# Mass-Radius relation of a Neutron star

by Tuugii
 P: 15 Hey all, I need a help to determine the Mass-Radius relationship for a neutron star. I've done it for a white dwarf, but for a neutron star I need to know the Neutron degeneracy pressure expression, can anyone please help me to solve it? I am thinking that if I have the n.deg.pressure expression then I can use the hydrostatic equilibrium, and assume the masses of proton and neutron to be exactly equal; I am not sure, but I might also need the density ratio? is it correct? for instance for a white dwarf, I have [ro_c]/[ro_mean] = 5.99, I don't know the value for a neutron star. please help me, thanks, T
 P: 1 for non relativistic case the degeneracy pressure varies as:- p=k*(density)^(5/3). The 'k' here you can easily calculate my first calculating the total energy of degenerate neutron gas and then differentiating it w.r.t volume to get pressure.
 P: 991 The neutron star mass-radius relation is dependent on a particular neutron star model, however the mass-radius relation for my model based upon the Proton charge radius and Tolman mass equation solution VII: $$m_n = 1.6749272928 \cdot 10^{-27} \; \text{kg}$$ - Neutron mass $$r_p = 0.8757 \cdot 10^{-15} \; \text{m}$$ - Proton charge radius Proton charge radius neutron density: $$\rho_n = \frac{3 m_n}{4 \pi r_p^3}$$ Neutron star core density equivalent to Proton charge radius neutron density: $$\rho_c = \rho_n$$ Total Tolman mass equation solution VII: $$M_0(R) = \frac{8 \pi \rho_c R^3}{15} = \frac{8 \pi R^3}{15} \left( \frac{3 m_n}{4 \pi r_p^3} \right) = \frac{2 m_n R^3}{5 r_p^3}$$ Total mass-radius equation for the Tolman solution VII: $$\boxed{M_0(R) = \frac{2 m_n R^3}{5 r_p^3}}$$ Mass of a 10 km radius Tolman VII neutron star: $$\boxed{M_0(10 \; \text{km}) = 9.976 \cdot 10^{29} \; \text{kg}}$$ Note that the lower limit for total radius R, is equivalent to the Schwarzschild radius and the upper limit for total mass M(R), is equivalent to the Tolman-Oppenheimer-Volkov mass limit. Reference: Neutron - Wikipedia TOV #39 - Orion1 TOV #47 - Orion1 Schwarzschild radius - Wikipedia Tolman-Oppenheimer-Volkoff mass limit - Wikipedia
P: 991