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The following rough approximation seems to show a neutron star at its Tolman–Oppenheimer–Volkoff limit of 2.17 solar masses and 12km radius, has gravitational binding energy on the same order as its relativistic mass-energy:

$$\frac{3}{5} \frac{(2.17*1.989*10^{30}kg)^2G}{12km}=6.2*10^{48}J$$ $$2.17∗1.989∗10^{30}kg *c^2=3.88*10^{48}J$$

While numbers and formula I've used are very rough -- which would explain the apparent 60% excess -- it does seem as though once a TOV star has settled down through radiation, just about all of its mass has been effectively converted to energy. Is this correct? If not, how much of its primordial (H, He, etc.) mass has been converted to energy?

Qmechanic
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James Bowery
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1 Answers1

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When a neutron star is at the TOV limit then the particles within it (predominantly neutrons, though there may be phase changes into heavier hadronic matter or quark matter) are becoming, or are, highly relativistic. As a result, their mass-energy is dominated by kinetic energy and not rest mass.

However, the mass that you are putting into your formula (for gravitational potential energy, not gravitational binding energy) is the gravitational mass. This is the mass that an external observer judges the neutron star to have on the basis of its gravitational effects. This is going to be (approximately) the sum of the rest mass of the particles that make up the neutron star plus the mass-equivalent of the gravitational binding energy (which is negative). i.e. It is lower than the rest mass of the particles that make up the neutron star.

ProfRob
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