Editing Gravitational binding energy (section)

==Non-uniform spheres==
Planets and stars have radial density gradients from their lower density surfaces to their much denser compressed cores. Degenerate matter objects (white dwarfs; neutron star pulsars) have radial density gradients plus relativistic corrections.

Neutron star relativistic equations of state include a graph of radius vs. mass for various models.<ref>[http://www.ns-grb.com/PPT/Lattimer.pdf Neutron Star Masses and Radii] {{Webarchive|url=https://web.archive.org/web/20111217102314/http://www.ns-grb.com/PPT/Lattimer.pdf |date=2011-12-17 }}, p. 9/20, bottom</ref> The most likely radii for a given neutron star mass are bracketed by models AP4 (smallest radius) and MS2 (largest radius). BE is the ratio of gravitational binding energy mass equivalent to observed neutron star gravitational mass of ''M'' with radius ''R'', 
<math display="block">BE = \frac{0.60\,\beta}{1 - \frac{\beta}{2}}</math>
<math display="block">\beta = \frac{G M}{R c^2} .</math>

Given current values
*<math>G = 6.6743\times10^{-11}\, \mathrm{m^3 \cdot kg^{-1} \cdot s^{-2}}</math>{{physconst|G|ref=only}}
*<math>c^2 = 8.98755\times10^{16}\, \mathrm{m^2 \cdot s^{-2}}</math>
*<math>M_\odot = 1.98844\times10^{30}\, \mathrm{kg}</math>
and the star mass ''M'' expressed relative to the solar mass,
<math display="block">M_x = \frac{M}{M_\odot} ,</math>

then the relativistic fractional binding energy of a neutron star is

<math display="block">BE = \frac{885.975\,M_x}{R - 738.313\,M_x}</math>