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Could a neutron star be a bit like our Sun or any other star?

Just to have a different scale:

A neutron star fusioning not hydrogen and other elements but neutrons and quarks in its core at a stable, constant speed, would release energy, and the neutron star would therefore be larger than just packed neutrons would be.

It would shine with very high frequency electromagnetic waves like gamma rays and far beyond. Finally it would be burning all its "material" into someting weird and collapse. Would it be possible for the neutron star to explode as a supernova or gamma-ray burst?

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No. The mechanisms by which Neutron Stars work are pretty well understood as far as the processes happening there in relation to physics. "Fusing" neutrons doesn't give you any new fundamental particles. Instead, it's easier to think of it as a giant neutron by itself (with added stuff).

Here are a few additional links on Neutron Stars that may prove helpful:

Neutron Stars as Quantum Systems (PDF).

Neutron Stars and Pulsars (Penn State).

Neutron Stars and Black Holes (University of Northern Iowa).

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Last time I looked there was considerable debate over how much heavy quark content to expect in neutron stars (and thus what the equation of state could be expected to be). Some light can be thrown on the equation of state question by finding the mass limit. –  dmckee Nov 21 '11 at 22:40
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Also, the question of time scales renders the exercise moot. Stellar fusion is ruled by the weak force, with correspondingly long reaction times. Hypothetical neutron star fusion would be ruled by the strong force, and would therefore proceed much, much faster. Whatever cool/exotic stuff is created at the center of a NS (like dmckee said) is there right from the beginning. –  Andrew Nov 24 '11 at 11:15
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