Pulsar gravitational binding energy?

Question

A Newtonian homogeneous density sphere has gravitational binding energy in Joules $U = -(3/5)(GM^2)/r$, G=Newton's constant, M=gravitational mass, r=radius, mks. The fraction of binding energy to gravitational mass equivalent, $U/Mc^2$, is then (-885.975 meters)(Ms/r), Ms = solar masses of body, c=lightspeed.

This gives ratios that are less than half that quoted for pulsars (neutron stars), presumably for density gradient surface to core and General Relativity effects (e.g., billion surface gees). Please post a more accurate (brief?) formula acounting for the real world effects.

Examples: 1.74 solar-mass 465.1 Hz pulsar PSR J1903+0327, nominal radius 11,340 meters (AP4 model), calculates as 13.6% and is reported as 27%. A 2 sol neutron star calculates as 16.1% and is reported as 50%. There is an obvious nonlinearity.

Thank you.

Answer 1

Dear Uncle Al, your formula with the coefficient $3/5$ only holds in the Newtonian physics. But obviously, the Newtonian physics is not appropriate for the description of hugely heavy objects such as black holes or neutron stars that are not far from them. You need to use a more complete theory developed by the physicist named Albert Einstein.

It's called the special theory of relativity. Special relativity implies the $E=mc^2$ law - among other things you didn't mention in your question - while general relativity is needed to study very heavy objects that strongly curve the space such as the neutron stars.

Sorry if I am telling you something that you have already heard but it is far from clear according to the text of your question.

With proper definitions of the "gravitational bound energy" in general relativity, black holes may be said to store 100% of their energy in the gravitational binding energy. Again, the latter notion is ambiguous in general relativity. Neutron stars are typically very close from the radius-mass relationship that apply to black holes. So before they reach the tipping point where the collapse to a black hole is inevitable, the ratio will jump above your classical formula. That's where a nonlinearity comes from.

Moreover, real neutron stars are really not uniform, either, so the formula at the top wouldn't apply even in the classical physics. The density closer to the center is larger. That effectively puts the matter closer together which increases the (absolute value of) the (negative) binding energy. The more massive neutron stars one considers, the more they tend to clump near the middle, so this is another source of the nonlinearity.