# Mass-Radius relation of a Neutron star

• Tuugii
In summary, the conversation discusses the need for help in determining the Mass-Radius relationship for a neutron star, specifically the Neutron degeneracy pressure expression and the density ratio. The conversation also mentions the Proton charge radius and Tolman mass equation solution VII as a possible model for the mass-radius relation. The summary also includes the total mass-radius equation for the Tolman solution VII and a reference to further information on the topic.

#### Tuugii

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.

thanks,
T

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.

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

$$\rho_n = \frac{3 m_n}{4 \pi r_p^3}$$

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:
http://en.wikipedia.org/wiki/Neutron" [Broken]
https://www.physicsforums.com/showpost.php?p=1718805&postcount=39"
https://www.physicsforums.com/showpost.php?p=1792334&postcount=47"
http://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_limit" [Broken]

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## What is the Mass-Radius relation of a Neutron star?

The Mass-Radius relation of a Neutron star is the relationship between the mass and radius of a neutron star, which is a type of compact star that is made up mostly of neutrons. This relation describes how the mass of a neutron star affects its size or radius.

## How does the Mass-Radius relation of a Neutron star differ from that of other stars?

The Mass-Radius relation of a Neutron star differs from that of other stars because neutron stars are much more compact and dense than other types of stars. Due to their high mass and gravity, the radius of a neutron star is much smaller than other stars of the same mass.

## What factors affect the Mass-Radius relation of a Neutron star?

The main factors that affect the Mass-Radius relation of a Neutron star are the mass and composition of the star. The more massive a neutron star is, the smaller its radius will be. Additionally, a neutron star's composition, which is mostly made of neutrons, also affects its size and density.

## Why is the Mass-Radius relation of a Neutron star important?

The Mass-Radius relation of a Neutron star is important because it helps scientists understand the properties and behavior of these unique objects in space. It also provides insights into the nature of matter at extremely high densities and can help in the study of other astrophysical phenomena.

## How is the Mass-Radius relation of a Neutron star measured and studied?

The Mass-Radius relation of a Neutron star is measured and studied through a combination of observational data and theoretical models. Scientists use techniques such as pulsar timing, X-ray spectroscopy, and gravitational wave observations to gather data on neutron stars and use mathematical models and simulations to interpret and understand this data.