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The official limits for a neutron star is $1.4 - 3.2\;M_\odot$. But I read that the limit depends on the particular structure of a star to estimate which mass it must have. I also read that neutron stars with less than $1.4\;M_\odot$ were observed. Given this information, I wonder if we can be sure that our Sun has definitely not enough mass to become a neutron star. Are there absolut limits (without the need of further information) for a star to become a neutron star or a black hole ?

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Can you provide a link to observations of neutron stars with a mass of less than 1.4 times the solar mass. –  John Rennie May 20 at 11:25
    
John, see table 1 of The Nuclear Equation of State and Neutron Star Masses. There are several examples. –  DavePhD May 20 at 12:02
    
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There are two questions here, namely about the limits on neutron star masses, and about the possibility of our sun becoming one. I'll try to argue that they are different questions, viz. the first about the stability and the second about the formation of such objects.

1) DavePhD's reference in the comments (here, for completeness) answers it completely. There is a lot of room for neutron star masses, because it depends intrinsically on the equation of state of nuclear (and possibly sub-nuclear) matter. Since we don't know the correct equation of state is hard to give strict boundaries. Without an equation of state one could have a mass as large as desired, just by increasing radius. So qualitatively the best one can do depends on the interplay between mass and radius, or density if you will.

The strictest limit comes from Schwarzschild radius, that is if you make too dense a star it would generate an event horizon and collapse into a black hole. Next to this, one notes that the speed of sound escalates with density, so if you try to make too dense a star it will have speed of sound greater than the speed of light, violating causality. This gives a limitation in the different equations of state possible. The upper bounds of about 3.5 solar masses comes from this consideration. On the other hand there has never been observed a neutron star without rotation. Taking the lowest rotation ever observed we can see that there is an lower mass limit, because a neutron star of low mass will be very tiny, and in this limit the rotation will significantly shed its mass by virtue of centrifugal forces. This gives a theoretical limit of about 0.1 solar masses, although the paper does not have an observation with such low mass. You'll find all this more deeply discussed in the aforementioned paper. The summary is in Figure 3, page 51.

2)Somewhat independently of the previous discussion, we can be pretty sure that the sun will never become a neutron star, no matter what equation of state is correct. This is because gravitational collapse of a star is a highly non-linear process, that besides the different nuclear fusion cycles, will generate shock waves. Therefore it will not proceed adiabatically, on the contrary this processes will shed most of a star's mass. Therefore to produce a neutron star we need to start with a very heavy one, typically of the order of tens of solar masses. This is the reason we attribute neutron star formation to supernova events.

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Observed neutron stars range from $1.0 \pm 0.1 M_{\odot}$ to $2.7 \pm 0.2 M_{\odot}$ according to table 1 of The Nuclear Equation of State and Neutron Star Masses, which lists dozens of examples. Keep in mind that the mass of the neutron star is typically substantially smaller than the mass of its progenitor star; late in the stellar life cycle a lot of mass is blown away, for instance a star that goes though an AGB phase may lose >50% of its mass. So our $1M_\odot$ Sun is likely to end up as a stellar remnant with $M < 1M_\odot$, probably a white dwarf.

According to Structure of Quark Stars, the mass is the only parameter to consider for neutron stars (but not hypothetical quark stars), although I would think rotation rate would be a factor.

This reference also states that neutron stars can be as small as $0.1 M_{\odot}$, but this does not imply that the sun will actually become a neutron star.

According to Possible ambiguities in the equation of state for neutron stars, it is the theory (equation of state) of neutron stars that is causing the current uncertainty about the limits of neutron stars.

Also, it is unknown whether or not neutron stars may become quark stars before becoming black holes. There is a term "quark nova" for such a hypothetical event.

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+1, and added a mention of the distinction between the stellar remnant mass and the stellar progenitor mass, which seems to be a point of confusion in the question. –  Kyle May 20 at 19:32
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