of them - the limit is close to M = 1.
That sounds strange. What is a temperature which would remove degeneration in the cores of neutron stars? For any temperature much below that scale the temperature should not matter a lot.

Let me know why I am wrong.

#### tzimie

AFAIK our current understanding of the equation of state of neutron star matter does not rule this out; I believe it allows neutron stars as light as about 1/10 of a solar mass to be stable. But our current understanding of the equation of state of neutron star matter is not very good.
So when neutron star becomes lighter and lighter (in a thought experiment) the radius would increase, like for the white dwarfs?

Staff Emeritus
Let me know why I am wrong.
Sorry, that's - posting nonsense and saying "prove me wrong!" - is not what we do here. Anyway EOS has a temperature dependence to it, and for a neutron star this is important because of chemical potentials and phase transitions. Hot NS core matter behaves differently than cold NS core matter just like ice behaves differently than steam.

Staff Emeritus
That might be because all of the neutron stars we have observed are fairly young ones
That's not actually the case. There are young pulsars, like the Crab and Vela, that are in the process of shedding their initial angular momentum. But there are other pulsars (so-called millisecond pulsars, although this term is misleading) that have been spun back up. Dating them is not easy; all we can do is date their companion stars. Typical ages are in the billions of years.

There are about 200 of these known. None have a mass below 1.1-1.2. So if there are smaller NSs out there, why haven't we seen any spun up?

As I mentioned earlier, there's no known formation mechanism. This may not adequately explain the difficulty. To make one of these, you need to squeeze it without heating it. Gravity will not do that for you. You need a non-gravitational squeezing mechanism that can radiate energy faster than thermal radiation. Good luck finding one of those.

Some fraction of these objects will be in binaries, so you have the possibility of SNa 1a equivalents. Nothing like that is visible.

#### PeterDonis

Mentor
when neutron star becomes lighter and lighter (in a thought experiment) the radius would increase, like for the white dwarfs?
Yes.

#### PeterDonis

Mentor
there are other pulsars (so-called millisecond pulsars, although this term is misleading) that have been spun back up. Dating them is not easy; all we can do is date their companion stars. Typical ages are in the billions of years.
Hm, interesting.

there's no known formation mechanism. This may not adequately explain the difficulty. To make one of these, you need to squeeze it without heating it. Gravity will not do that for you. You need a non-gravitational squeezing mechanism that can radiate energy faster than thermal radiation. Good luck finding one of those.
Yes, good point.

AFAIK our current understanding of the equation of state of neutron star matter does not rule this out; I believe it allows neutron stars as light as about 1/10 of a solar mass to be stable. But our current understanding of the equation of state of neutron star matter is not very good.

See above.
Yes, it is true, but the equation of state that provides this low mass neutron stars is the superior limit ( something like ρ= 6-7 x 10^14) of the equation of state to White dwarfs, BBP, which i belive is well know.

#### PeterDonis

Mentor
the equation of state that provides this low mass neutron stars is the superior limit ( something like ρ= 6-7 x 10^14) of the equation of state to White dwarfs
Do you have a reference for this? AFAIK the equations of state of white dwarfs and neutron stars are quite different, because of the different interaction between the particles (strong interaction for neutron stars vs. electromagnetic for electrons).

#### snorkack

There is one natural process to remove mass from a neutron star. When the matter from the surface of neutron star starts to spill over to another neutron star or a black hole, and escape on the other side.

But because the neutron stars expand on removal of mass, as a neutron star starts losing mass it expands causing further mass to spill over. The timescale of that process? A neutron star is in the region of 10 km across. Speed of sound is in the region of 100 000 km/s. So the timescale is hundreds of microseconds.

How does the timescale of neutron star expansion compare against timescale of beta decay? Urca process cooling?

What precisely has a neutron star merger to reveal about equation of state of nuclear matter, cold and hot?

The smaller neutron star is cooled by adiabatic expansion.

And decay of neutrons necessarily releases both antineutrinos and heat.

Heat may go to adiabatic expansion, electromagnetic radiation or Urca process... which unlike beta decay produces both antineutrinos and neutrinos.

In the milliseconds after GW170817 started spilling over, the bigger neutron star must have immediately got hot and started emitting both thermal electromagnetic and Urca process. On the other hand, a black hole is, well, black, and therefore will NOT produce Urca process neutrinos.

How efficient is the expanded, spilt-over neutron matter in producing neutrinos by Urca process, as opposed by producing antineutrinos by beta decay?

Also, a black hole has no hair. A neutron star has hair. Oscillations of an actual matter neutron star with no event horizon should produce gravity waves completely different from the ringdown of a structureless black hole, and sensitive to the actual equation of state of the interior.

Mentor

#### snorkack

Do you have any references for this?
How else are you claiming a neutron star merger takes place?
Yes.
Note: while gravity waves come from bulk movements of mass (thermal production of neutrinos is not negligible, thermal gravitons are negligible even in hot neutron stars) and the escaping electromagnetic radiation comes from surfaces, neutrinos do probe neutron star interior conditions.

#### PeterDonis

Mentor
How else are you claiming a neutron star merger takes place?
Merging two neutron stars doesn't require any mass to "spill over" from one to the other as you describe (much less to "escape out the other side"). It just requires them to merge.

If you think "spilling over" takes place as you described, you ought to be able to provide a reference that gives a model of such a merger that shows it taking place. That's what I was asking for.

#### snorkack

Merging two neutron stars doesn't require any mass to "spill over" from one to the other as you describe (much less to "escape out the other side"). It just requires them to merge.
A neutron star might spiral in and fall into a black hole without being torn apart.
However, such a process could not produce R-process and extended optical afterglow.
The observed kilonova does precisely require mass escaping out the other side.

#### PeterDonis

Mentor
@snorkack, either give references to back up your statements or stop posting.

Do you have a reference for this? AFAIK the equations of state of white dwarfs and neutron stars are quite different, because of the different interaction between the particles (strong interaction for neutron stars vs. electromagnetic for electrons).
Black Holes, White Dwarfs and Neutron from Shapiro and Teukosky, at chapter 8. You may prefer to read the sumary at page 227.

#### PeterDonis

Mentor
Black Holes, White Dwarfs and Neutron from Shapiro and Teukosky, at chapter 8. You may prefer to read the sumary at page 227.
This talks about the cold equation of state above neutron drip, but the discussion of how this relates to the minimum mass of a neutron star doesn't occur until early in Chapter 9 (pp. 252-253). The minimum mass appears to be just below 1/10 of a solar mass, and to occur at a central density of $7 \times 10^{12} \text{g / cm}^3$. In the regime between $\rho_{\text{drip}}$ and this density, there are no stable solutions; so I'm not sure I would describe the minimum mass neutron star solution as a "superior limit" of a white dwarf solution. But I agree that the equation of state in this regime appears to be well understood, so the cold minimum mass limit given here should be reliable. (But as @Vanadium 50 has pointed out, the cold limit actually is not very useful practically, since the neutron stars we observe are not cold.)

This talks about the cold equation of state above neutron drip, but the discussion of how this relates to the minimum mass of a neutron star doesn't occur until early in Chapter 9 (pp. 252-253). The minimum mass appears to be just below 1/10 of a solar mass, and to occur at a central density of $7 \times 10^{12} \text{g / cm}^3$. In the regime between $\rho_{\text{drip}}$ and this density, there are no stable solutions; so I'm not sure I would describe the minimum mass neutron star solution as a "superior limit" of a white dwarf solution. But I agree that the equation of state in this regime appears to be well understood, so the cold minimum mass limit given here should be reliable. (But as @Vanadium 50 has pointed out, the cold limit actually is not very useful practically, since the neutron stars we observe are not cold.)
This mass curve is obtained for HW equation of state, if one uses BPS-BBP the curve continues and has a minimum at a higher central density. I dont know if there is this curve plotted on this book, but these diferences of EoS are debated in page 48 figure 2.2. I have BPS-BBP mass curve plotted by myself also.

#### PeterDonis

Mentor
This mass curve is obtained for HW equation of state
I'm not sure which mass curve you're referring to, but the minimum neutron star mass I referred to, discussed on pp. 252-253 of the reference you gave, is clearly stated to be derived using the BBP equation of state. Since this thread is about the minimum mass for a neutron star, that would seem to be the relevant one.

"Low mass limit of a neutron star"

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