# Mass-Radius relation of a Neutron star

1. ### Tuugii

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

2. ### Tarunks

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.

3. ### Orion1

990
neutron star mass-radius relation...

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

Last edited: Oct 18, 2008
4. ### Orion1

990

Tuugii, what is your mass-radius equation for a white dwarf?

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