I What is the minimum mass of a neutron star?

[bbbl67]

We just discovered the maximum mass of a neutron star, discovered after the recent neutron star merger event back in Aug. They say that the maximum mass of a neutron star is approximately 2.16 solar masses.

So I always assumed that the lowest mass for one is 1.4 solar masses, the Chandresekhar Limit, but I'm not sure anymore? I'm hearing that there are neutron stars less than that size. Anyone know what the lowest theoretical limit actually is?

 

[jedishrfu]

Yes, I think you're right about the Chandresekhar limit being the smallest value:

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

The article says that below that value and the stars are considered white dwarf stars in general:

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

 

[bbbl67]

I've heard of some neutron stars being 1.1 or 1.2 solar masses. I can't find the references right now.

 

[|Glitch|]

The pulsar PSR J0348+0432 is the most massive observed neutron star at 2.01 ± 0.04 M_⊙. The pulsar PSR J0453+1559 is the least massive observed neutron star at 1.174 ± 0.004 M_⊙.

According to the calculations made by Oppenheimer and Volkoff in 1939, neutron stars have a theoretical mass range between 0.7 and 3.0 M_⊙. Rhoades and Ruffin calculated the maximum mass of a neutron star in 1974 to be 3.2 M_⊙. In 2002 Nauenberg and Chapline calculated the upper mass limit of a neutron star to be 3.6 M_⊙. Since different assumptions result in different mass limits for neutron stars, it is still unknown what the actual range might be.

Some consider a theoretical lower mass limit for neutron stars smaller than the Chandrasekhar Limit to be a flaw, but as we have been recently finding out the Chandrasekhar Limit is not fixed at 1.44 M_⊙ either. Depending on the density and rotational speed of the object the range limits appear to vary considerably, and there will be overlap between the maximum mass of a white dwarf and the minimum mass of a neutron star.

Sources:
http://www.mpia.de/homes/fendt/Lehre/Vorlesung_CO/1939_oppenheimer_volkoff.pdf - Physical Review, Volume 55, February 1939
Maximum Mass of a Neutron Star - Physical Review Letters, 32, 324, February 1974
The Maximum Mass of Neutron Stars - The Astronomy & Astrophysics Review, Volume 11, Issue 1, May 2002
On the Maximum Mass of Neutron Stars - International Journal of Modern Physics E, Volume 22, Issue 7, July 2013 (free preprint)
Neutron Star Mass and Radius Measurements from Atmospheric Model Fits to X-ray Burst Cooling Tail Spectra - Astronomy & Astrophysics Review, Volume 608, December 2017 (free preprint)

 

[PAllen]

We just discovered the maximum mass of a neutron star, discovered after the recent neutron star merger event back in Aug. They say that the maximum mass of a neutron star is approximately 2.16 solar masses.

So I always assumed that the lowest mass for one is 1.4 solar masses, the Chandresekhar Limit, but I'm not sure anymore? I'm hearing that there are neutron stars less than that size. Anyone know what the lowest theoretical limit actually is?

No, the neutron star merger produced a black hole after a 10 to 100 milliseconds. This is the lightest BH known, not the heaviest neutron star. It establishes that the momentarily formed neutron star could not be stable, and very quickly collapsed to a BH.

See the discussion from p.9 on in

https://arxiv.org/abs/1710.11576

summarized on p.12.

 

[Chronos]

We don't really know either the lowest or highest possible mass for a neutron star. It is reasonable to assume the observed mass range is statistically representative of the general population. It is unrealistic to assume electron degeneracy pressur dictates the low end because a supernova event could contirbute enough compressive force to produce degenerate stars below the Chandrasekhar mass limit, as evidenced by PSR J0453+1559. We don'yet know enough to predict a minimum mass, although it is reasonable to assume the lowest observed mass is probably in the ball park

 

[bbbl67]

The pulsar PSR J0348+0432 is the most massive observed neutron star at 2.01 ± 0.04 M_⊙. The pulsar PSR J0453+1559 is the least massive observed neutron star at 1.174 ± 0.004 M_⊙.

According to the calculations made by Oppenheimer and Volkoff in 1939, neutron stars have a theoretical mass range between 0.7 and 3.0 M_⊙. Rhoades and Ruffin calculated the maximum mass of a neutron star in 1974 to be 3.2 M_⊙. In 2002 Nauenberg and Chapline calculated the upper mass limit of a neutron star to be 3.6 M_⊙. Since different assumptions result in different mass limits for neutron stars, it is still unknown what the actual range might be.

After the recent neutron star merger event, studies have now fixed the upper limit on neutron star mass to be 2.16 solar masses.

http://physicsworld.com/cws/article...vation-helps-pinpoint-neutron-star-mass-limit

 

[PAllen]

After the recent neutron star merger event, studies have now fixed the upper limit on neutron star mass to be 2.16 solar masses.

http://physicsworld.com/cws/article...vation-helps-pinpoint-neutron-star-mass-limit

Ah, ok. They are not saying the remnant was a 2.16 mass neutron star (consensus is that it is a ballpark 2.7 solar mass BH), but that using data from the the merger event they are able to justify this constraint. Good to know.

 

[|Glitch|]

After the recent neutron star merger event, studies have now fixed the upper limit on neutron star mass to be 2.16 solar masses.

http://physicsworld.com/cws/article...vation-helps-pinpoint-neutron-star-mass-limit

Actually, that should be "study," not "studies," since it was Dr. Luciano Rezzolla who wrote the one and only study which was published this month. The actual paper is provided below. What makes you think this paper is correct and everyone else is wrong?

Source:
Using Gravitational-wave Observations and Quasi-universal Relations to Constrain the Maximum Mass of Neutron Stars - The Astrophysical Journal Letters, Volume 852, Number 2, January 2018 (free preprint)

 

[mfb]

And even if we take this single study, there is quite some uncertainty in the number. It is not a fixed 2.16.

 

[bbbl67]

Actually, that should be "study," not "studies," since it was Dr. Luciano Rezzolla who wrote the one and only study which was published this month. The actual paper is provided below. What makes you think this paper is correct and everyone else is wrong?

Because I read that there were other groups working independently that also came to the same conclusion, in a different article.

"This study is a good example of how theoretical and experimental research can coincide to produce better models ad predictions. A few days after the publication of their study, research groups from the USA and Japan independently confirmed the findings. Just as significantly, these research teams confirmed the studies findings using different approaches and techniques."

https://www.universetoday.com/138283/astronomers-set-limit-just-massive-neutron-stars-can/

 

[bbbl67]

Ah, ok. They are not saying the remnant was a 2.16 mass neutron star (consensus is that it is a ballpark 2.7 solar mass BH), but that using data from the the merger event they are able to justify this constraint. Good to know.

Yeah, they have used the data from that merger event to tie down the upper limit completely. It's awesome that just a couple of years after the technique was perfected, Gravitational Waves are answering questions not previously answerable.

Also the final product of a 2.7 solar mass BH would probably make it the smallest known black hole yet, I suppose? I had heard the previous smallest record holder was 2.8 solar masses.

Now we need to tie down the lower limit completely too. The Oppenheimer-Volkoff limit seems to be very rough, it came up with an upper limit of 3.0 solar masses, but now we've got an actual upper limit of 2.2, so it stands to reason that the its lower limit is also pretty inaccurate.

 

[mfb]

White dwarf mergers could be interesting for the lower bound.

We'll get a lot more events in the next few years, so all these models will get much more precise.

 

[bbbl67]

White dwarf mergers could be interesting for the lower bound.

We'll get a lot more events in the next few years, so all these models will get much more precise.

Well, a white dwarf merger is a Type Ia supernova, which leaves nothing behind, 100% matter release.

 

[PAllen]

Yeah, they have used the data from that merger event to tie down the upper limit completely. It's awesome that just a couple of years after the technique was perfected, Gravitational Waves are answering questions not previously answerable.

Also the final product of a 2.7 solar mass BH would probably make it the smallest known black hole yet, I suppose? I had heard the previous smallest record holder was 2.8 solar masses.

Now we need to tie down the lower limit completely too. The Oppenheimer-Volkoff limit seems to be very rough, it came up with an upper limit of 3.0 solar masses, but now we've got an actual upper limit of 2.2, so it stands to reason that the its lower limit is also pretty inaccurate.

Yes, the remnant is now considered the lightest known BH. I heard a talk by the lead author of the paper I linked earlier, and he said at the start we have a major record coming out of the merger - either the heaviest neutron star or the lightest BH. The question is which. He presented the work of his team, along with others to support that the result was the lightest known BH.

 

[bbbl67]

Elsewhere on this forum, I'm reading that the lightest theoretical neutron star possible is about 0.1 solar masses! That's for cold neutronium, as opposed to hot neutronium. Any truth to this?

 

[mfb]

Well, a white dwarf merger is a Type Ia supernova, which leaves nothing behind, 100% matter release.

Hmm, good point.
Then maybe a nearby supernova, although their signals tend to be smaller than merging objects.

Elsewhere on this forum, I'm reading that the lightest theoretical neutron star possible is about 0.1 solar masses! That's for cold neutronium, as opposed to hot neutronium. Any truth to this?

Where?

 

[|Glitch|]

Yeah, they have used the data from that merger event to tie down the upper limit completely. It's awesome that just a couple of years after the technique was perfected, Gravitational Waves are answering questions not previously answerable.

Also the final product of a 2.7 solar mass BH would probably make it the smallest known black hole yet, I suppose? I had heard the previous smallest record holder was 2.8 solar masses.

Now we need to tie down the lower limit completely too. The Oppenheimer-Volkoff limit seems to be very rough, it came up with an upper limit of 3.0 solar masses, but now we've got an actual upper limit of 2.2, so it stands to reason that the its lower limit is also pretty inaccurate.

One very questionable event does not establish a standard, no matter how much you wish it to be true.

Oppenheimer and Volkoff's calculations are based upon certain assumptions about the properties of a neutron star, as are all the other estimated neutron star mass ranges. Just because you now have a different mass range based upon completely different set of assumptions does not invalidate any of the other calculations until you can show that their assumptions were incorrect.

Oppenheimer and Volkoff also calculated that the theoretical lower end for a neutron star was 0.7 M_⊙ and we know of at least one neutron star that is below the Chandrasekhar Limit. Considering white dwarfs and neutron stars are cores of dead stars that came about as a result of two completely different processes, it makes sense that one would have absolutely nothing to do with the other. A white dwarf does not become a neutron star as soon as they exceed 1.44 M_⊙, nor does a neutron star cease to be just because it is below the Chandrasekhar Limit.

Keep in mind that the Chandrasekhar Limit is also based upon certain assumptions. We have observed both sub- and super-Chandrasekhar Limit Type Ia SN, so those Chandrasekhar Limit assumptions are not the only thing that produce Type Ia SN. The same is undoubtedly true for neutron stars. Be wary of locking in on just one particular assumption and assuming that only it can be true and everything else must be false.

 

[|Glitch|]

White dwarf mergers could be interesting for the lower bound.

We'll get a lot more events in the next few years, so all these models will get much more precise.

Or possibly a neutron core that either ejected part of its core during the deflagration, or lost some mass as a result of a collision, resulting in a mass smaller than the Chandrasekhar Limit. After all, it is the density that determines a neutron star, not its mass.

 

[PAllen]

One very questionable event does not establish a standard, no matter how much you wish it to be true.

On what basis do you say the neutron star merger observed by LIGO and multiple other sites is "very questionable"?

Oppenheimer and Volkoff's calculations are based upon certain assumptions about the properties of a neutron star, as are all the other estimated neutron star mass ranges. Just because you now have a different mass range based upon completely different set of assumptions does not invalidate any of the other calculations until you can show that their assumptions were incorrect.

Oppenheimer and Volkoff also calculated that the theoretical lower end for a neutron star was 0.7 M_⊙ and we know of at least one neutron star that is below the Chandrasekhar Limit. Considering white dwarfs and neutron stars are cores of dead stars that came about as a result of two completely different processes, it makes sense that one would have absolutely nothing to do with the other. A white dwarf does not become a neutron star as soon as they exceed 1.44 M_⊙, nor does a neutron star cease to be just because it is below the Chandrasekhar Limit.

There are many neutron stars below the Chandresekhar limit. Some tenths of solar mass overlap is expected.

Please reference a white dwarf over the currently accepted Chandresekhar limit. None exist, so far as I know.

Keep in mind that the Chandrasekhar Limit is also based upon certain assumptions. We have observed both sub- and super-Chandrasekhar Limit Type Ia SN, so those Chandrasekhar Limit assumptions are not the only thing that produce Type Ia SN. The same is undoubtedly true for neutron stars. Be wary of locking in on just one particular assumption and assuming that only it can be true and everything else must be false.

Clearly, in producing the lowest mass BH so far observed, the NS merger event adds substantial information on the upper bound of Oppenheimer-Volkoff limit.

 

[|Glitch|]

On what basis do you say the neutron star merger observed by LIGO and multiple other sites is "very questionable"?

Other than LIGO and Virgo detector to which "multiple other sites" are you referring?

Please reference a white dwarf over the currently accepted Chandresekhar limit. None exist, so far as I know.

Clearly, in producing the lowest mass BH so far observed, the NS merger event adds substantial information on the upper bound of Oppenheimer-Volkoff limit.

As a result of these superluminous Type Ia SN discoveries being made since 2003 there have been new assumptions made about white dwarfs. Some place the new limit at almost double the Chadrasekhar Limit, at 2.58 M_⊙. While others have suggested that magnetic fields may result in superluminous Type Ia SN that exceed the Chadrasekhar Limit.

Sources:
The Type Ia Supernova SNLS-03D3bb from a super-Chandrasekhar-mass White Dwarf Star - Nature, Volume 443, September 2006 (free preprint)
New Mass Limit for White Dwarfs: Super-Chandrasekhar Type Ia Supernova as a New Standard Candle - Physical Review Letters, Volume 110, Issue 7, February 2011 (free preprint)
Significantly super-Chandrasekhar limiting mass white dwarfs as progenitors for peculiar over-luminous type Ia supernovae - arXiv : 1509.09008

 

[PAllen]

Other than LIGO and Virgo detector to which "multiple other sites" are you referring?

All the optical observations of the same event. There were many of these. What exactly do you mean by this event being very doubtful?

As a result of these superluminous Type Ia SN discoveries being made since 2003 there have been new assumptions made about white dwarfs. Some place the new limit at more than double the Chadrasekhar Limit, at 2.58 M_⊙. While others have suggested that magnetic fields may result in superluminous Type Ia SN that exceed the Chadrasekhar Limit.

Sources:
The type Ia supernova SNLS-03D3bb from a super-Chandrasekhar-mass white dwarf star - Nature, Volume 443, September 2006 (free preprint)
New Mass Limit for White Dwarfs: Super-Chandrasekhar Type Ia Supernova as a New Standard Candle - Physical Review Letters, Volume 110, Issue 7, February 2011 (free preprint)
Significantly super-Chandrasekhar limiting mass white dwarfs as progenitors for peculiar over-luminous type Ia supernovae - arXiv : 1509.09008

Thanks, I had been thinking of white dwarfs we have observed directly, for which I still see no sign of overmass dwarf. However, the inference of overmass dwarfs from events these papers analyze is, admittedly, quite convincing.

 

[PAllen]

Ok, the really interesting question about these inferred overmass dwarfs is the possibility you may have a white dwarf more massive than the lightest BH; the inferred masses are already well over the heaviest ever observed neutron star.

 

[PeterDonis]

the inference of overmass dwarfs from events these papers analyze is, admittedly, quite convincing.

An alternative hypothesis, which appears to be briefly mentioned in some of these papers but not discussed in detail, is a white dwarf merger.

 

[GeorgeDishman]

There are many neutron stars below the Chandresekhar limit. Some tenths of solar mass overlap is expected.

I'm looking for a paper I can cite that gives a lower credible limit for a neutron star mass, specifically to rule out an NS being below 0.5 Msun which I think should be possible. Do you have a specific source for your statement, it would be very helpful.

Thanks.
George

 

[PAllen]

I'm looking for a paper I can cite that gives a lower credible limit for a neutron star mass, specifically to rule out an NS being below 0.5 Msun which I think should be possible. Do you have a specific source for your statement, it would be very helpful.

Thanks.
George

I don’t know of a paper that gives a strong theoretical limit for the lowest mass neutron star we should see. But observation of neutron stars a little below the Chandrasekhar limit are easy to find.

For example: https://academic.oup.com/mnras/article/443/3/2183/1077087

 

[jedishrfu]

I think this limit is the Chandrasekhar limit as mentioned by @PAllen so you could refer to his paper on it.

https://en.m.wikipedia.org/wiki/Chandrasekhar_limit

 

[PAllen]

I think this limit is the Chandrasekhar limit as mentioned by @PAllen so you could refer to his paper on it.

https://en.m.wikipedia.org/wiki/Chandrasekhar_limit

No, the Chandresekhar limit sets a theoretical maximum mass for a white dwarf, not a minimum mass for neutron stars. The paper I just linked above discusses observation of neutron stars a little below the Chandresekhar limit. This phenomenon is expected from theory, but what I don't know are any results saying how much smaller a neutron star can be given plausible formation scenarios.

 

[jedishrfu]

But wouldn’t a neutron star of insufficient mass revert back to an ordinary star since the gravity is insufficient to force the electrons into the nucleus? and this then would indicate that the Chandrasekhar limit defines a kind of minimum neutron star stability.

 

[PeterDonis]

wouldn’t a neutron star of insufficient mass revert back to an ordinary star since the gravity is insufficient to force the electrons into the nucleus?

No, because that would require it to do a huge amount of work against its own gravity, to expand to the radius of a normal star (or even a white dwarf).

his then would indicate that the Chandrasekhar limit defines a kind of minimum neutron star stability.

AFAIK it's quite possible for a neutron star of mass significantly less than the Chandrasekhar limit to be stable on theoretical grounds, if one were somehow formed. I don't have my copy of Shapiro & Teukolsky handy, but I believe they derive a minimum stable neutron star mass on the order of about half a solar mass. (Oppenheimer and Volkoff actually did the first crude estimate of this back in 1939 and came out with a value in the same range.) The issue is how such a neutron star, with mass much less than the Chandrasekhar limit, could form in the first place, since the collapse of any star or star-like object of such mass would stop at the white dwarf stage unless it were exceptionally rapid and violent, and the only such collapse we know of is a supernova, which requires a star much more massive.

 

[Vanadium 50]

I don't have my copy of Shapiro & Teukolsky handy

I do. They give 0.18 solar masses under Harrison-Wheeler EOS. The Oppenheimer and Volkoff EOS has no lower bound. The reason they don't spring back to white dwarfs is, as you say, that this requires work, and the energy to do that work has already left the system.

 

[GeorgeDishman]

I've found this paper which probably meets my needs:

https://iopscience.iop.org/article/10.1088/0004-637X/778/1/66
Equation (3) gives a range at formation for an isolated NS as:

M_birth∼1.08–1.57M_☉

Presumably a star exceeding the Chandrasekhar limit would start to collapse but the implosion of the core could eject some surface material to leave a remnant slightly below the limit.

In binary systems, mass transfer raises the mass of the first NS formed as the companion sheds so they have a different distribution with a higher mean (see figure 2) but I don't see any way the mass could be reduced significantly after formation.